WO1992004701A1 - Visual imaging construction system - Google Patents

Abstract

A class of devices, named picture elements, or pixels, for the production of two-dimensional (planar) and three-dimensional (spatial) visual images for both picturing and sculpturing and methods by which such visual images can be constructed from said picture elements. The design of the picture elements simultaneously facilitates the construction of the two necessary elements of every visual image, the image itself and its supporting medium. Being substantially flat, the picture elements are shaped to provide for a solid and stable support medium in the form of a single, flat, double-surfaced opaque or transparent sheet or a three-dimensional structure made of one or more of such sheets. The design of the pixels further facilitates well defined color contours or the blending of colors along their contours. In the case of the opaque sheet or sheets having two surfaces with different coloring schemes thereon, the picture elements are designed to facilitate the simultaneous creation of two images, one on each surface of the constructed sheet. Some rules are provided for generating new designs of such picture elements.

Description

Visual Imaging Construction System

DESCRIPTION

BACKGROUND OF THE INVENTION

The present invention relates to a new form of art and craft based on a self-supporting system of picture elements having different coloring schemes.

The following terminology is used throughout the specification:

Imaging. Any method, system or technique of creating visual images such as painting, drawing, quilting, printing, mosaic, paint-by-the-numbers, computer graphics, sculpting, photography, cutting and pasting. Note that solving a jig-saw puzzle, the user merely reassembles a pre-existing image which has originally been produced by some imaging; no creation is taking place. Thus, solving a jig-saw puzzle is not imaging.

Digital. Discrete gradation of colors, curves, etc. Examples, mosaic is a form of digital imaging. In a sense, stained glass may also be considered digital imaging.

Picture Element or Pixel - The smallest image detail that an imaging is capable of producing, may be defined as a picture element of that particular imaging. A picture element alone contains no information which may be uniquely linked to the image to which it belongs. However, each pixel of an imaging system is unique only within the system itself and it is generic to all of the images that can be produced by that system. The picture element is the most primitive building block, the atom, of the visual imaging system.

Analog. Continuous gradation of colors, curves, etc. Painting is a form of analog imaging.

Friction Fit. A property that facilitates assembly. Tight manufacturing tolerances of the dimensions of the pieces to be assembled provide tight fitting joints, i.e., provides the necessary friction to hold them together. Interloclάng or coupling. A method that facilitates the formation of stable joints between individual elements for the purpose of assembling and constructing objects. In an interlocking joint that is stable in only one dimension, the two elements can be easily pulled apart, but none can be moved along the edge which it shares with the other. Together with friction, the one- dimensional interlocking joint makes nails useful.

A two-dimensionally stable interlocking joint, on the other hand, is formed by a peninsula on a male edge of one component that is inserted into a bay in a female edge of the other component such that forces, acting on the joint within the plane that is common to both elements, will not separate them. To assemble or disassemble such an coupling joint, the force(s) must act in the third dimension that is perpendicular to the plane of the joint. If the two flat elements form a three-dimensional joint, that is the two components do not share the same plane, then an angular force must act on one of the elements (i.e., one element has to be rotated) around its axis that lies in its own plane and intersects the plane of the other element. For example, dovetail joints are used to join flooring panels and two perpendicular surfaces such as in furniture.

Cementing. Any method that facilitates joining individual elements for the purpose of assembly and construction of an object by increasing the friction coefficient between the joining elements as high as possible. Of interest, in particular, are imaging construction systems which utilize the bonding properties of their imaging materials or of additional materials, specifically used for this reason, to adhere the image to a surface and/or to other elements of the image. For example, in mosaic cementing bonds the pieces both to the surface and to each other. In stained glass the pieces are bonded to each other. For self-adhesive tiles the cementing is a layer of material that has been permanently applied to the tiles. In "COLORFORM" static electricity is used to adhere high-static vinyl pieces to each other and to the carrying surface.

Completeness. An imaging system that requires no additional tools, devices and/or materials whatsoever is said to be complete. In contrast, an imaging system that requires some additional tools, devices and/or materials, even if they are provided together as an integral part of such a system kit, is not complete. For example, painting, drawing, mosaic, paint by the numbers, cut-and-paste and stained glass are not complete while LEGO, being a construction set, is.

Color. A general term for any color or color gradation (see below) including black, white, silver, gold, skin tone, etc. A color combination may be composed of various gradations of a single color, such as in a gray scale, or of several colors, such as in a rainbow. It may be created in discrete steps as in a gray scale, or in continuous gradation as in a rainbow. Opaque. Any surface which is not transparent. This includes light reflecting as well as light emitting surfaces.

Color Gradation. A generic term for any type of color density, intensity, hue, shade, tone, etc. A color-gradation scale may have discrete incremental steps or a single continuous change in its range. Whether it is digital or analog, the scale may be between two values of a single color or it may be between the values of two or more colors (e.g., in this sense a rainbow is a linear spectral scale and the CIE Standard Chromaticity Diagram is a two-dimensional spectral scale.) In the first case we shall say that it is a monochromatic color-gradation scale and in the second case it is a spectral color-gradation scale. In any case, a single gradation, that is, a single hue, shade or tone of a color, is any discernible step or segment within its range.

Pattern. A generic term for any repetitive or non- repetitive arrangement of geometric elements (dots, lines, dashes, squiggles, etc.) in any color, color gradation or color combination.

Color Contour. The imaginary line that is formed where two colors or two patterns butt. Under certain conditions, such as where the distinct colors of two adjacent areas blend together, there is no well defined color contour.

Solution. Let a subject be mapped into a visual image. If a lay viewer can visually recognize the correspondence between the original subject and the image that is the result of the mapping, then this mapping is a correct solution. Solution is used (for lack of a better term) to describe situations in which several images of the same model, all closely resembling the original and yet each is slightly different from all the others, having its own perceptual quality, may be produced. In this sort of a case, multiple correct solutions exist. Clearly this is universally true for every image producing system. However, if certain restrictions are placed on the imaging system, this may no longer be true.

The fundamental principle of all visual-image production methods is the arrangement of some light transmitting or light reflecting elements, in consideration of the complex human vision system and psychology of perception, such that the desired visual image will be perceived. These elements may transmit the light directly, being its source, as the light bulbs of a scoreboard or the fluorescent dots of a television screen. Alternatively, the light may be transmitted through these elements as is the case of stained-glass images or the light may be reflected off of these elements as is the case of most visual-image production techniques like drawing, painting, sculpting, etching, curving, writing, printing, mosaic, photography and quilting. Each perceived color or shape is affected by the other colors and shapes that are in view. In order to produce the desired visual image every visual-image producing technique must satisfy two necessary conditions. First, it must be capable of arranging some light generating, transmitting or reflecting elements in a desired composition. Second, it must provide the arrangement with some support by which the arrangement is positioned in space so that the arrangement can be completely produced and, once that has been accomplished, perceived. Arranging paint pigment is not sufficient to produce a painting. A canvas onto which the pigments are to be placed and a method by which the pigments will adhere to the canvas are also necessary. If the paint is too thin, it will drip and "run" and the painter will not be able to paint, let alone display his or her artwork.

Most visual-image production methods, including painting, mosaic, drawing, printing, photography, tiling, video, curving, etching, laserium, quilting and others, utilize a medium onto which the image is applied for decorative and/or utilitarian purposes. Most often the medium is a surface as the drafting paper, the tiled wall, the mosaic floor, the television screen, the etched glass, the projection screen or the quilted blanket. The most common and popular methods cover the surface with the image and in most cases, the image is supported by its medium by cementing. Other methods embed the image in the surface, superficially alter the surface or project the image onto the surface. Very few imaging methods utilize a medium which is three dimensional. In air writing, which is produced by aircraft smoke, for example, the atmosphere is the canvas, so to speak.

A few visual image production systems require no preexisting medium. These techniques manipulate raw materials to produce the image and construct its support simultaneously. Such systems including sculpting, stained glass and glass blowing. Another system involves constructing shapes with "LEGO".

"LEGO" employs a unique joint to achieve stable and solid construction. The "LEGO" elementary construction unit is a three-dimensional block and the joint utilizes the principle of nailing. The "LEGO" pieces are fastened to each other as the prongs of one piece are friction fit into the other. In the direction of the insertion, stability of the joint is achieved by the friction fit. Stability in the plane that is perpendicular to this direction is achieved by the presence of the prong or prongs of one piece in the cavity or cavities of the other. Rotational stability (i.e., preventing rotation) is accomplished whenever it is desired by having two or more prongs of one piece inserted into the other. For the sake of simplicity in the construction of three dimensional objects, the "LEGO" pieces have a single male surface and a single female surface and the two are opposite each other. This arrangement provides for only one- axis joints, commonly it is the Z or the vertical axis. Structural strength in the X and Y axes is achieved by staggering the vertical joints. One of the properties of every visual image production method is its resolution. Resolution may be defined as the limit beyond which the imaging technique is incapable of producing fine details. The resolution of painting is a function of such variables as the size of the brush, the number and thickness of its hair, the shape of its tip, the size of the color pigments, the size of the canvas fiber and the painter's skill. A viewer's capability to discern fine details is a function of several variables, including the viewer's distance from the observed image, the relative stability of the image with respect to the viewer, and the image illumination conditions. Thus, under the right viewing conditions, the perceived resolution of a low resolution image may be equal to that of an image with higher resolution. For example, from the proper distance, curves and colors appear continuous on a television screen. Using low resolution may also have a desired visual effect beyond the viewer's capability to discern fine details. The impressionists like Van Gogh and Cezanne applied paint to their canvas in loose and coarse strokes, they dabbed, splattered dots and avoided minutely precise painting techniques.

As stated, the smallest image detail that a visual image production technique is capable of producing, may be defined as a picture element, or pixel, of that particular method. A picture element alone contains no information which may be uniquely linked to the image to which it belongs. However, each picture element of a visual-imaging system is unique only within the system itself and it is generic to all of the images that can be produced by the system.

There are different types of application of picture elements. The most primitive picture element is the kind of which all properties, including its dimensions, that is, its size and shape, its color and its spatial positioning scheme, are all fixed and unchangeable. An image is formed by either placing such a picture element or not. The combination of the presence and absence of picture elements produces an approximation of the desired details, such as curves and colors. This technique is common in such popular applications as newsprint, silk screening, computer applications like dot-matrix printers and scoreboards.

Many systems vary some or all of the properties of their picture elements. For instance the image area can be divided into a grid of some sort and the picture elements are positioned in it so that they cover the image without overlapping and with minimal gaps. Such a system contains a finite number of different picture elements, differing in the size of grid-cell area that each one covers. The arrangement of these picture elements and the combination of the covered area and its background produces feature details and color gradations. The technique of producing images by using alphanumeric characters and U.S. Patent Nos. 3,987,558 and 4,398,890 belong to this category. Some techniques provide infinitely many different picture elements. Such are mosaic and Cezanne's painting. Further, some systems utilize picture elements which themselves are conglomerations of more primitive picture elements. For example, computer graphics may make use of the alphanumeric characters to form images, utilizing the complete shape and density of the characters; at the same time each character is composed (on a video screen or by a dot-matrix printer) of an array of dots. Such a computer graphic system does not attempt to use the individual dots to achieve a finer degree of resolution.

Some systems utilize three-dimensional picture elements to produce two dimensional images. Among these are mosaic, tiling, and the techniques based on U.S. Patent Nos. 3,987,558 and 4,398,890. Other systems utilize three-dimensional picture elements to produce three-dimensional structures. The most common of these are the use of bricks to construct buildings and structures and the "LEGO" construction toys; the bricks and the "LEGO" pieces are their respective picture elements.

The process of visual perception is very complex. Developing the present invention took into consideration the recent scientific discoveries and the current descriptions of the visual perception and, in particular, that of color and form. The brain processes the color and the form elements of the perceived separately; the results are integrated, affecting one another in the totality of the wholly perceived image. For example, using paint-by-the-numbers it is not necessary to perfectly color to the border lines; the viewer's brain will form the image by adjusting miscoloring to the perceived form. The bottom line is that using picture elements to approximate an image, one can cause a viewer to actually perceive the image (which the creator used as a model.) Moreover, the same model may inspire multiple images, created from picture elements, all correct solutions yet perceptually different from one another. Images, illusionary at times, can be recognized when only a few scant details are presented.

During the past two decades computer scientists have studied closely all aspects of visual perception in their quest to produce high-quality computer graphics. Since computer images are digital and limited by the worst resolution that is dictated by the hardware and software, methods were developed to improve the quality of the image without actually improving the resolution itself. One particular method, known as anti-aliasing, enables the user to generate curves that are perceived to be smoother than they really are. This is achieved by spreading picture elements of varying lower intensity along the main body of the curve that is composed from picture elements of the most highest intensity desired. A special algorithm determines the location and the gradation of the additional picture elements. This method requires that the picture elements are available in a gradation scale. Another method achieves a similar result by using very fine pixels only wherever the ordinary, more coarse pixels cause curves to appear discontinuous, having a "staircase" visual effect. There are important mathematical aspects to picture elements that deserve consideration. Of utmost value are the geometric properties of the picture elements, among which are symmetry and asymmetry, tessellation and lattice.

A one-to-one mapping PQ --> Pi from all the points in a plane (or in a space) to all the points in the same plane is called transformation. In any transformation, every point PQ — > Pi that is mapped to P such that PQ = P_j, if any, is called invariant point. A transformation is called an isometry, or congruent transformation, if it preserves length; for every line PøQo ^tnat is mapped to PiQi, P()Qθ ⁼

For example, translation, or parallel displacement, is an isometry with no invariant points. Rotation of the plane about point P is an isometry with P its only invariant point. A reflection is an isometry in which all the invariant points coincide with the line of reflection, or the mirror. A set of transformation is said to form a group if it contains the inverse of each and the product of every two. A figure is said to be symmetrical if there exists an isometry which maps every point PQ on the figure to point P_j on the same figure. The four fundamental symmetry operations are reflection, rotation, translation and glide reflection. Glide reflection is the product of reflection in a line and translation along the same line. There are seventeen symmetry groups.

Tessellation is the property of any arrangement of two- dimensional regions (e.g., polygons, any area bounded by a simple closed curve) fitting together without leaving gaps or without overlapping so as to cover the whole plane. A picture element that can by itself, in its plurality, cover a whole plane without overlapping and without leaving gaps between individual picture elements, is said to be tessellating, or to have a tessellating shape. The basic tessellating polygons are the triangle, the quadrangle and the hexagon. Every triangle and every quadrangle, including scalene (all sides are of unique lengths) are tessellating. The circle, the equilateral pentagon and the equilateral octagon are not tessellating shapes. If a finite number of different shapes may be used to cover the plane, then that set of the unique shapes is said to be tessellating. The set of an equilateral octagon of a given size and the diamond that is formed between such four tangent octagons and the set of a circle of given radius and the equilateral triangle with concave sides that is formed between such three tangent circles are two tessellating sets. There is no finite set of circles that is tessellating. Tessellation is fundamental to tiling. The regular repetitive arrangement of tessellating shapes or tessellating sets of shapes is a lattice or grid.

All of the symmetry groups are important tools in generating tessellating picture elements. However, bilateral symmetry and rotational asymmetry are of critical concern to the present invention. A figure has bilateral symmetry if and only if there exists at least one line that bisects the figure such that for every point in one section of the figure there exists one and only one corresponding point on the other section of the figure, and the following holds true: (a) Any line between any pair of such points is perpendicular to the bisecting line, (b) The distances of the two points of any such a pair from the bisecting line are equal. This symmetry is often referred to as symmetry by reflection or mirror symmetry and the bisecting line is its line of reflection. A figure has rotational symmetry if and only if there exists at least one point around which the figure can be rotated by an angle alpha, such that alpha is greater than 0 degrees smaller than 360 degrees and the figure in its new orientation can be precisely superpositioned over itself in its original orientation. Such a point is called axis of rotation. A figure that has no rotational symmetry is called rotational asymmetric.

Symmetries preserve one another. For example, if a picture element is both mirror and rotational symmetric then its mirror image is also rotationally symmetric around the same axis of rotation and to the same degree of rotation and vice versa. Similarly, if a picture element is mirror symmetric and rotationally asymmetric then its mirror image also has no rotational symmetry.

The tessellating quality of a picture element is unchanged by any of the symmetric operations. Thus, if a picture element is tessellating, then any picture element that can be derived from the original one by an isometry is also tessellating. Such is the case with mirror and rotational images whether they are symmetric or not.

Both the practical and artistic purposes of tiling are to cover a surface with periodic or nonperiodic patterns. For tiling, tessellation and the existence of symmetric and asymmetric designs are, therefore, essential. Being a complete visual imaging system, this invention is of a general purpose. Thus, neither tessellation nor the existence of symmetric or asymmetric designs is of intrinsic value beyond that which exists in a particular implementation. Tessellation and symmetric and asymmetric geometric relationships are used by specific implementations of this invention as tools and guidelines for the design of picture elements.

The imaginary well-defined line that is formed where a color butts against the colors that surround it is the color contour of this color. A color contour may be incomplete if the color which it surrounds butts against the edge of the picture. Under certain conditions, such as two colors blending into one another, there may be no well-defined color contours. Color contours have an important role in the way images are perceived. For instance, careful arrangement of color contours may create the illusionary presence of image features that are not really present. Some image producing systems readily facilitate color contours. Among these are paint-by-the- numbers, stained glass and mosaic. With other methods it is easier to blend adjacent colors than to keep them distinct from each other.

The present invention provides a class of devices called picture elements, or pixels, to be used in their plurality for the production of two- and three dimensional visual images for decorative and utilitarian applications. The unique feature of this invention is the fact that it is a complete imaging system. That is, the pixels, substantially flat physical objects, facilitate the simultaneous construction of the support structure of the image while creating the work of art. No additional tools or materials are necessary. The invention also provides methods for producing visual images and structures in two and three dimensions from these picture elements.

A fundamental principle of the present invention is the introduction of a variations on tessellating shapes according to a specific set of objectives. Some pixel designs provide for all the primary objectives of the invention, that is, self-support, completeness, reversibility, dual imaging, flexibility in color contouring and two- and three-dimensional imaging. Other pixel design provide for some combination of said objectives. In addition, the invention exploits tessellation by making use of its principles but not adhering to them. In particular, there is no restriction against gaps between picture elements.

The elements of this invention are described below. These elements are divided into two types: main objectives which are necessary for all configurations and applications and secondary objectives which are necessary for some configurations or applications.

Self-Support and Structural Stability. An objective of this invention is to provide a picture element that alone, in its plurality, or a small finite set of picture elements in their plurality, facilitate for both the production of at least one image and simultaneously the construction of a stable support structure for said image, comprised of at least one, substantially flat medium. In this sense, the pixels are said to be self supporting. Unlike most visual imaging systems, the support structure of the invention is an integral part of the visual medium. Other imaging systems such as stained glass and sculpting are similar in this manner. For example, while tiling covers an existing surface, the imaging system of the invention creates a new independently stable structure with at least two surfaces of which at least one displays an image.

Self support and structural stability free this imaging construction system from all of the common physical constraints that other imaging systems have to contend with. Self support is the result of the application of a combination of interlocking and friction fit. Interlocking prevents any pair of picture elements from being disjointed by any linear or angular forces which may act on them in their plane or planes. To achieve interlocking every picture element of this imaging construction system has to have at least one male edge or at least one female edge or at least one edge that has both a male element and a female element. A male edge is composed of the normalized edge of the body of the picture element from which a protrusion extends outwardly. This protrusion is named peninsula. The female edge is composed of the normalized edge of the body of the picture element from which an intrusion extends inwardly. Said intrusion corresponds in all of its dimensions to the said protrusion. Said intrusion is named bay. Thus, the peninsula and the bay are complimentary to each other. A non-gender edge includes at least one peninsula and at least one bay. Since the peninsula is composed of the material of the pixel and the bay is composed of the absence thereof, each is the inverse shape of the other. For all practical purposes the dimensions and the shapes of the bay and the peninsula are identical; some manufacturing technologies (e.g., due cutting) provide that, any time a peninsula is produced, a corresponding bay is also produced, and vice versa.

The coupling scheme of a pixel or of a set of pixels is defined as the species of said pixel or set of pixels. If two pixels have different interlocking schemes, that is, any variation in shape or dimensions which prevents stable coupling between said pixels, then said pixels belong to different species. On the other hand, if two picture elements of different shapes (e.g., a square and a triangle) can be interlocked since they have a matching male and female edge, then these pixels are of the same species.

A stable two-dimensional (planar) coupling joint can exist if the interlocking scheme is of either the bottleneck type (Figure 12a-i), the gooseneck type (Figure 13a-e S-shape type (Figure 13f).

Strictly speaking bottleneck joints are a subset of the gooseneck joints. The distinction is made since all bilateral symmetric bottleneck joints are capable of stable and solid spatial interlocking. Gooseneck joints that do not belong to the set of bottleneck joints cannot have bilateral symmetry and, therefore, are incapable of forming stable and solid three-dimensional coupling.

A male or a female edge is generally referred to as an coupling edge. A non-interlocking edge is a plain edge. All the edges of every general-purpose pixel are coupling edges. Special- purpose pixels may have plain normalized edges, e.g., a framing pixel may have a straight or curved edge or two or more straight or curved adjacent edges. Pixels with a single interlocking edge are also useful. For instance, each of the three framing triangular pixels at each of the corners of a triangular image has only a single coupling edge. To maximize the usefulness of pixels, it is preferable, but not necessary, to have an equal number of male and female edges, unless, of course, the pixel has an odd number of edges. In this case, one gender of edges necessarily has at least one more edge than the other gender. Of course, non-gender edges may be combined with male and/or female edges.

Friction fit provides stability to every interlocked pair of pixels against a disjointing force(s) that act on them along a line(s) or a curve(s) that is perpendicular their common plane. Friction fit is accomplished by a combination of precise manufacturing technologies, achieving precise tolerances, and a careful selection of materials. Perfect matching between bays and peninsulas of the same species regardless to the product edition, the age and the amount of previous use of the pixels.

Together interlocking and friction fit facilitate the stable interconnection of every two picture elements of the same species of this imaging system. The significant outcome of this fact is the user's capability to generate arrangements of such picture elements without any restriction (other than the resolution constraint) on the size or the shape of the arrangement. Other systems also depend on the properties of their support platforms (e.g., U.S. patent no. 4,398,890 depends on the size and shape of the support tray; painting depends on the size and shape of the canvas or sheet of paper; U.S. patent no. 3,987,558 depends on the distance between the pegs or cavities in the platform and other properties thereof.)

The design of the interlocking scheme must take into consideration the tessellating property of the normalized body of the pixel. For, if the geometric shape of the body of the pixel is tessellating, reshaping it with the addition of one or more peninsulas and the subtraction of one or more bays, may or may not render it non-tessellating. Similarly, one must consider the effect of the shape of the coupling edge on the reversibility of the pixel, i.e., on the ability to interlock the pixel with another which is shaped precisely as its mirror image.

In order to facilitate the interlocking of two picture elements which are placed in intersecting planes (Figure 24d), the coupling scheme should have bilateral symmetry. This restriction limits the variety of the interlocking designs to the bottleneck joints. In order to form a three-dimensional joint between two flat picture elements, one must do the following in the prescribed order:

Color Schemes and Color Gradation. Once a support for the visual image can be constructed, coloring and patterning schemes facilitate its visual creation. In the case of this invention, as, for example, is the case of sculpting, the two processes (that of the construction of the image support and that of the image creation) proceed simultaneously. Each surface of every opaque picture element is provided with a homogeneous (not necessarily uniform) color, shade, hue or tone of color. The term coloring scheme refers to any of the finite, but quite large, number of combinations of colors, color graduations, analog and digitals, monochromatic and spectral, in various ranges, number of gradations, pattern combinations, etc.

It is an objective of this invention to provide pixels which facilitate for color gradations with respect to the other pixels with which they may be interconnected. The gradation scale may be digital or analog; it may be monochromatic or it may be spectral. Similarly, textural gradation and the gradation of geometric patterns, both with respect to the density of the textures and the patterns and with respect to their colors, may be facilitated by the pixels. Combinations of any of such gradations are also possible.

By definition, the color scales are designed to facilitate both variations in color intensities, densities, shades, tones and hues. The color scales may vary in the number of gradations per scale to accommodate different applications. They may also vary in the incremental (or decremental) changes between the different gradations of the same scale or of similar scales. For example, non-professionals, such as children, may be satisfied with a small number of gradations per scale while professionals and serious artists may require much finer details. This invention does not impose any limit in this respect, one way or the other.

Color Contours. In addition to providing pixels with coloring schemes to satisfy a large spectrum of needs, it is also an objective of this invention to utilize the shape of the pixels to affect the perception of the color contours in the produced images. The pixels can facilitate either well defined color contours, or blending of colors along their contours, or displaced colors.

Rotationally asymmetry facilitates flexibility and versatility in the approximation of curves and of color contours. Therefore, it is another objective of this invention to provide pixels that can be interlocked (whether or not they tessellate) with their rotational asymmetric counterparts. Tessellating sets of pixels may be composed of rotationally symmetric pixels. In such a case, some collective shapes that are formed by interconnecting some or all of the pixels of the set is rotationally asymmetric. Since in most cases (the few exceptions are circle, ellipse, etc.) every shape that is rotationally symmetric has only a finite number of axes of symmetry, it is possible to position a pixel in a rotationally asymmetric orientation thereby forcing a rotational transformation on the whole image which may alter the alignment of some, or even all, of the color contours. This objective may conflict with the objective to facilitate two- and three-dimensional images and structures. Thus, it is not necessary that a tessellating picture element or a tessellating set of pixels satisfy both of these objectives.

Dual Imaging. It is an objective of this invention to provide opaque pixels which facilitate for the simultaneous production of more than a single image. That is, since every opaque sheet of pixels has two surfaces, an image may be created on each surface. If only a single image is produced at a time, it is by the user's choice, not a necessary result of the design of the picture element. No symmetry is necessary in order to provide for dual imaging. Three- dimensional picture elements facilitate for the simultaneous production of at least four images.

Each of the two surfaces of opaque pixels is colored. Thus, every opaque sheet has a single monochromatic or spectral, digital or analog color scale on each of its surfaces. The combination of the two color scales, one per surface of the opaque pixels, is designed to further maximize the effectiveness of the pixels. Having a different color per surface, facilitates the simultaneous creation and display of two images, one per surface. In particular, one image may be the mirror image of the other. When opposing color scales are selected, then one image may be the "negative" of the image on the opposing side. That is, images with the same contour lines but with different coloring are "automatically" formed. Or, given "unrelated" color combinations, it is possible to create an image on one surface and, totally disregarding the other surface, end up with a random mosaic. Using the similar "unrelated" color combinations, it will be very challenging to create two different and unrelated images, one per surface. Similarly, a sheet of transparent or light emitting pixels has a single either digital or analog, either monochromatic or spectral color scale. So, a transparent or light emitting image (note that light emitting pixels may have the properties of either opaque or transparent pixels) may be viewed from either side. Transparent images can be formed and be displayed with rear illumination. Dual imaging has further significance when this imaging system is used to create three-dimensional images.

Without its self-support quality, dual imagining requires special support system or is out right impractical. For example, a cut-and-paste imaging may be used for dual imaging. But the pasting must be done onto both surfaces of the paper, the adhesive on one surface must not interfere with the image on the opposite surface, the artwork must be hung to dry so nothing else adheres to any of its surfaces, and no "automatic" dual -imaging is possible.

Reversibility. In order to facilitate maximum flexibility in the use of the pixels, it is an objective of this invention to provide pixels that can be interlocked with their mirror images.

In other words, the front surface of one pixel may be interconnected with the back surface of another identical pixel. Bilateral symmetry is a sufficient but not necessary condition for complete interchangeability of the front and the back surfaces of pixels. That is, the front surface of one pixel can be interconnected with either the front or the back surface of another pixel. Using bilateral symmetry and asymmetry, it is possible to design pixels that interconnect with either their own duplicates (i.e., translated images) or with their mirror images or with both.

If both surfaces of a pixel are of the same species, the pixel is said to have left-right symmetry. A pixel that has no left-right symmetry is said to have one left-hand surface and one right-hand surface. The choice of which surface is the right-hand and which is the left- hand is arbitrary. Left-right symmetry exists if and only if bilateral symmetry exists. That is, a pixel is left-right symmetric if and only if each of its surfaces has at least one line of reflection which is the axis of this surface's bilateral symmetry. Left-right symmetry is desirable; both surfaces of a left-right symmetric pixel can be used in the creation of an image. Namely, a left- right symmetric pixel can contribute either of its colored surfaces to the image, not only one of them, as is otherwise the case.

Two- and Three -Dimensional Image Construction. It is an objective of this invention to provide picture elements that provide for the interconnecting of intersecting surfaces, which were formed from the same picture elements, such that the joint between the surfaces is stable. The interlocking of surfaces provides for the production of multi-surface visual images and/or other three-dimensional structures.

Note that a flat sheet that is constructed from coplanar picture elements is structurally stable so that forces can be applied on it so it becomes a three-dimensional curved surface. In order to preserve such a contorted shape of a flat sheet, the edges of the sheet must be affixed in their relative position in space. That can be achieved with additional picture elements.

To facilitate the construction of a stable three-dimensional corner, that is, where three surfaces intersect, it is necessary that: (a) at least one more surface is constructed, such that the fourth surface intersects with at least two of the first three surfaces; and (b) every two pixels, between which there exists an intersection, must be interlocked. Said surfaces may be planar or curved.

Completeness. It is an objective of this invention that no additional means whatsoever, other than its pixels, are necessary for accomplishing the tasks of producing the image and constructing its support. That which produces the image simultaneously constructs its structural support and maintains its stability. Furthermore, it is the design of a single feature of the pixel(s) which provides the pixel the unique combination of self support and structural stability, reversibility and the capability to manipulate color contours. This feature is the shape of the pixel. Therefore, this system of producing images from physical pixels is said to be complete. If the primitives of painting are the paints, the paint diluting materials, the paint mixing tools, the brushes and the canvas, if the primitives of drawing are the pencils, the paper and the eraser, if the primitives of mosaic are the stones, the stone dying materials and tools, the chisels, the cement and its application tools and the floor or wall onto which the mosaic is applied, then the one and only primitive of the image producing system according to the invention is its picture element.

From its completeness and all the features of the picture element which are listed above, the following features are derived according to the invention.

Limitless variation of Image Size and Shape. A further objective of this invention is to readily facilitate any size and shape image (larger than a single pixel and within the constraints of the shape of the picture elements). For example, to produce a single painting in the shape of the letter "A" requires a customized canvas; this invention readily provides for an A-shaped self-supporting image.

Changeability. Another objective of this invention is that at any time, during the production of an image and after its completion, any part of the image may be modified. Any individual pixel or a group of pixels can be separated from the structure of the image and replaced by another or others without adversely effecting the rest of the image. The replaced pixel(s) may be reused immediately or at any time in the future in the same image at any place or in any other image. This invention is thereby tolerant to errors and facilitates trial-and-error experimentation. Furthermore, a completely formed image can always be disassembled down to its individual picture elements so they may be reused to produce a variation of the disassembled image or another totally different image.

Simplicity and Ease. It is a further objective of this invention to require no special skill for its use. Producing an image and the construction of its support, replacing pixels in an image with others and disassembling an image are all simple and easy tasks. Moreover, the technique by which the pixels are used to produce images and construct their support is intuitive. The skills, such as dexterity, and intuition of a young child suffice. Even some persons with limited physical capabilities such as older or some physically or mentally disabled persons who are not able to paint or draw will be able to create works of arts using imaging systems provided by this invention.

Durability. The durability of the pixels is substantially independent of and is unaffected by their use. Unlike using paint pigments and paper, mosaic stones on walls, and other such imaging methods, using pixels does not substantially alter their properties or significantly increase their wear and tear. The pixels require no maintenance and are resistant to the normal conditions that exist in the normal household, office and studio environments. The pixels attract no foreign elements as, for instance, adhesive or static electricity attract dust.

No Special Environmental Conditions Required. Since no additional materials, chemicals or tools are necessary, this invention provide for a predetermined safe environment where images can be produced. For example, parents can therefore be satisfied that in such an environment children are safe. The additional safety needs of very young children can be addressed by especially designed picture elements. A complete environment for the use of this invention is provided at any location that contains a user of this invention, some picture elements and nothing else.

Versatility. A further objective of this invention is to provide the gamut of users, from the professional artist to the lay child, with a choice of pixels and sets of pixels for the gamut of applications. For the lay person, this imaging method compensates for the lack of the mechanical skill that is so often a prerequisite for other imaging methods. The professional artist, on the other hand, will find very few restriction to his or her imagination. For instance, this invention provides a wide range of resolutions both in pixel size, color definition and curve approximation.

Tessellation is not a necessary condition for the development of any picture element or any set of picture elements for this imaging construction system. An image created by this imaging system may contain gaps between picture elements and/or overlapping picture elements. The acceptance or rejection of gaps and/or overlapping, depending on a particular implementation of this invention, is the choice of the user (Figures 12d, 12g and 12i). Tessella¬ tion is used only as a guideline in designing picture elements. For example, a circular pixel (as pixel 17a in Figures lla-b) can be used quite effectively; true, a planar image will have one gaplδa between every three interlocked pixels 17a, shaped as a triangle with concave sides. Or, if the circular pixel is of the type of pixel 17c (Figure llf), then there is a single larger gap 18b between every four interlocked pixels. But, if an artist desires a visual affect or if the perceptual affect of such gaps is negligible due to the resolution of the image, then the imaging system does not impose any restriction against having such gaps.

A picture element that has some planar asymmetry may be placed at the same lattice point in different orientations. Two cases are described as follows:

1. Reorienting some pixels leaves the rest of the pixels in the lattice unaffected.

In this case a pixel is repositioned such that in each orientation variant its main body is congruent to all of its other orientation variants. The variations are achieved only

reorienting the male and the female edges in differed ways (Figures 22k-p).

There are several ways to connect every two pixels. As more and more pixels are added to an image, the number of possible combinations grows. However, the interrelationship between the pixels may dictate the orientation of other pixels, or even which one may be used at all. For example, after several picture elements have been interlocked, the orientation of their male and female edges starts to dictate the orientation of some future adjacent pixels. Any attempt to avoid such a dictated orientation will result in a chain reaction - some of the previ¬ ously placed pixels must be reoriented.

The arrangement of pixels is governed by mathematical rules. Of particular importance are the laws of combinatory and group theory. Examination of these factors will advance one's creativity. For example, educational products based on this invention can be used to teach group theory and the study of combinatory.

For instance, there are six basic square pixels (Figure 22a-f). Every other square pixel can be generated from one of these by rotation. For instance, if pixel la is rotated by 90-degree angles, three additional pixels are generated (Figure 22g-j). This pixels can tessellate the plane and the next set of drawings illustrates the progression of arranging this pixel in several possible two-by-two arrays. It can be seen how the possible combinations can grow rapidly. Orientation of the fourth pixel, however, is dictated by the previous three pixels (Figure 22k-p).

This effect is significant with picture-element species which have area ratios that result with the blending of color contours.

2. Reorienting even a single pixel causes the reorientation of all pixels in the lattice. Given a lattice of pixels, pull one of them out of the arrangement and rotate to a new orientation in which its body is no longer congruent with the way it was inside the existing lattice. Namely, disregarding the male and female edge, the main body of the pixel cannot be placed back in the existing arrangement. If we want to keep this pixels in its new orientation, all the other pixels in the lattice must also be rotated with respect to their existing orientations at the same rate and direction as was the first pixel was rotated with respect to its original orientation. (Figure 21.)

Since the image is digital, that is, color contours are only approximations of continuous curves, the new orientation of the lattice will necessarily result with a different digitalization of the same color contours. For example, rotating a square picture element clockwise by 45 degrees from its initial north /south and east /west orientation will result in a northeast / south¬ west and north-west / south-east lattice orientation. (Figure 21.) This effect is significant when the created image includes digitized curves or various angles are being approximated, regardless of the perceptual effect of the area ratio. For example, consider the face shown in Figure 25. It is constructed from square picture elements having their edges positioned parallel the edges of the sheet of paper. Rotating one of these pixels by 45 degrees in either direction will force the reorientation of all other pixels in a similar manner. As a result, the approximation the original face will have to be modified. For example, the jaw lines are approximated to appear in 45 and 135 degree angles with respect to the horizontal mouth. After rotating the pixels the jaw lines may be constructed in the proper angles and the horizontal mouth will have to be approximated.

This imaging construction system facilitates more then the reorientation of the whole lattice. Since it imposes no restriction on the overall shape of the image, it provides the freedom to maintain the orientation of the shape of the image while its underlying lattice is reoriented. In other words, the overall shape of the image - say, a vertical rectangle such as in a portrait - can be maintained regardless to the orientation of the lattice that forms it. Note that this is not the case for imaging systems that depend on a support platform. To reorient the lattice of such a system (e.g., those described by Patent No. 4,398,890 and 3,987,558) means to reorient the support platform. Or the edge picture elements must be significantly altered (e.g., reorienting the lattice of tiles which has matched a floor perfectly, will require breaking some of the tiles where the floor meets the walls.) Thus, in such other imaging systems, the outer edges of the image are also reoriented; or the image must be sufficiently smaller than its supporting platform so it is possible to avoid the limitations the platform imposes on its size and overall shape. This invention imposes no such limitation.

The invention as described here is a general purpose visual imaging system. Like any other imaging system, it has many and versatile applications such as artistic, decorative, educational and practical applications.

The following is an example of a practical application which takes advantage of several specific qualities of this invention. According to this invention, a line of do-it-yourself products for tiling walls, floors and other surfaces can be developed. While tiling a surface with an existing method renders the surface is unusable for the duration of the process. This is not the case for a tiling system based on this invention. In fact a family may take its time experimenting with various designs, tiling a kitchen counter, while the kitchen remains fully functional. Food can be cut and processed on the interlocked tiles; if something is spilled, tiles can be removed, cleaned and replaced. Only when the desired result has been achieved, the tiles can, if so desired, be set permanently. This invention describes methods for generating picture elements or sets of pixels that fulfill its objectives as well as methods for using these pixels or sets of pixels for creating works of art.

The pixels of the invention can conveniently be manufactured and distributed in sheets. A sheet may be composed of at least a single group of pixels, each of which is interlocked with its adjacent pixels. If the sheet is framed, then the frame is interlocked to the outer pixels. Such a frame is an integral part of the sheet of pixels. If a sheet contains several groups of pixels, then within each group, the pixels are interlocked with their adjacent pixels. Further, if an integral frame is part of such a sheet, then the outer pixels of each group can also be interlocked with the frame that surrounds them.

Patterning Schemes. The coloring scheme of the picture elements and their sheets may be of two types. A solid coloring scheme is one in which the resolution of the color applied to the surfaces (or, in the case of transparent picture elements, to the body picture elements themselves) is sufficiently high to allow no discernible patterns of any sort, random or otherwise. A patterned coloring scheme, on the other hand, is one in which the resolution is sufficiently low to permit clearly visible patterns, random, geometric or of any other kind. Patterning may be produced in one color-gradation scale on the background of another color or another color scale. This invention does not impose any limit with respect to patterning, one way or the other. By mixing pixels having different patterns or by mixing pixels having pattern with pixels having no pattern, it is possible to create new patterns which are of coarser resolution than the size of the pixels themselves. Indeed, this imaging system leaves the artist or creator in charge; given sufficient demand for a particular pattern design, it can be manufactured (by the individual who desires it or by anyone else).

Dual-Surface Functionality. A pixel may be designed such that its body and the arrangement of its male and female edge facilitate the coupling of it with its mirror image. In other words, such a pixel may be turned over to be interlocked with its clone that has not been turned over. The functionality of each pixel is thereby doubled. The result is that, if N is the number of desired color scales, then if N is even, then it is necessary to produce only N/2 sheets of pixels and, if N is odd, then it is necessary to produce only N/2 + 1 sheets.

Multi-Layered Image Construction. Given sufficient thickness and/or sufficiently high friction coefficient for the friction fit, then it is possible to couple regular picture elements in a staggered arrangement, with respect to their thickness, thereby generating multi-layered images. A step further will be to provide picture elements especially designed for such construction. Such picture elements can be created as if two picture elements were cemented to each other along their flat surfaced. These picture elements, in combination with the simple single-layer picture elements, open a new realm of multi-layered imaging (Figure 24a-g).

Although double layered pixel may be designed as such, it is simpler to describe its design and behavior as if it is composed of two single-layered pixels "sandwiched" together. Such two single-layered pixels may or may not be identically shaped or even of the same species. If they are identical, then they most likely be positioned in a non-congruent orientation with respect to each other; otherwise they form a single double-layered pixel having an identical shape. The most common design of double-layered pixels is to combine two single-layered pixels having the same normalized body shape and being of the same species but not being identically shaped.

Multi-level imaging is the most obvious goal of these double-layer pixel. The combination of reflecting, transparent and light emitting pixels with other pixels having same or similar optical properties is a most powerful application. Such a combination can generate new color combinations and visual effects.

Several factors contribute to the selection of the optimal size of pixels for a specific application. The first is the image resolution. Clearly, the smaller the picture element, the greater is the resolution. Dexterity and the user's capability to manipulate the picture elements is the second consideration. Some users, such as young children or physiologically disabled, may be limited in their ability to manipulate small objects. Thirdly, manufacturing constraints, including tooling, manufacturing precision, material limitations, etc. limit the smallness of the pixels or the smallness of the bays and peninsulas. Fourthly, is safety. For instance, pixels to be used by very young children and infants, are required by law to be sufficiently large so they cannot be swallowed. The first constraints conflicts with the second and with the third ones.

It is possible to manufacture pixels which have peninsulas and no bays. These special peninsulas, called islands, are removable such that, when a peninsula is removed, a bay is left in its place. This lets the user determine the interlocking scheme of each pixel so constructed. If sufficiently high friction coefficient is provided between the islands and the bodies of the pixels, then an island may be reinserted into a pixel from which it has been removed. In a sense, such island become "Sub-pixels" - they are atomic image elements which contain no bay or peninsula. However, independent use of islands does not provide for the strength and stability of ordinary pixels.

Other types of picture elements include special purpose pixels for framing or constructing three-dimensional images. Framing strips are only one pixel wide. They have one long edge that is made of only convertible pixel edges. The other long edge is straight or in the shape of any curve, such as a sinusoid, that forms a finished edge. Pixels from such a strip can be fitted to the jagged edge of a completed image in order to provide the image with a finished edge. Corner pixels are also possible. These are picture elements with two adjacent finished, i.e., non-interlocking, edges. The designs of corner picture elements match the designs of the available framing strips. Pictorial pixel are also possible. As M.C. Escher has demonstrated, it is possible to design tiles that have pictorial images other than pure geometric shapes. For various reasons (e.g., educational promotional and artistic) it is, therefore, possible to design pictorial pixels or sets of pixels.

As described earlier, it is possible to construct three-dimensional structures from the flat two-dimensional picture elements. In particular, it is possible to generate three-dimensional pixels from the two-dimensional pixels. For example, six flat square pixels of the same species can form a cubical pixel; and four triangular picture elements of the same species (not necessarily the same as that of the squares') will generate a pyramidal picture element. Such picture elements can then be interlocked with others of the same three-dimensional species.

Unlike existing three-dimensional construction systems that utilize blocks, these pixels can be interlocked along every edge of every surface. For example, only two of the surfaces of every "LEGO" piece can be used for coupling. In contrast, a cubical picture element made of six flat square pixels (the type shown in Figure 22c) can be interlocked along each and every one of its surfaces (as shown in Figure 24h); moreover, coupling is possible along the four edges of each and every surface. Since the three-dimensional pixels are actually a composition of flat two-dimensional ones, whenever a corner is created by an arrangement of such cube cubes in all three axes, it is sufficient to use only three additional flat square pixels to "fill" the corner, as if an additional cubical pixel is used.

Two equilateral right angle triangular pixels, a first pixel with a female edge along its hypotenuse and a second pixel with a male edge along its hypotenuse, can generate a single square pixel. In addition to the five angles that can be generated by a square pixel - 0, 90, 180,

270 and 360 degree (see Figure 20e-f) - the set of right angle pixels mentioned here facilitates polygons with four additional angles: 45, 135, 225 and 315 degrees. However, the hypotenuse is of an odd length. Although a male edge of the hypotenuse length may be interlocked with a female side edge of the triangle, and vice versa, gaps or an irregular edge may occur. A set of two equilateral triangular picture elements (Figure 20b-c) having two female edges and one male edge, or two male edges and one female edge. Further, it is possible to generate any regular hexagonal picture element from some combination of these two triangular picture elements (Figure 23a-j.) The value of the hexagonal picture element is in its close approximation of a circle. The set of square pixels and the set of equilateral triangular picture elements is enough to generate a complete imaging system. However, an imaging system that combines both of them, that is, both being of the same species, will be more versatile; it will provide a greater accuracy approximating curves and angles. A square picture element and an equilateral triangular pixel facilitate 11 angles: 0, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, and 360 degrees. And if the square is replaced with a set of two isosceles right- angle triangular picture elements then it is possible to generate ten additional angles: 45, 105, 135, 165, 195, 225, 255, 285, 315 and 345 degrees. (Figure 20.)

However, since this visual imaging system is infinitely versatile, to some extent as versatile as paint and paper and to some extent even more, any selection of picture elements is only for the purpose of the initial introduction of products based on this system to all potential users. Once that is done, the users, amateurs and professionals, children and adults, and the different applications into which they will put the imaging system to use, will dictate additional designs, including customizing for special applications.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be described with reference to the accompanying drawings, in which:

Figures la-b show various designs for square picture elements of the invention.

Figures 2a-d show various designs for triangular picture elements of the invention.

Figures 3a-c show various shapes for hexagonal picture elements of the invention.

Figure 4a shows a picture element of the invention and its back side, being the mirror image thereof, and figure 4b shows an assembly of the picture elements of Figure 3a.

Figure 5a shows another picture element of the invention and Figure 5b shows an assembly thereof.

Figure 6a shows a picture element of the invention and its back side, being the mirror image thereof, and figure 6b shows an assembly of the picture elements of Figure 6a.

Figure 7a shows another embodiment of the picture element of the invention and Figure 7b shows an assembly thereof. Figure 8a shows another embodiment of the picture element of the invention and Figure 8b shows an assembly thereof.

Figures 9a-c show a set of picture elements according to the invention wherein the picture elements have different shapes as shown in Figures 9a and 9b, respectively and their assembly.

Figure 10a shows a picture element of the invention, having a single bay and a single peninsula along each edge, and its back side, being the mirror image thereof, and 10b shows a staggered assembly of the pixels in Figure 10a.

Figure 11a shows another picture element of the invention and Figure lib shows an assembly thereof in a non-tessellating arrangement.

Figures llc-e show a set of picture elements according to the invention which are arranged in a tessellating arrangement as shown in Figure lie the set of picture elements including the respective shapes shown in Figure lie and Figure lid, respectively.

Figure llh shows an assembly of four pixels shown in Figure llf and a gap that is formed within said assembly may or may not be filled with a pixel shown in figure llg.

Figures 12a-i show various shapes of peninsulas and bays of the bottleneck type according to the invention.

Figures 13a-f show various shapes of peninsulas and bays of the gooseneck type according to the invention. Figures 13f shows a peninsula and a bay having an S shape.

Figures 14a-c show arrangements for peninsulas and bays which are outside the scope of the invention.

Figures 15a-c show how the peninsulas and bays are interconnected in a vertical direction.

Figure 16a shows a picture element according to the invention and Figure 16b shows a "LEGO" building block according to the prior art.

Figures 17a-c show how prior art "LEGO" type blocks are assembled.

Figure 18a shows a pixel arrangement having blending color contours.

Figure 18b shows a pixel with well defined color contour, the pixel having peninsulas and bays with an area ratio similar to the area ratio of the pixel in Figure 18a and the area of said bays and peninsulas is displaced close to the edges thereof. Figure 18c and 18d show pixels with well defined color contour, the pixel having peninsulas and bays with a small area ratio and the area of said bays and peninsulas of pixel 18c being displaced far from the edges thereof and the area of said bays and peninsulas of pixel 18d being displaced close to the edges thereof.

Figure 18e shows blending color contours wherein the bays and peninsulas having moderately sized area ratio.

Figures 18f shows well defined color contours wherein the bays and peninsulas having small area ratio.

Figures 19a-h show a sequence of eight assemblies of square pixels illustrating the gradual change from displaced color contour through a range of blending color contours to well defined color contours.

Figures 20a-i show different sets of pixels capable of constructing different angles, approximating curves to different degrees, and their assemblies.

Figures 21a-b show the effect of shifting the lattice or grid of an image with respect to its model.

Figures 21c-d show the effect of reorienting a lattice of square pixels on the capability to cover an area enclosed between two curves with said pixels.

Figures 21e-f show the effect of lattice orientation using triangular shaped picture elements according to the invention on the capability to approximate a fixed set of circles and curves.

Figures 22a-p show various shapes and arrangements of picture elements having a square basic shape.

Figures 23a-j show how two picture elements having triangular basic shapes can be used to construct every possible hexagonal picture element.

Figures 24a-c show different arrangements of double-thickness pixels having single- and double-thickness peninsulas and bays.

Figures 24d-e show how the picture elements of Figure 24a-c can be arranged in a two- dimensional structure as shown in Figure 24d or a three-dimensional structure as shown in Figures 24e. Figure 24f shows an assembly of pixels most of which are transparent and some of which may be opaque, five of the pixels shown having a double-thickness and one having single- thickness pixel.

Figure 24g shows an assembly of a single-thickness pixel and a multi-thickness pixel.

Figure 24h shows how flat picture elements according to the invention can be arranged in a three-dimensional structure.

Figures 25 shows a visual image produced by interconnecting picture elements according to the invention.

Figures 26a-c show a single sheet of square pixels having a digital gradation of either monochromatic or spectral color scheme and including a frame which encloses each group of pixels having a common color scheme separating it from the adjacent group(s) of pixels. Figure 26a shows one side of said sheet. Figure 26b shows a section of the other side of the same sheet shown in Figure 26a having the color but with the gradation scale changing in the reverse order thereof. Figure 26c shows a section of the other side of the same sheet shown in Figure 26a having a different color and with the gradation scale changing in the same order thereof.

Figure 27 shows a sheet of square pixels having a continuous gradation of either a monochromatic or spectral color scheme.

Figure 28 shows a triangular sheet of triangular pixels having a continuous spectral color scheme which is based on the chromaticity diagram of the CIE triangle.

Figures 29a-b show assemblies illustrating two color schemes having color contours that are independent of the coupling schemes thereof. Black and white patterns are used to represent solid color schemes or, alternatively, patterned color schemes.

Figures 30a-b show pixels which are of different species and have different overall size and pixels of the same species and having different overall size.

Figure 31 shows an assembly which incorporates pixels of two different species with a special type of pixel called a transition pixel bridging between the two species.

Figures 32 show a pixel having the shape of a winged horse.

Figures 33a-e show special pixels having non-interlocking edges, the purpose of such pixels being to terminate the edges of a work of art. Figure 34 shows an assembly of complex coupling of the first level wherein a bay is formed by two bay segments each of which is part of a separate pixel.

Figure 35 shows an assembly of coupling of the second level wherein bays are embedded in peninsulas and peninsulas are embedded in bays.

Figure 36 shows a sheet of picture elements, the picture elements including square basic shapes and islands in the shape of double peninsulas.

Figure 37 shows an assembly composed of set of especially designed pixels displaying the image of a car, and the assembly may include different numbers of general purpose pixels.

BRIEF DESCRIPTION OF THE EMBODIMENTS

An object of the present invention is to provide a picture element for simultaneously constructing a support structure while creating a work of art. In one embodiment, the pixel comprises a substantially flat piece of material having different color schemes on opposite faces thereof and includes means for connecting the pixel to other pixels having the same or a different basic shape with the same or different coloring schemes on opposite faces thereof. The connecting means permits detachment of the pixel from the other pixels in a vertical direction perpendicular to the faces of the pixel and prevents detachment of the pixel from the other pixels when the pixel is pulled away from the other pixels in any direction that lies within the plane of the faces of the pixel.

The pixel preferably, but not necessarily, has bilateral symmetry or rotational symmetry.

In particular, the pixel has a basic shape which affects visual perception of color contours of a visual image produced by connecting a plurality of pixels having the same or different basic shapes as the basic shape of the pixel. The basic shape can provide well defined color contours or blending of colors along contours thereof. The pixel can further include means for approximating curves in the visual image produced by connecting a plurality of pixels having the same or different basic shapes as the shape of the pixel, the curve approximating means comprising the basic shape of the pixel being asymmetric, such that rotation of the pixel to different angular positions at which the pixel can be connected with at least one other pixel changes color contours between the pixel and an adjacent pixel. In general, the connecting means can comprise at least one peninsula which protrudes from at least one edge of the pixel and at least one bay extending through the opposite faces of the pixel, which intrudes from at least one edge thereof. The connecting means thus provides interlocking and a friction fit between the pixel and another pixel.

In an embodiment allowing three-dimensional structuring, the connecting means comprises a first bay extending part way into but not through one face of the picture element and a second bay extending part way into but not through an opposite face of the pixel. The connecting means can also comprise first and second peninsulas having thicknesses less than that of the pixel, the first peninsula protruding from one edge of the pixel and having one surface thereof coplanar with one face of the pixel. The second peninsula protruding from another edge of the pixel, and has one surface thereof coplanar with an opposite face of the pixel.

In a preferred embodiment, the pixel includes a plurality of edges extending between the opposite faces, at least one of the edges comprising either a male edge which includes a peninsula protruding therefrom or a female edge which includes a bay intruding therefrom. For instance, one of the edges can comprise a male edge which includes a peninsula protruding therefrom and a second one of the edges can comprise a female edge which includes a bay intruding therefrom. The peninsula can include first and second sections lying in the same plane, the first section being closer to the one edge than the second section and the first section having a width which is more narrow than that of the second section. Alternatively, opposite side edges of the first section can be concave in shape and opposite side edges of the second section can be convex in shape. The first section can have a straight side edge which intersects with a straight side edge of the second section at an angle between 0 degrees and 180 degrees. For each peninsula shape there is a corresponding bay shape. For instance, the peninsula and the bay can have various shapes such as a keystone, a rhomboid, a T-shape, a C-shape, a nipple-like shape or a mushroom-like shape.

In another embodiment the peninsula has three sections, one side edge of the first section is concave in shape and an opposite side edge thereof is substantially rectilinear, one edge of the second section is convex in shape and joins the concave shaped side edge of the first section at an inflection point, an opposite edge of the second section is substantially rectilinear and coincident with the rectilinear side edge of the first section, and the third section is dome- shaped and one side edge of the third section is coterminous with the rectilinear side edge of the second section. Alternatively, the peninsula can comprise a single section having two opposite side edges which are concave in shape and a third side edge which is concave in shape and extending between the two concave side edges. In another embodiment of a two section peninsula, the first section can be rectangular in shape and the second section can be triangular in shape, one side edge of the triangular shaped second section being joined to an outermost side edge of the first section such that the first section and second sections form an arrow- shaped peninsula. The peninsula can also include two spaced-apart concentric, arc-shaped side edges, the arc- shaped side edges forming segments of a circle, the center of which is located either inside or outside an outer periphery of the pixel.

The picture element can have many different basic shapes. For instance, the pixel can include four rectilinearly extending edges, the edges forming a square basic shape or the pixel can include six rectilinearly extending edges, the edges forming a hexagon basic shape. The pixel can also include three rectilinearly extending edges, the edges forming a triangular basic shape. Another possibility is for the pixel to have a generally quadrilateral shape with one inner angle being greater than 180 degrees.

The picture element could also include four rectilinearly extending edges of unequal lengths, at least one pair of the edges forming an obtuse angle therebetween and at least one other pair of the edges forming an acute angle therebetween. The pixel also can include four rectilinearly extending edges of unequal lengths, at least one pair of the edges forming a right angle therebetween.

Another picture element includes eleven rectilinearly extending edges with at least one pair of the edges forming a V-shaped recess in the pixel. The pixel can also include a plurality of non-rectilinear edges, each edge including a concave portion and a convex portion joined to the concave portion at an inflection point. The pixel can also include a plurality of non- rectilinear edges, two of the edges being convex in shape and one of the edges being concave in shape, the concave edge extending between one end of each of the convex edges and an opposite end of each of the convex edges being joined together.

In another embodiment, the pixel includes a plurality of non-rectilinear edges, two of the edges being concave in shape and one of the edges being convex in shape, the convex edge extending between one end of each of the concave edges and an opposite end of each of the concave edges being joined together. The pixel can also include a single edge so as to provide a circular pixel. The pixel can also include three edges, each of which is concave in shape and is formed by an arc having a constant radius for each of the edges.

The locking means can comprise at least one bottleneck joint and/or at least one gooseneck joint. The bottleneck joint can have bilateral symmetry. The pixel can also include at least one plain edge which is a non-coupling edge. The pixel can include an even number of interlocking edges or an odd number of coupling edges. Another objective of this invention is to provide the imaging system with a choice of pixels based on the perceptual eifect of the bays and peninsulas on the constructed image. From perceptual viewpoint, the bays and peninsulas are nothing but deviations from the normalized edges of the basic shape of the body of the pixel. The perceptual effect of such a deviation is directly proportional to the extent of such a deviation. First, consider the effect of the size of the interlocking bay and peninsula on the perception of the color contours. If, A is the area of the normalized body of this pixel and B is the area of a single deviation from the normalized edge thereof, then, the area ratio is defined as the ratio between the area displaced from a pixel by a single deviation from the normalized edge thereof and the area of the normalized body thereof and can be expressed by B/A. As the value of B approaches zero, the limit of the area ratio, is zero. Namely, as the size of the deviation is reduced, so does its visual effect on the color contour. When their area shrinks to nothing, their visual effect on the color contour is nil. On the other hand, as the value of B grows toward A, the limit of the area ratio approaches infinity. Indeed, at this point the normalized body of the pixel under consideration is totally displaced by one or more peninsulas of the adjacent pixels which fill the bays thereof. And the peninsulas of said pixel have contributed to the displacing of its adjacent pixels. If the color of said pixel is different from the color(s) of its adjacent pixels, then the perceptual effect of this displacement is quite visible. This aspect of the invention offers the following possibilities.

Case l

The components of the pixels which facilitate the construction of the supporting structure for the image leave the color contours well defined. Their existence has no significant perceptual effect on the basic shape of the pixel. In this embodiment the pixels have small area ratio or, if the area ratio is large, then the areas of the bays and peninsulas are distributed as close as possible to the coupling edge of said bays and peninsulas.

Case 2

The color of one picture element is almost completely displaced by the colors if the peninsula(s) which intrude into it from some adjacent pixel(s) while the color of its own peninsula(s) intrude into other adjoining pixels. Perceptually, the basic shape of the pixel is no longer discernable. The color contours are said to be displaced. In this embodiment, the pixel must have both large area ratio and the areas of the bays and peninsulas are displaced significantly away from their respective interlocking edges. Case 3

The range of sizes of bays and peninsulas from the very small ones in Case 1 and the maximum possible size in Case 2 forms a continuum. Between these two extremes lies a range of perceptual ambiguity. That is, when the proportion of the size of the bay(s) and of the peninsula(s) with respect to the size of the basic body of the pixel is in this range, some people perceive well-defined contour lines (as in Case 1) while others no longer perceive the basic shape of the pixel (as in Case 2). For most people, the effect is as if the colors along the color contours simply blend into each other. This is the range of ambiguity. The perceptual effect is similar to that of such famous optical illusions of alternating images as the old lady and young maiden or the vase and the two facial profiles. In a sense, in this embodiment, the invention provides for mixing of colors using areas that are indeed smaller than the size of an individual pixel. As a result, it is possible with this embodiment to fine tune the color contours more than it is possible with the resolution of a single pixel.

Case 4

In another embodiment of this invention, the color scheme of each pixel has a composition of several areas each of which has its own coloring scheme. This coloring scheme facilitates color contours which are independent of the edges of the pixels and their coupling schemes. In one variation of this embodiment, the basic area of the pixel has one color scheme and each of the peninsulas has another. For example say pixels are provided in either colors each of which is divided into an eight-gradation color scale. Thus, the color of the basic body of each pixel and the color of each peninsula may be one of the 64 different color shades. Therefore, if a pixel design has two peninsulas, such a color scheme may have up to 262,144 (=64^) different color combinations for each pixel. This embodiment provides for color contours, well defined, displaced or blending, which for perceptual considerations, are completely independent of the interlocking scheme. Depending only on the color schemes of the pixels, the user is free to generate color contours as desired.

The pixel can have a shape which provides tessellation such that an image created with the pixel does not include gaps between the pixel and other pixels. Alternatively, the pixel can have a shape which provides non-tessellation such that an image created with the pixel includes at least one gap between the pixel and other pixels.

The pixel can have a thickness which is large enough to provide a friction fit between a peninsula or bay of the pixel and a corresponding bay or peninsula of another pixel in a staggered arrangement which generates a multi-layered three-dimensional image. The pixel can also have a first basic shape excluding peninsulas and bays on one side thereof which is different from a second basic shape excluding peninsulas and bays on an opposite side thereof, the first basic shape having a thickness equal to about one half the thickness of the pixel and the second basic shape having a thickness equal to about one-half the thickness of the pixel. The first basic shape can have a small area ratio of the total area of the bays and peninsulas thereof to the overall area of the first basic shape excluding peninsulas and bays and the second basic shape can have a large area ratio of the total area of the bays and peninsulas thereof to the overall area of the second basic shape excluding peninsulas and bays.

The picture element can be a convertible pixel having convertible means called an island for selectively providing a male or female edge. The convertible means can comprise a removable double peninsula which when removed from the pixel forms a bay in the pixel. The pixel can include a peninsula and a perforation in the shape of the bay, the perforation being adjacent the peninsula and being a mirror image thereof.

Another object of the invention is to provide a set of pixels comprising a plurality of first and a plurality of second pixels for simultaneously constructing a support structure while creating a work of art, each of the pixels comprising a substantially flat piece of material having different color schemes on opposite faces thereof and means for connecting the set of pixels together, the connecting means permitting detachment of two of the pixels from each other in a vertical direction perpendicular to the opposite faces and preventing detachment of the two pixels in a direction parallel to the opposite faces. For example, in one embodiment, the first pixel has a basic shape of a square and the second pixel has a basic shape of a right isosceles triangle, the hypotenuse of the triangle being equal in length to the diagonal of the square.

The first and second pixels can have tessellating shapes and each surface of the first and second pixels can be completely covered with a single homogeneous color. The first pixel can include at least one bay or peninsula and the second pixel can include at least one bay or peninsula, a total area of all bays and peninsulas of the first pixel being equal to a total area of all bays and peninsulas of the second pixel, each of bays of the first pixel having a different shape than each of the bays of the second pixel and each of the peninsulas of the first pixel having a different shape than each of the peninsulas of the second pixel.

At least one of the picture elements can be transparent.

One of the pixels can be a framing pixel having one edge which is a male or female edge or both and other edge which is not a male edge or a female edge. For instance, the other edge can be straight, or curved. Also, one of the pixels can be a corner pixel having two adjacent finished edges which are not male or female edges. In accordance with the invention the pixels can be provided in the form of a sheet of pixels for simultaneously constructing a support structure while creating a work of art, the sheet of pixels including at least a first group of identical pixels, each of which has different coloring schemes on opposite faces thereof and including connecting means for connecting one of the pixels to another one of the pixels having the same or different coloring schemes on opposite faces thereof. The sheet can include a frame having at least one plain edge, the frame including connecting means connecting the frame with the pixels. The sheet can include at least one second group of identical pixels, each of the second group of pixels having a shape which is different from a shape of each of the first group of pixels, each of the second group of pixels having different coloring schemes on opposite faces thereof and including connecting means for connecting one of the pixels to another one of the pixels having the same or different coloring schemes on opposite faces thereof. A first color scale can be provided on one face of the sheet and a second color scale can be provided on an opposite face of the sheet. For instance, one face of the pixel can be covered with a dark color and an opposite face of the pixel can be covered with a light color thereby creating an image on one side of the work of art and a negative of the image on an opposite side of the work of art.

Another object of the invention is to provide a method of picturing and sculpturing with pixels which are interconnected to form a two-dimensional or a three-dimensional self- supporting visual image, the method being carried out without the use of tools, devices and/or materials other than the pixels, comprising: (a) interconnecting a plurality of pixels having different color schemes on faces thereof such that a peninsula protruding from an edge of one of the pixels extends into a bay intruding from an edge of an adjacent one of the pixels and the color scheme of the one pixel is different from the color scheme on the adjacent pixel on faces thereof visually observable from a first direction thereby creating an area of confusion along the adjoining edges of the pixels to a person visually observing an image created by the connection between the one pixel and the adjacent pixel; (b) repeating step (a) until a desired visual image is produced.

The pixels can include the male and female edges having a first color scheme and facing certain directions with respect to male and female edges of other pixels having a second color scheme so as to create different perceptual effects. For instance, male edges of a group of pixels can be oriented in the same direction so as to create a perceptual effect with respect to another group of adjacent pixels having different color schemes.

If the picture elements are square and include two adjacent male edges and two adjacent female edges, the method can comprise a step of orienting all of the pixel such that each of the male edges extend in only two directions such as +X and +Y. To form a three-dimensional image with flat pixels, a first pixel can be fitted in the bay of a second pixel by aligning a wider section of the peninsula with a length of the bay and an opening formed by the bay, inserting the peninsula through the bay, rotating the first pixel about a bilateral axis thereof until the wide section is aligned with a narrow section of the bay and the first pixel cannot be pulled away from the second pixel. Alternatively, a set of pixels can comprise first and second types of equilateral triangular shaped pixels, the first type having two female edges and one male edge and the second type having two male edges and one female edge, the image being created by connecting six of the pixels including at least one each of the first and second types so as to form a hexagonal shape, forming additional hexagonal shapes in the same manner and connecting the hexagonal shapes together to form the desired image.

The method of the invention can be carried out by mapping the visual image to be produced by determining a pattern of the color schemes of the pixels which will correspond to a visual image of an original subject whereby a two-dimensional self-supporting image is formed or a three- dimensional self-supporting image is formed. For instance, the method can include a first step of mapping the visual image to be produced when the pixels are viewed by a person from the first direction by determining a first pattern of the color schemes of the pixels which will correspond to a visual image of a first original subject and a second step of mapping a visual image to be produced when the pixels are viewed by a person from a second direction which is opposite to the first direction, the second mapping step being performed by determining a second pattern of the color schemes of the pixels which will correspond to a visual image of a second original subject which is different from the first original subject. The pixels can be interconnected with gaps therebetween or without any unfilled gaps therebetween. Step (a) can be repeated until the pixels form a self-supporting structure which has an overall shape in two-dimensions other than a quadrilateral or a parallelogram. Also, one or more of said pixels can be replaced with pixels having different color schemes on opposite sides thereof without disturbing remaining ones of the pixels forming the desired visual image.

DETAILED DESCRIPTION OF THE DRAWINGS

Figure la shows one configuration of a picture element 1 having peninsulas 2 and bays 3 according to the invention. In particular, the picture element la has a square basic shape and has bilateral symmetry about the axis A. The picture element la also has two male edges 4 which are adjacent each other and two female edges 5 which are adjacent each other. The square picture element lb shown in Figure lb has bilateral symmetry about two axes, that is about axis B and about axis C. The picture element lb includes two male edges 4 which are opposite to each other and two female edges 5 which are opposite to each other. The picture element shown in Figure la does not have any rotational symmetry whereas the picture element shown in Figure lb has two 180-degrees rotational symmetries.

Figures 2a-d show another form of a picture element 6 having a triangular basic shape. The picture element 6a shown in Figure 2a includes two male edges 4 and one female edge 5 having one axis of bilateral symmetry D. The picture element 6b shown in Figure 2b includes two female edges 5 and one male edge 4 having one axis of bilateral symmetry E. The picture element 6c shown in Figure 2c includes three male edges 4 having three axes of bilateral symmetry F, G and H. The picture element 6d shown in Figure 2d includes three female edges 5 having three axes of bilateral symmetry I, J and K. The triangular picture elements shown in Figures 2a and 2b have no rotational symmetry whereas the triangular picture elements shown in Figures 2c and 2d have three 60-degree rotational symmetries.

Figure 3a shows a form of a picture element 7 having a hexagonal basic shape. The picture element 7a includes three male edges 4 and three female edges 5, each of the male edges 4 being between two of the female edges 5. The picture element 7a has bilateral symmetry about three axes, that is about axis L, about axis M and about axis N.

The hexagonal picture element 7b shown in Figure 3b includes six male edges 4 whereas the hexagonal picture element 7c shown in Figure 3c includes six female edges 5. The pixel 7b has six axes of bilateral symmetry O, P, Q, R, S and T and the pixel 7b has six axes of bilateral symmetry U, V, W, X, Y and Z. The picture element shown in Figure 3a has three 120-degree rotational symmetries. The pixels shown in Figures 3b and 3c have six 60-degree rotational symmetries.

Figure 4a shows a picture element 8 having a substantially triangular basic shape with a male edge 4, a female edge 5 and a combination edge 9 which includes a male and female edge in a non-linear arrangement. The picture element 8b shown in dashed lines in Figure 4a, being the mirror image of the picture element 8a, is the reverse side of pixel 8a. An assembly of the picture elements 8a and 8b in the form of a sheet is shown in Figure 4b. It can be seen that this arrangement is a tessellating arrangement, that is, neither gaps nor overlaps are provided between the picture elements.

Figure 5a shows a picture element 10 having four rectilinear combination edges 9. The combination edge 9 shown is linear. Figure 5b shows an assembly of the picture elements 10, this arrangement being a tessellating arrangement. Pixel 11 is an example of having a peninsula lie protruding from a vertex of the edges of the basic shape of the picture element, rather than from somewhere along an edge. Pixel 11a interlocks with pixel lib which, being the reverse side thereof, has the shape that is the mirror image of the shape of pixel 11a. In this assembly peninsula lie of pixel 11a interlocks with bay llg of pixel lib and peninsula llj of pixel lib interlocks with bay lid of pixel 11a. Each pixel 11a and lib also has two vertices without a protruding peninsula. These vertices, lie, llf, llh and Hi are notched so that no overlap occurs when several pixels are interlocked in an assembly of as shown in Figure 6b.

Figure 7a shows an 11 sided picture element 12. Seven of these sides are plain edges meaning they do not have a peninsula or bay therein. Two of the edges are female edges 5 and two of the edges are male edges 4. The edges of the picture element 12 are rectilinear and five pairs of these edges are parallel to each other. Pixel 12 is a variation of pixel la (Figure la) having the square edges thereof bent and their bays and peninsulas shifted from their central locations along said edges. As shown in Figure 7b, the picture elements 12 can be arranged in a tessellating arrangement.

Figure 8a shows a picture element 13 having four curved edges, that is, two curved male edges 4a and two female edges 5a. These curved edges each comprise a convex part and a concave part which meet at an inflection point. Figure 8b shows a tessellating assembly of pixel 13.

Figure 9a shows a picture element 14 having three curved edges, two of the edges being concave male edges 4b and the other edge being a convex female edge 5b. Figure 9b shows a mating picture element 15 which also includes three edges, two of the edges being concave male edges 4c and the other edge being a convex female edge 5c. Figure 9c shows how the picture elements 14 and 15 can be assembled in a tessellating arrangement. As shown in Figure 9c, the picture elements 14 can include an optional peninsula 2a and a corresponding bay 3a.

Figure 10a shows a picture element 16a having combination edge 9 which includes a a bay and a peninsula in a linear arrangement. The picture element 16b shown in dashed lines in

Figure 10a, being the mirror image of the picture element 16a, is the reverse side of pixel 16a.

A tessellating staggered "brick-layered" assembly of the picture elements 16a and 16b is shown in Figure 10b. Pixel 16a alone can in its plurality form a tessellating arrangement.

Figure 11a shows a picture element 17 having a single edge. In particular, the picture element 17a shown in Figure 5a includes alternating bays and peninsulas, there being three bays 3 and three peninsulas 2. Figure lib shows an assembly of the picture elements 17a which provides a non-tessellating arrangement with gaps 18a between adjacent picture elements 17a.

Figure lie shows another form of the picture element 17. In particular, a picture element 17b includes six bays 3. Figure lid shows a picture element 19 which interconnects the picture elements 17b. The picture element 19 includes three concave male edges 4c. Figure He shows how the picture elements 17b and 19 can be arranged in a tessellating arrangement.

Pixel 17c in Figure llf can be interlocked in its plurality to construct images. Figure llh shows an assembly constructed of six pixels 17c. Between these size pixels two gaps 18b are formed. Hence, pixel 17c alone, in its plurality is not tessellating. Pixel 20 in Figure llg has the precise shape to fill the gap 18b in Figure llf without overlapping. Therefore, the set of pixels 17c and 20 a is tessellating.

Figures 12a-i show various configuration for the peninsulas and bays, the connection therebetween being a bottleneck type connection. The various shapes of the peninsulas are shown at 3b-j and the various shapes of the bays are shown at 2b-j.

Figures 13a-f show various arrangement of a gooseneck type connection according to the invention. In this case, each of the peninsulas 2k-p and bays 3k-p, respectively, includes a first section 21 and a second section 22, the first section having a particular relationship to the second section. Figure 13b shows how an interior angle xl on one side of the bay or peninsula and the exterior angle yl on the other side of said bay or peninsula both have their vertices pointed in the same direction. Figure 13c shows a modified version of the arrangement shown in Figure 13b.

Figure 13d-e shows an arrangement wherein the peninsula and bay are formed by curved edges having different radii, the center of which is at a point 26 located within one picture element or the other. With the arrangement shown in Figure 13e, the picture elements will not separate in the X-Y plane whereas the picture elements shown in Figure 14c could separate in the X-Y plane since the center of the arcs 27 is located at abutting corners of the picture elements. Figure 13d shows an arrangement wherein the center of the arcs of the bay are located within the same picture element whereas Figure 13e shows the center of the arcs located outside the picture element. Conversely, the center of the arcs 26 for the peninsula in Figure 13d is located outside the picture element having the peninsula whereas Figure 13e shows the center of the arcs 26 being located within the same picture element having the peninsula.

Figure 13f shows a first curved section 21' and a second curved section 22', the curved edges of the first section 21' meeting the curved edges of the second section 22' at an inflection point 23. Figure 13f shows a first curved section 21' and a second curved section 22', such that an angle x2 is formed between an interior tangent to the curved edge of section 21' and an interior tangent to the curved edge of section 22'. Also, an angle y2 is formed between an exterior tangent to the opposite curved edge of section 21' and the exterior tangent to the opposite curved edge of section 22', both angles having their vertices pointed in the same direction.

Figures 14a and 14b show arrangements wherein the peninsulas and bays are capable of separating in the X-Y plane. The pairs of angles x3 and y3 and xl and y2 violate the requirement set the description of Figure 13b. Accordingly, peninsulas 2q-r and corresponding bays 3q-r are outside the scope of the present invention.

Figures 15a-c show how the picture elements are connected together by moving them along the Z axis. Figure 15b shows how the picture elements cannot be separated in the X-Y plane. Figure 8c shows how the picture elements are separated by applying a force F in the Z direction and applying a rotational component of the force F2 or F3 which separates the picture elements.

Figure 16a shows a picture element 1 according to the invention compared to a "LEGO" block shown in Figure 16b according to the prior art. Figures 17a-c show how the "LEGO" blocks are assembled. As shown in Figure 17b the blocks 28 are assembled by pressing them together in the Z direction but unlike the picture elements of the invention, the blocks 28 cannot be connected together in the same X-Y plane. Instead, as shown in Figure 17c, the blocks 24 must be assembled in a layered arrangement in order to connect the blocks 28 together in the X-Y plane.

Figures 18a through 18f show how color contours are affected by the area ratios and the distances of displacement of bays and peninsulas. In Figure 18a the bays 3 and the peninsulas 2 have a moderate area ratio and their displacement is far from the edge thereof resulting with blending color contours. Some viewers perceive pixels 29a and 29b having well-defined color contours while other viewers perceive these two pixels as blending one into the other. Or the same viewer may alternately perceive well-defined color contours between pixels 29a and 29b at some times and blending color contours between pixels 29a and 29b at other times. Figure 18b shows that if the area ratio remains the same as in Figure 18a but the displacement of the area of the peninsula is not far enough from the edge thereof, the color contour appears well defined. Figure 18c shows that when the area ratio is small and the displacement of the area of the peninsula is far, the color contours appear well defined. Figure 18d shows that when the area ratio is smaller than that which is shown in Figure 18b and the displacement of the area of the peninsula is the same as shown in Figure 18b, the color contour appears even more well defined. Having differently shaped bays and peninsulas, Figures 18e and 18f are equivalent to Figures 18a and 18d, respectively.

Figures 19a-h show eight identical assemblies of nine square pixels, having a shape that is a variation of pixel lb (Figure lb), four of which are gray and four of which are black. The normalized bodies of all pixels in all eight figures are of exactly the same size and shape. The bays and peninsulas in Figure 19a are of a first size, the largest within said set of figures, and the bays and peninsulas in Figure 19h are of an eighth size, the smallest within said set of figures. Each consecutive figure in the sequence from Figure 19a through 19h shows a discrete and uniform decrease in the size of the bays and peninsulas thereof with respect to the size of the bays and peninsulas in the preceding figure. Figures 19a-b show a displaced color contour. Figure 19c through 19f show blending color contours. Figures 19g-h show well defined color contours.

Figure 20a shows a square pixel with a side "a" and a diagonal "c". Figures 20b and 20c show equilateral triangular pixels with a side "a". Forming a set, the three pixels in Figures 20a-20c, in their plurality, can be combined to form a wider range of angles as illustrated, for example, in Figure 20d. Figure 20d shows angle zl = 120 degrees, and angle z2 = 30 degrees. Angle z2 cannot be filled by any pixels of this set. Hence, once angle z2 is formed, a gap or an overlap necessarily exists. Figures 20e and 20f show isosceles right-angle triangles with the side "a" and the hypotenuse "c". Forming a set with the pixel shown in Figure 20a, or without it, these two or three pixels, in their plurality, can form such angles as angle z4 = 135 degrees, angle z4 = 45 degrees shown in Figure 20g (two other angles of 225 degrees are also formed). A gap may be filed with overlapping as shown by the dashed line representing the pixel of Figure 20h. Figure 20i shows that when the pixels in Figures 20b, 20c, 20e and 20f are combined into a set (the pixel in Figure 20a may or may not be included), additional angles such as angle z5 = 120 degrees, angle z6 = 165 and angle z7 = 195 degrees can be formed. Figure 20h illustrates that since the edge of the size of hypotenuse "c" is longer than edge "a", a gap the length of "d" may be created if an edge of length "a" is interlocked with an edge of length "c". The gap cannot be filed without overlapping. The pixel shown in Figure 20i, being rectangular and having a first side of length "a" and having a second side of length "c", can be added to the aforementioned set to avoid this latter problem. However, having a side of length "c", the pixel of Figure 20i only pushes the problem toward the edge of the image. Also note that the angle z8, being 75 degrees, cannot be completely filled; a partial gap or an overlap necessarily exits. Some combination of a square pixel, an equilateral triangular pixel and isosceles right-angle triangular pixel can form all the following angles 0, 45, 60, 90, 105, 120, 135, 150, 165, 180, 195, 210, 225, 240, 255, 270, 285, 300, 315, 330, 345 and 360 degrees. Figure 21a shows how an area below a curved line was digitized with dark square pixels and the area above said curved line was digitized with light square pixels. In Figure 21b, the grid of the image from Figure 21a was first shifted downward with respect to said curved line by a distance that is equal to approximately half the length of a single edge of the square pixel thereof. Then the area above said curved line was digitized with a light square pixel and area below said curved line was digitized with a dark square pixel. Although both figures illustrate images based on the same model, the two images are different. A viewer can recognize the resemblance between each image and the original model. However, the judgement which image better resembles the original model is subjective for each viewer.

Figures 21c and 21d show how the orientations of the picture element affect the color contours and their ability to approximate curved lines in the original model. In particular, the model is outline 33 of a figure-8. The image is digitized with a plurality of gray square pixels 34. Creating the image in Figure 21d, the first pixel to be placed in the image was rotated by 45 degrees with respect to its orientation in Figure 21c. As a result the whole lattice of pixels in Figure 21d has been rotated with respect to the lattice of Figure 21c and more pixels are necessary to approximate the same model. The judgement which image better resembles the original model is subjective for each viewer.

Figure 21e-f show how the orientation of the picture elements affect other picture elements in the lattice. In particular, the picture elements 35 and Figure 21e are triangular in shape and are arranged such that at least one edge thereof is horizontal. The pixels 36 shown in Figure 21 f are arranged such that at least one edge thereof is parallel to the vertical direction. Figures 21e-f show two digitizations of a face using curved lines. The effects of rotating the pixels in Figure 21 f by 30 degrees with respect to the picture elements in Figure 21e can be easily appreciated from these figures. For instance, in Figure 21f, the eyes are spaced wider apart, the nose is angled and the mouth is comprised of two pixels instead of three pixels as in Figure 21e.

Figures 14a-f show six variations of a square basic shape picture element 1. In particular, picture element lc includes four male edges, picture element Id includes three male edges and one female edge, picture element la includes two male edges which are adjacent each other and two female edges which are adjacent each other, picture element lb includes two female edges which are opposite each other and two male edges which are opposite each other, picture element le includes three female edges and one male edge and picture element If includes four female edges.

Figures 22g-j show how picture element la can be oriented in four different positions, that is laa, lab, lac and lad. Figures 22k-p show how the picture elements 1 can be assembled with various orientations. For instance, Figures 22k-l include only picture elements in orientation laa whereas Figure 22m includes two picture elements in orientation laa and two picture elements in orientation lad. Figure 22n shows one picture element in orientation laa and another in orientation lab. Figure 22o includes two picture elements in orientation laa and two picture elements in orientation lab. Figure 22p includes a picture element in each orientation laa-ad.

Figures 23a-k show various arrangements for triangular shaped basic pixel 7. In particular, the picture elements 6a and 6b shown in Figures 23a and 23b, respectively can be arranged in various combinations to form various hexagonal picture elements 7. In particular, Figures 23c-k show how the triangular shaped picture elements 6a and 6b can be arranged in the hexagonal picture elements 7a-i.

Figure 24a shows a pixel 37 having a double thickness compared to the earlier described pixels. The pixel 37 includes peninsulas 2 having a thickness one-half that of the pixel 37. Likewise, the bays 3 are formed by recesses in the pixel 37, the recesses extending to a depth one-half the thickness of the pixel 37. The pixel 37 can be assembled as shown in Figures 24d- e. In particular, Figure 24d shows a single layer whereas Figure 24e shows a staggered layer. Figure 24e shows a cross section of a multi-layer assembly wherein the upper and middle pixels 37a, 37b, 37c and 37d are transparent and the lower pixels 37e and 45 f each is either transparent or opaque. Pixel 37f is of a first thickness, called single thickness. Pixels 37a through 37e are of a second thickness, called double thickness, which is twice the size of the first thickness. Pixel 37f is used if it is desired for the bottom face of the image to have a single coplanar surface. (Note that in these figures hatching is used only to illustrate cross sections. No particular material is suggested by any choice of the hatching pattern.)

Like pixel 37 in Figure 24a, pixel 38 in Figure 24b is a double-layer pixel, having bays and peninsulas which are single-layer. Unlike the bays and peninsulas of pixel 38 which are aligned above each other, the bay and peninsulas of pixel 38 are independent of each other. In particular, two edges have one peninsula 2 each with no bay, two edges have only one bay 3 each with no peninsulas.

Pixel 39 in Figure 24c is of a double-layer. Its edges 5 and 4 have single thickness peninsulas and double-thickness bays respectively. Figure 24g shows an assembly of a single- thickness pixel 40a with a multi-thickness pixel 40b. A single layer peninsula and a multi-layer bay form coupling 41 with sufficient space for another pixel layer.

Figure 24h shows how picture elements la can be assembled in a three-dimensional arrangement forming a cube. Figure 25 shows show a visual image produced by interconnecting pixels la.

Figure 26a shows one face of a sheet of square picture elements and Figure 26b and 26c show the opposite face of said sheet. The color scheme of each surface of said sheet has several discrete colors or discrete shades of the same colors, each of which is represented by a different density of dot patterns. In Figure 26b the reverse order of the dot densities on the opposite faces of said sheet represents the fact that if both sides have the same color scales, then these scales are reversed in order. In Figure 26c the same order of the dot densities on the opposite faces of said sheet represents the fact that both sides have the same color scales whether said surfaces have the same colors or not.

The pixels are enclosed within a frame 42 having the same color scheme as the pixels therein. A segment of the frame 44 extends from one side of the sheet and across the sheet to the other side to separate each set of pixels 43 having the same color scheme from the pixels around it having different color schemes thereof. The color scheme of each set of pixels having the same color scheme extends half way width-wise into said extensions 44 of the frame. This frame can be used to restore unused pixels and as a color-scale indicator.

Figure 27 shows an example of a sheet of square picture elements. The continuous gradation of the density of the random pattern represents a continuous gradation in a monochromatic or spectral color scheme.

A triangular sheet of triangular picture elements is shown in Figure 28. The color of pixel 45a is yellow-green having a wavelength at its top vertex of approximately 520 millimicron. The color of pixel 45b is purple having a wavelength at its left vertex of approxi¬ mately 400 millimicron. The color of pixel 45c is red having a wavelength at its right vertex of approximately 770 millimicron. The color around point 46 is white. This spectral sheet of pic¬ ture elements approximates the chromaticity diagram as shown in a standard CIE triangle of colors. The color graduation between individual pixels in this sheet may be either discrete or continuous along both the X and the Y axes.

Figures 29a-b show assemblies illustrating two color schemes having color contours that are independent of the coupling schemes thereof. Black and white patterns are used to represent different solid color schemes or, alternatively, different patterned color schemes. This color scheme generates color contours that are independent of the interlocking scheme of said pixels.

The rectilinearly extending edges of the normalized bodies of pixel la-xl in Figure 30a and of pixel 47 in Figure 30b are three times as large as the rectilinearly extending edges of pixel la in Figure 30b. The bay and peninsulas of pixel la-xl were enlarged with respect to the bays and peninsulas of pixels la by the same proportion in which the normalized body of pixel la-xl is larger than the normalized body of pixel la. Pixels la-xl and la are of different species as they cannot be interlocked. Pixel 47 has an arrangement of bays and peninsulas along the edges thereof such that it can be interlocked with pixel la. Pixels 47 and la are of the same species.

Transition picture elements are shown in Figure 31. Pixel la is of a first species having well-defined color contours. Pixel 50 is of a second species having displaced color contours.

Pixels 48 and 49 have at least one edge of the first species and at least one edge of the second species. Pixels 48 and 49 facilitate an assembly which may include pixels of both the first and the second species.

Figure 32 shows an asymmetric pixel having various a symmetric peninsulas and bays. For example, peninsula 2z and bay 3z are of the gooseneck type.

Variants of edge or framing picture elements are shown in Figures 33a-e. The pixels in Figure 33a through 33d terminate in a work of art with a straight edge. For. example, Figure 30a-30c show pixels 51-57 each of which has one non interlocking edge. Figure 33d shows an L-shape pixel having a length of several pixels and a width of a single pixel. Figure 33e shows a similar pixel as in 33d but it terminates a work of art with a sinuous line. Pixel 51 of Figure 33a is a corner pixel having a beveled corner.

Figure 34 shows a set of three pixels that illustrates a complex coupling scheme of a first level. Two Pixel 58 together form a bay that interlocks with the peninsula of pixel la. Each pixel la can interlocked in this manner with up to six picture elements that surround it (not shown), four are pixel 58 and two are pixel la. Each pixel 58 interlocks with six pixels that sur¬ round it, four are pixel la and two are pixel 58.

Figure 35 shows picture element 60 having a basic square shape 61 which illustrates a complex interlocking scheme of a second level. In particular, interlocking is established by peninsulas 2 that protrude from second level bays 59 and are coupled with bays 3 which intrude into second level peninsulas 62. Second level bays 59 are formed by horizontally adjacent pixels 60a.

Figure 36 shows a sheet of picture elements including square picture elements 63 that is composed of a body 64 and double peninsula 65. The double-peninsulas 65 are islands in the sense that they can be separated from the body 64 on both sides, becoming independent pixels, or remain attached to a particular pixel thus forming a peninsula on the picture element. The island 65 are connected to the bodies 64 by perforated material providing easy detachment. Figure 37 shows a special purpose set of picture elements 66-73 that is designed to gener¬ ate images of cars. Using different numbers of pixels la, the car may be stretched or shrunk.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

An objective of the invention is to produce an imaging system which can provide the following advantages:

(1) Imposes no special requirements on the person who uses it. It requires no special skill such as eye-hand coordination, or dexterity, etc., and no special training. It provides for anyone, from a child of approximately 2-3 years old to the elderly, including people with certain disabilities (which disable them from using existing imaging systems such as painting and drawing), with very limited instructions, will intuitively know how to and be capable of making use of said imaging system. Further, the user of said imaging system should need no tool and no additional material whatsoever in order to create a work of art with this imaging system. (Therefore, there is no need to have access to any tool or material and to have the skill and knowledge of using any additional tool or material.)

(2) Is very versatile. It imposes no restriction on either the size or on the shape of the work of art which is created with said imaging system. That is, an image may be as small as a few picture elements (a single-pixel image is possible but trivial) or as large as a billboard, a football field or even larger. The overall shape of the work of art may be rect- angular, circular, ellipsoid, or any other shape the creator wants it to have. It should have no intrinsic restrictions on the selection of color or patterns. There should be no intrinsic restriction on the manner of displaying works of art created with this imaging system. These images may be hung or placed on top of or against flat horizontal or vertical surfaces such as walls and floors, desktop and even ceilings, and non-flat images may be hung or cover matching non-flat surfaces.

(3) The imaging system should be able to produce opaque images (reflecting light), transparent images (transmitting light), and light emitting images.

(4) The imaging system must be inexpensive, within the financial reach of all (except, perhaps, for light radiating imaging systems). (5) The imaging system may be used to produce works of art, or images, for a large variety of utilitarian, recreational, educational and decorative applications. Further, provided the proper material is used in the fabrication of the pixels, it should be possible to put the images into the use to which they were designed while the creation work is still in progress. For example, if one is creating a place mat, a mat for hot pots, a coaster, or a tiled counter top or a floor, it should be possible to use these assembled objects as intended even before the design has been finalized (not while actual work is performed).

Another object of the invention is to provide an imaging system which includes a substantially flat picture element, in one embodiment, or a small, finite set of substantially flat picture elements, in another embodiment, for simultaneously constructing a supporting structure while creating a work of art. These picture elements are to provide the user with a choice of the following features:

(1) The pixel may have a tessellating shape (e.g., square) or it may have a non- tessellating shape (e.g., square with rounded corners or circle). If a set of pixels is used, it may be a tessellating or a non-tessellating set. In one embodiment, a first pixel can be combined with a second pixel having the mirror image shape of the first pixel. (See Figures la-b; 2a-d; 34; 35). In another embodiment, a first pixel must be capable of coupling with a second pixel having the mirror image of the first pixel (See Figures 4a-b, 6a-b; lOa-b). Yet in another embodiment, it is not possible to combine a first pixel with a second pixel having the mirror image shape of the first pixel. (See Figures 32; 37).

(2) In every embodiment, the model for the image can be shifted with respect to the lattice (or grid) of the art work along one or more of the grid's X, Y and Z axes, the X and Y axes being in the grid's plane and the Z axis being perpendicular to the grid's plane, such that, following said shift, no pixel's new position in the image precisely overlaps any pixel's position in the image prior to said shift. (See Figures 21a-b).

The basic, normalized shape of the pixel may have at most a small finite number of rotational symmetry axes, such that rotating a single picture element in its position within any constructed work of art to an angle that is rotationally asymmetric with its original orientation, necessitates an identical rotation of all other pixels in said image without altering the overall shape of the image. (See Figures 21c-f). Also, rotating a pixel into one of the rotationally symmetric orientations of its basic shape may affect its color contours. Therefore, if the basic shape of a pixel has some axes of rotational symmetry and if some, or all, of these symmetries are violated by the presence of the bay(s) and/or peninsula(s), then this pixel provides for more variation of color contours) than other pixels which lack such design. (See Figures 22g-p). Another aspect of the invention is that any picture element within a constructed image may be disengaged from its surrounding pixels with no adverse effect on itself, on its surrounding pixels or on the overall image. (See Figure 15c). Another aspect is that the picture elements having different normalized shapes can be combined to improve the approximation of curves by the color contours (when compared to such curves as produced by pixels having a single overall shape). (See Figures 20a-h). A further aspect is that there are no intrinsic restrictions on the size and shape of the picture elements except as dictated by the material that is used for manufacturing the pixels and by the fabrication and manufacturing technology. The picture elements may be manufactured from a variety of materials such as but not limited to cardboard, chipboard, plastic, wood, particle board, clay, ceramic, glass, metal, or from some compositions or lamentations of such materials. (See Figs 15a-c). According to another aspect, the normalized shape of the picture element(s) may be geometric (e.g., triangle, quadrilateral, hexagon for tessellating pixels or circle, ellipse, pentagon for non-tessellating pixels) with straight or curved (e.g., sinusoid) edges. Or the basic shape of the pixel(s) may be in some artistic form (See Figures la-llc; 30a-b; 32; 33d-e; and 34, 35, 37).

In all embodiments of this invention, the property of symmetry or, conversely, the property of asymmetry may be used in the design of a picture element. When symmetry or asymmetry is discussed, it is done with reference to any one of the following four elements of the picture-element design or it is done with reference to any combination of these four elements, including all four of them together:

(1) The normalized body of the picture element may be symmetric or asymmet¬ ric. Example: a pixel of a normalized square shape (Figure 4a) versus a pixel of a normalized quadrilateral shape with all of its edges having different lengths (Figure 5a).

(2) Each edge of a pixel may be symmetric or asymmetric. For example, the symmetric edges of an eleven edged pixel (Figure 7b) versus the asymmetric edges of a four edged pixel (Figure 5b).

(3) Each bay and peninsula may be symmetric (Figures 12a, c, e, f, h, i) or asymmetric (Figures 12b, d, g).

(4) The overall shape of the picture element, that is, the pixel's normalized body as affected by its edges and their bays and peninsulas, may be symmetric (Figures la; 8a, 9a,

10a) or asymmetric (Figures 4a; 7a).

In all embodiments of this invention picture elements are designed to couple with each other. If the interlocking scheme of two picture elements, regardless to their basic (normalized) shape, facilitates the interlocking of the one with the other, then the two pixels are said to be of the same species. If the two pixels cannot be interlocked, then they are said to be of different species. For all practical purposes, any pair of pixels of the same species can be interlocked with each other regardless to manufacturing batch, generation, etc. Further, interlocking is a transitive property: If edge A interlocks with both edge B and edge C, and if edge D interlocks with edge C, then edge D also interlocks with edge B.

The pixels of a given imaging system provide for the construction of a substantially flat work of art, or image, in a single planar surface and/or of artworks in a curved surface. In one embodiment, the bays and peninsulas which provide for the pixel interlocking, extend thickness-wise from one surface of the pixel all the way to the other surface. These type of pixels are very versatile; they provide for the construction of images on a planar sheet, of images on a curved sheet, and, if they are sufficiently thick, they provide for the construction of images that are composed of multi-layered intersecting sheets.

In a variation of the latter embodiment, the thickness of the pixel is sufficient to provide for staggered, layered coupling. That is, two pixels, not necessarily of the same basic shape but necessarily of the same species, may be arranged such that (a) one surface of a first pixel butts against one surface of a second pixel; and (b) one bay of a first pixel is perfectly aligned with one bay of a second pixel such that the two bays couple with a single peninsula of a third pixel. Then, each bay has half of its outer thickness (the portion away from the abutted pixel) free from being interlocked with the peninsula; thereby, a fourth pixel with a peninsula may be aligned with the first peninsula of the third pixel on either side thereof. In this manner pixels may form a staggered interlock to form a multi-layered work of art. (Figures 24a-g).

In another embodiment, the bay and peninsula, which provide for the pixel coupling, extend from one surface toward the other surface only partially. That is, the bay and peninsula have the thickness of an ordinary picture element, called single thickness, while the thickness of the normalized body of the pixel is twice as thick, called double-thickness. The thickness of the peninsula is identical to the thickness of the bay. (Figures 24a, 24c-f). Or, in a variant of this embodiment, the peninsula is of a single thickness while the bay is of a double thickness. This embodiment provides for two pixels with butting surfaces, each of which has such a peninsula, to be interlocked with a single pixel with such a bay. (Figures 24a-g). Or, in a variant of this embodiment, both bays and peninsulas are of a single thickness and the positioning of a bay along the edge of the pixel is independent of the positioning of the peninsula along the edge of the pixel. (Figure 24b). Or, in another variant of this embodiment, the bay and peninsula, both of a single thickness, are aligned "on top of each other" along the edge of the pixel. (Figure 24a). In one type of embodiment, each interlocking edge of a picture element has either a single protruding peninsula or a single intruding bay but not both. In another type of embodiment, each interlocking edge may have two or more protruding peninsulas or two or more intruding bays. If an interlocking edge has one or more peninsulas and no bay, it is said to be a male edge and, if it has one or more bays and no peninsulas, it is said to be a female edge. In a third type of embodiment, an interlocking edge may include both one or more protruding peninsulas with one or more intruding bays. Some embodiments which utilize pixels with multiple bays and/or peninsulas along some edges can provide for the staggering of pixels within the plane of construction in a "brick-laying" fashion. (Figures 5a-b; 6a-b; lOa-b; 24a; 32; 33D). In case of a circular or an ellipsoidal picture element the whole circumference is taken into account. For instance, Figures llc-d show female pixels whereas the pixels shown in Figures lla-b, f and h, have no gender.

In one type of embodiment, the bay and peninsula are centered along their edges. (Figures la-4b). In another type of embodiment, the bay and peninsula are displaced from the center of their edges in an equal distance such that they can be interlocked. In such an embodiment, two interlocked pixels may be aligned or they may be staggered in a brick-laying fashion. In particular, if pixel A can be interlocked with pixel B and also with B' which is the mirror image of B, then if A and B are interlocked and aligned then A and B' are interlocked and staggered. (Figure lOa-b). In another type of embodiment, the peninsula or bay may be located at a vertex where two edges of the normalized body of the picture element intersection. (Figure 6a). In another type of embodiment, referred to as complex coupling of the first level, only a portion of a bay or only a portion of a peninsula exists along an edge of a picture element. More than a single pixel is necessary to form a whole bay or a whole peninsula. To provide a stable coupling, the picture elements that together form such a composite bay must interlock with each other and their composite bay must couple with a single peninsula. (Figure 34). Conversely, such a composite peninsula must couple with a single bay. In yet another type of embodiment, referred to as complex interlocking of the second level, the peninsula of a first design is sufficiently large to accommodate one or more bays of a second design and the bay of the first design is sufficiently large to accommodate one or more peninsulas of the sec- ond design. (Figure 35).

Another objective of this invention is to provide the imaging system with substantially flat picture elements for the creation of visual images, such that:

(1) If the pixel is opaque, it is colored on both of its surfaces. The coloring scheme of one surface may be dependent on the coloring scheme of the opposite surface. While creating an image on one surface, another image that has the same general design but with different colors (e.g., negative or inverse colors) is being generated simultaneously on the opposite surface. (Figures 26a-c). The coloring scheme of one surface may be independent of the coloring scheme of the opposite surface. A user of this embodiment can simultaneously create two independent images, one on each surface. (Figures 26a-c).

(2) The pixel may be transparent. (Figure 24f).

(3) The pixel may be light emitting. That is, it may be fluorescent or contain the necessary modular circuitry (including power source) and light emitting devices. Some pixels are light emitting (e.g., fluorescent) on both of their surfaces while others (e.g., electric) emit light from only one surface.

(4) Regardless of whether the pixel is opaque, transparent, or light emitting, any color scheme may be composed of either a solid color (26a-c); or some geometric pattern, peri¬ odic or non-periodic, having one color scheme for said pattern and another color scheme for the background of the pattern; or some combination of solid colors or of geometric patterns. (Figures 29a-b).

(5) If the color of one pixel surface is different from the color of another surface, and both surfaces have either a solid color or the same pattern or, in the case of transparent or light emitting picture elements, the color of one pixel is different from the color of another pixel, and both pixels have either a solid color or have the same pattern, then the graduation scale between the two surfaces or the graduation scale between the two pixels is: either digital (Figures 26a-c, 28) or analog (Figures 27, 28); and either monochromatic (Figure 25) or spectral (Figures 26a-c, 27, 28).

Another objective of this invention is to provide the imaging system with a choice of pixels based on the perceptual effect of the bays and peninsulas on the constructed image. From perceptual viewpoint, the bays and peninsulas are nothing but deviations from the normalized edges of the basic shape of the body of the pixel. The perceptual effect of such a deviation is directly proportional to the extent of such a deviation. First, consider the effect of the size of the coupling bay and peninsula on the perception of the color contours. If, A is the area of the normalized body of this picture element and B is the area of a single deviation from the normalized edge thereof, then, the area ratio is defined as the ratio between the area displaced from a pixel by a single deviation from the normalized edge thereof and the area of the normalized body thereof and can be expressed by B/A. As the value of B approaches zero, the limit of the area ratio, is zero. Namely, as the size of the deviation is reduced, so does its visual effect on the color contour. When the area shrinks to nothing, its visual effect on the color contour is nil. On the other hand, as the value of B grows toward A, the limit of the area ratio approaches infinity. Indeed, at this point the normalized body of the pixel under consideration is totally displaced by one or more peninsulas of the adjacent pixels which fill the bays thereof. And the peninsulas of said pixel have contributed to the displacing of the bodies of its adjacent pixels. If the color of said pixel is different from the color(s) of its adjacent pixels, than the perceptual effect of this displacement is quite visible. (Figures 19a-h). The following arrangements are possible according to this embodiment:

Case l

The components of the picture elements which facilitate the construction of the supporting structure for the image leave the color contours well defined. Their existence do not significantly perceptually alter the basic shape of the pixel. In this embodiment the picture elements have small area ratio or, if the area ratio is large, then the areas of the bays and peninsulas are distributed as close as possible to the coupling edge of said bays and peninsulas. (Figures 18b-d; 19g-h; 20b; 25).

Case 2

The color of one pixel is almost completely displaced by the colors if the peninsula(s) which intrude into it from some adjacent pixel(s) while the color of its own peninsula(s) intrude into other adjoining pixels. Perceptually, the basic shape of the pixel is no longer discernable. The color contours are said to be displaced. In this embodiment, the pixel must have both large area ratio and the area of the bays and peninsulas are distributed significantly away from their respective interlocking edges. (Figures 19a-b).

Case 3

The range of sizes of bays and peninsulas from the very small ones in Case 1 and the maximum possible size in Case 2 forms a continuum. Between these two extremes lies a range of perceptual ambiguity. That is, when the area ratio and the area distribution together are in this range, some people perceive well-defined contour lines (as in Case 1) while others no longer perceive the basic shape of the pixel (as in Case 2). For most people, the effect is as if the colors along the color contours simply blend into each other. This is the range of ambiguity. The perceptual affect is similar to that of such famous optical illusions of alternating images as the old lady and young maiden or the vase and the two facial profiles. In this embodiment, the invention provides for mixing of colors using areas that are indeed smaller than the size of an individual picture element. As a result, it is possible with this embodiment to fine tune the color contours more than it is possible with the resolution of a single pixel. (Figures 18a; 19c- f). The various coloring schemes in all three cases are independent of the coupling schemes.

In particular, even if bays and peninsulas are shaped such that no firm interlocking and no stable construction is possible, the color contours will still be perceptually affected from these bays and peninsulas. This invention takes the two principles of the constructions and of the perceptual affect on the color contours and combines them.

Case 4

In another embodiment of this invention, the color scheme of each pixel has a composition of several areas each of which has its own coloring scheme. This coloring scheme facilitates color contours which are independent of the edges of the picture elements and their coupling schemes. In one variation of this embodiment, the basic area of the pixel has one color scheme and each of the peninsulas has another. For example say pixels are provided in eight colors each of which is divided into an eight-gradation color scale. Thus, the color of the basic body of each pixel and the color of each peninsula may be one of the 64 different color shades. Therefore, if a pixel design has two peninsulas, such a color scheme may have up to 64^ (=262,144) different color combinations for each pixel. This embodiment provides for color contours, well defined, displaced or blending, which, for perceptual considerations, are completely independent of the interlocking scheme. Depending only on the color schemes of the pixels the user is free to generate color contours as desired. (Figures 29a-b).

Another objective of this invention is to provide an imaging system with transition picture elements. Such a picture element has at least one interlocking edge of one species and at least one interlocking edge of another species. The use of a transition pixel provides for the change from one coupling scheme into another. For example, in one section of the constructed image, the coupling provides for well defined color contours and in another section of the constructed image the interlocking provides for blending color contours. Transition pixels are used in order to connect the two sections. (Fig. 31).

Another objective of the invention is to provide the imaging system with finishing picture elements. That is, picture elements with one straight edge or with two adjacent straight edges, forming a comer, without any peninsula and without any bay, may be generated to form the outmost edge of a triangular, quadrilateral, hexagonal (etc.) image. Similarly, pixels which form an outer curved edge, such as sinusoid, (with comers, if needed, to match) should also be possible. (Figures 33a-e).

Another objective of this invention is to provide the imaging system with convertible picture elements. These picture elements have only peninsulas and for each peninsula, where a bay would have been present, penetrating the main body of the pixel, a perforation, outlining of the bay is present. The user, at his or her choice may leave the peninsula undisturbed or punch out the perforation, leaving a bay in its place. (Fig. 36).

Another objective of this invention is to provide a plurality of picture elements in the form of sheets. The sheets may be framed, such that: (1) regardless to the color scheme, monochromatic or spectral, and regardless to its gradation, digital or analog, the frame can be used as a color gradation scale, or (2) the frame may be used repeatedly to store unused pixels.

Figures 26a-c use digital variations in the dot densities. They may represent, for illustrating some aspects of this invention, a monochromatic color scale or, for illustrating other aspects of this invention, these different dot densities represent different colors or patterned colors. In Figure 27 the analog variation in the density of the pattern represents a monochromatic color gradation; for other embodiments, it represents spectral color gradation. Fig. 28 represents either an analog or digital spectral color scale.

Yet another objective of this invention is to provide picture elements of different overall sizes. In one variation of this embodiment, the larger picture elements have proportionally enlarged bays and peninsulas. Such size variation render the pixels of different sizes to belong to different species. For instance, compare pixel la-xl in Figure 30a to pixel la in Figure 30b.

In another variation of this embodiment, regardless to the overall size difference between various picture elements, their bays and peninsulas have identical shapes and sizes rendering them to belong to the same species. (Figure 30b).

Another objective of this invention is to provide special purpose picture elements. Such pixels are designed to better approximate specific shapes. For example, a kit that is designed for imaging cars may include pixels in the basic shape of car wheels, bumpers, etc. (Figure 37).

Another objective of this invention is to provide a method of creating works of art from such an imaging system as described above. An image is assembled from individual picture elements. At any time any individual pixel or set of pixels may be disengaged from the assembled work of art and replaced by another without adverse effect on it, on its adjacent pixels or on the whole image. Examples of a method include the use of various coupling schemes which utilize the bottleneck, or the gooseneck, or the S-shape, interlocking edges. The method also includes the use of symmetric and asymmetric coupling schemes, utilizing the symmetry or asymmetry of the coupling shape, or the symmetry or asymmetry of the interlock¬ ing edge. The method also includes the use of a single bay or peninsula per interlocking edge, or the use of multiple bays or peninsulas per coupling edge, such edges including male/female edge or bay and peninsula combination. In addition, various coloring schemes are possible including digital or analog, monochromatic, or spectral with changes along one or two axes. The method also includes the use of well-defined color contours, or blending color contours or displaced color contours. The method can further include pattern making schemes.

Another objective of this invention is to provide a method for generating new picture elements or small finite sets of picture elements each of which can facilitate an imaging system that have all the features enumerated above.

Another objective of this invention is to provide a method for producing motion pictures animation.

Claims

THE CLAIMS:

1. A picture element for simultaneously constructing a support structure while creating a work of art, the picture element comprising a substantially flat piece of opaque, transparent or light-emitting material having different color schemes on opposite faces thereof, the picture element including means for connecting the picture element to other picture elements having the same or different basic shape thereof.

2. The picture element of claim 1, wherein the connecting means permits detachment of the picture element from the other picture elements in a vertical direction perpendicular to the faces of the picture element and prevents detachment of the picture element from the other picture elements when the picture element is pulled away from the other picture elements in a direction parallel to the faces of the picture element.

3. The picture element of claim 1, having bilateral symmetry or the basic shape thereof having a small number of rotationally symmetric orientations.

4. The picture element of claim 1, wherein the connecting means comprises a first bay extending part way into but not through one side of the picture element and a second bay extending part way into but not through an opposite side of the picture element.

5. The picture element of claim 1, wherein the connecting means comprises a first peninsula protruding from one edge of the pixel and zero or peninsulas protruding from another edge or edges of the pixel, the peninsulas each having a thickness less than that of the pixel, the first peninsula having one surface thereof coplanar with one face of the pixel and one or more of the other peninsulas having one surface thereof coplanar with an opposite face of the pixel.

6. The picture element of claim 5, wherein the connecting means comprises at least one bay along at least one edge, extending from one side of the pixel part way into but not all the way through to the opposite side of the pixel and zero or more bays extending from the opposite side of the pixel part way into but not all the way through to the first side of the pixel.

7. The picture element of claim 6, wherein the position of each bay along the edge of the picture element and the position of each peninsula along the edge of the picture element are matched such that they appear on top of each other, or alternatively, the position of each bay along the edge of the pixel and the position of each peninsula along the edge of the pixel are independent of each other, the pixel having an edge with one or more bays and with no peninsulas, or the pixel having an edge with one or more peninsulas with no bays, or the pixel having an edge with a combination of one or more bays with one or more peninsulas.

8. The picture element of claim 1, wherein the connecting means comprises at least one peninsula which protrudes from at least one edge of the pixel, extending through the opposite faces of the pixel.

9. The picture element of claim 1, wherein the connecting means comprises at least one bay which intrudes from at least one edge of the pixel, extending through the opposite faces of the pixel.

10. The picture element of claim 1, wherein the connecting means comprises at least one peninsula protmding outwardly from the circumference of the pixel, the peninsula having each of its surfaces coplanar with one surface of the body of the pixel and/or at least one bay intruding into the body of the pixel from the circumference of the pixel and extending through opposite surfaces of the pixel.

11. The picture element of claim 1, wherein the connecting means provides interlocking and a friction fit between the picture element and another picture element.

12. The picture element of claim 1, wherein the picture element includes a plurality of edges extending between the opposite faces, at least one of the edges comprising a male edge which includes at least one peninsula protmding outwardly therefrom.

13. The picture element of claim 12, wherein the peninsula includes first and second sections, the first section being closer to the edge than the second section and the first section having a width which is more narrow than that of the second section.

14. The picture element of claim 12, wherein the peninsula includes first and second sections, the first section having a straight side edge which intersects with a straight side edge of a second section at an interior angle smaller than 180 degrees and the first section having the opposite straight side edge which intersects with the opposite straight side edge of the second section at an exterior angle smaller than 180 degrees such that both angles have their vertices pointing in the same direction.

15. The picture element of claim 1, wherein the picture element includes a plurality of edges extending between the opposite faces, at least one of the edges comprising a female edge which includes at least one bay intruding inwardly therefrom.

16. The picture element of claim 15, wherein the bay includes first and second sections, the first section being closer to the edge than the second section and the first section having a width which is more narrow than that of the second section.

17. The picture element of claim 15, wherein the bay includes first and second sections, the first section having a straight side edge which intersects with a straight side edge of a second section at an interior angle smaller than 180 degrees and the first section having the opposite straight side edge which intersects with the opposite straight side edge of the second section at an exterior angle smaller than 180 degrees such that both angles have their vertices pointing in the same direction.

18. The picture element of claim 1, wherein the picture element includes a plurality of edges extending between the opposite faces, at least one of the edges comprising includes at least one peninsula protmding inwardly therefrom and at least one bay intruding inwardly therefrom.

19. The picture element of claim 1, wherein the picture element includes a plurality of edges extending between the opposite faces, at least a first one of the edges comprising a male edge which includes a peninsula protmding outwardly therefrom and at least a second one of the edges comprising a female edge which includes a bay intmding inwardly therefrom.

20. The picture element of claim 1, wherein the pixel includes four edges, the edges forming a quadrilateral basic shape.

21. The picture element of claim 1, wherein the pixel includes four straight edges, the edges forming a square basic shape.

22. The picture element of claim 1, wherein the pixel includes six edges, the edges forming a hexagon basic shape.

23. The picture element of claim 1, wherein the pixel includes three edges, the edges forming a triangular basic shape.

24. The picture element of claim 1, wherein the pixel includes three edges, the edges forming a equilateral triangular basic shape.

25. The picture element of claim 1, wherein the pixel includes four straight edges, a first pair of the edges having a first length and having an interior angle between them that is smaller than 180 degrees and a second pair of edges having a second length shorter than the first length and having an internal angle between them that is greater than 180 degrees, facing the internal angle of the first pair of edges.

26. The picture element of claim 1, wherein the picture element includes four rectilinearly extending edges of unequal lengths.

27. The picture element of claim 1, wherein the picture element includes eleven rectilinearly extending edges.

28. The picture element of claim 1, wherein the picture element includes a plurality of non-rectilinear edges.

29. The picture element of claim 1, wherein the picture element includes a plurality of non-rectilinear edges, the edges including a concave portion and a convex portion joined to the concave portion at an inflection point.

30. The picture element of claim 1, wherein the picture element is circular in shape and includes a single edge.

31. The picture element of claim 1, wherein the locking means comprises at least one bottleneck joint.

32. The picture element of claim 31, wherein the bottleneck joint has bilateral symmetry.

33. The picture element of claim 1, wherein the locking means comprises at least one gooseneck joint.

34. The picture element of claim 1, wherein the picture element includes at least one plain edge which is a non-interlocking edge.

35. The pixel of claim 1, wherein, if the pixel is opaque, then the coloring scheme on at least one face thereof is a solid coloring scheme in which resolution of the color is sufficiently high to allow no visually discernible patterns of any sort and if the pixel is transparent or light- emitting, then the coloring scheme thereof is a solid coloring scheme in which resolution of the color is sufficiently high to allow no visually discernible patterns of any sort.

36. The picture element of claim 1, wherein, if the pixel is opaque, the coloring scheme on at least one face of the picture element is a pattemed coloring scheme in which resolution of the color is sufficiently low to permit clearly visible patterns, and, if the pixel is transparent or light-emitting, the coloring scheme thereof is a pattemed coloring scheme in which resolution of the color is sufficiently low to permit clearly visible patterns.

37. The picture element of claim 1, wherein the picture element has a shape which provides tessellation such that an image created with the picture element does not include gaps in between the picture element and other picture elements and no pixel overlaps any other pixel.

38. The picture element of claim 1, wherein the picture element has a shape which provides non-tessellation such that an image created with the picture element includes at least one gap is formed when the picture element is combined with other picture elements.

39. The picture element of claim 1, wherein the picture element has a thickness which is large enough to provide a friction fit between a peninsula or bay of the pixel and a corresponding bay or peninsula of another picture element in a staggered arrangement which generates a multi-layered three-dimensional image.

40. The picture element of claim 1, wherein the pixel is a convertible picture element having convertible means for selectively providing a male or a female edge, said convertible means comprising a double-peninsula, comprising of a peninsula and a perforation in the body of the pixel, the perforation intersecting the edge of the body of the pixel precisely where the peninsula intersects said edge and the perforation being a mirror image of said peninsula with the normalized edge between them is the axis of reflection, such that when said double- peninsula is removed from the pixel a bay is formed in the pixel.

41. The picture element of claim 1, wherein the color contours are independent of the shape of the edges thereof.

42. The picture element of claim 1, further comprising a bay and/or a peninsula each having a thickness less than the thickness of the body of said picture element and another bay and/or another peninsula, respectively, each having a thickness identical to the thickness of the body of said picture element.

43. The picture element of claim 1, wherein the picture element includes three or more rectilinearily extending edges.

44. The picture element of claim 43, wherein each one of the rectilinearily extending edges thereof can include a combination of one or more peninsulas protmding outwardly from said edge and/or one or more bays intmding inwardly from said edge.

45. The picture element of claim 43, wherein one or more of the vertices formed between said rectilinearily extending edges thereof can include a peninsula protmding outwardly from the said edges of said picture element and/or a bay intmding inwardly from said edges.

46. The picture element of claim 1, wherein the non-coupling edge has the shape of a sine curve.

47. The picture element of claim 1, wherein the pixel comprises part of a sheet of such pixels, some of the pixels being identical in shape and in color schemes and other ones of the pixels being identical in shape but varying in their color schemes.

48. A picture element for the creation of a work of art, wherein the pixel is substantially flat and having N normalized edges thereof forming a basic shape, the edges of said picture element including deviations from the normalized shape thereof, wherein said deviations affect visual perception of color contours of a visual image produced by juxtapositioning a plurality of picture elements having the same or different basic shape thereof. The area of the normalized basic shape of said pixel, regardless to any deviation thereof, being equal to A and the area of a single deviation being equal to B provide the area ratio of said deviation with respect to the pixel thereof being equal to B/A.

49. The picture element of claim 48, wherein the deviations from the normalized shapes of the edges provide well defined color contours.

50. The picture element of claim 49, wherein the picture element simultaneously creates a support stmcture while creating the work of art, the picture element, having same or different opaque color schemes on opposite faces thereof or having a transparent or light-emitting color scheme, and the picture element includes means for connecting the picture element to other picture elements having the same or different basic shape thereof.

51. The picture element of claim 48, wherein the deviations from the normalized shapes of the edges provide blending color contours thereof, such that the deviations from the normalized shapes of the edges provide for perceptually ambiguous color contours, similar to some optical illusions, wherein some viewers perceive well defined color contours while other viewers perceive blending colors along the color contours or the same viewer perceives well defined color contours some times and blending colors along the color contours at other times.

52. The picture element of claim 51, wherein the picture element simultaneously creates a support stmcture while creating the work of art, the picture element having same or different opaque color schemes on opposite faces thereof or having a transparent or light-emitting color scheme, and the picture element includes means for connecting the picture element to other picture elements having the same or different basic shape thereof.

53. The picture element of claim 51, further including means for approximating curves in the visual image produced by assembling a plurality of pixels, the curve approximating means comprising the basic shape of the pixel and its edges together, in their totality being asymmetric, such that rotation of the pixel to different asymmetric positions at which the pixel can be connected with at least one other picture element changes the color contours between the pixel and an adjacent pixel.

54. The picture element of claim 53, wherein the pixel simultaneously creates a support stmcture while creating the work of art, the pixel having same or different opaque color schemes on opposite faces thereof or having a transparent or light-emitting color scheme, and the pixel includes means for connecting the pixel to other pixels having the same or different basic shape thereof.

55. The picture element of claim 48, further including means for approximating curves in the visual image produced by assembling a plurality of pixels, the curve approximating means comprising the basic shape of the pixel being rotationally asymmetric, such that rotation of the pixel to a different asymmetric position necessitates an identical rotation of all other pixels within the work of art in which said pixel is present, changing the orientation of the color contours between colored areas within the created work of art.

56. The picture element of claim 55, wherein the picture element simultaneously creates a support stmcture while creating the work of art, the pixel having same or different opaque color schemes on opposite faces thereof or having a transparent or light-emitting color scheme, and the pixel includes means for connecting the pixel to other pixels having the same or different basic shape thereof.

57. The picture element of claim 48, wherein the area ratio is small such that the protmsion of the peninsula from the body of said pixel or the intmsion of the bay into the body of said pixel becomes visually insignificant and the normalized edge is perceived undisturbed.

58. The picture element of claim 48, wherein the area ratio is of a moderate size such that the protmsion of the peninsula from the body of said pixel or the intmsion of the bay into the body of said pixel becomes visually affect the color contours such that adjacent colors appear to be blending.

59. The picture element of claim 48, wherein the area ratio is sufficiently large such that the protmsion of the peninsula from the body of said pixel or the intmsion of the bay into the body of said pixel visually displaces the normalized shape of said pixel and/or its adjacent one.

60. The picture element of claim 48, wherein the pixel has a peninsula with a thickness between opposite faces thereof equal to about one-half the thickness of the body of the pixel, a first face of the peninsula being coplanar with a first face of the pixel.

61. The picture element of claim 60, wherein the picture element has a bay adjacent the peninsula, the bay having a thickness equal to the thickness of the peninsula, the bay intersecting with the normalized edge of the body of the picture element precisely where the peninsula intersects with said edge, the bay being defined in part thereof by a recessed surface in the picture element which is coplanar with a second face of the peninsula.

62. The picture element of claim 48, wherein the pixel has a bay with a thickness between opposite faces thereof equal to about one-half the thickness of the body of the pixel, a first face of the bay being coplanar with a first face of the pixel.

63. The picture element of claim 48, wherein the deviations from the normalized shapes of the edges provide displaced color contours thereof.

64. The picture element of claim 63, wherein the picture element simultaneously creates a support stmcture while creating the work of art, the pixel having same or different color schemes on opposite faces thereof or having a transparent or light-emitting color scheme, and the pixel includes means for connecting the pixel to other pixels having the same or different basic shape thereof.

65. The picture element of claim 48, further including means for approximating curves in the visual image produced by assembling a plurality of pixels, the curve approximating means comprising the basic shape of the pixel and its edges together, in their totality being asymmetric, such that rotation of the picture element to different asymmetric positions at which the pixel can be connected with at least one other pixel changes the color contours between the pixel and an adjacent pixel.

66. The picture element of claim 65, wherein the pixel simultaneously creates a support stmcture while creating the work of art, the pixel having same or different opaque color schemes on opposite faces thereof or having a transparent or light-emitting color scheme, and the pixel includes means for connecting the pixel to other pixels having the same or different basic shape thereof.

67. The picture element of claim 48, wherein the pixel comprises part of a sheet of such pixels, some of the pixels being identical in shape and in color schemes and other ones of the pixels being identical in shape but varying in their color schemes.

68. The picture element of claim 48, including a peninsula of a first type having at least one intmding bay of a second type and a bay of a first type having at least one protmding peninsula of a second type.

69. A set of picture elements comprising a finite number of pixels for simultaneously constmcting a support stmcture while creating a work of art, each pixel comprising a substantially flat piece of material, each pixel having a different overall shape with respect to the normalized shape of every other pixel that is a member of said set, each pixel having means for connecting itself with every subset of pixels of said set, that is, no subset of pixels of the set may be isolated from (i.e., incapable of interlocking with) the other pixels of the set, the connecting means permitting detachment of the pixels from each other in a vertical direction perpendicular to opposite faces of the pixels and preventing detachment of said pixels in a direction parallel to the opposite faces.

70. The set of picture element of claim 69, wherein a first pixel includes a plurality of non-rectilinear edges, two of the edges being convex in shape and one of the edges being concave in shape, the concave edge extending between one end of each of the convex edges and an opposite end of each of the convex edges being joined together. And wherein a second pixel includes a plurality of non-rectilinear edges, two of the edges being concave in shape and one of the edges being convex in shape, the convex edge extending between one end of each of the concave edges and an opposite end of each of the concave edges being joined together.

71. The set of picture element of claim 69, wherein a first pixel is circular in shape and includes a single edge and a second pixel includes three edges, each of which is concave in shape and is formed by an arc having a constant radius for each of the edges, the three radii having the identical length as the radius of the circular edge of the first pixel.

72. The set of picture elements of claim 69, wherein the connecting means comprises a male edge and/or a female edge and/or a combination male and female edge, the male edge including a peninsula protmding therefrom, the female edge including a bay intmding therefrom and the combination edge including a peninsula protmding therefrom and a bay intmding therefrom.

73. The set of picture elements of claim 72, wherein a first one of the pixels has a basic shape of a square and second and third pixels have a basic shape of a right isosceles triangle, the hypotenuse of the triangle being equal in length to the diagonal of the square.

74. The set of picture elements of claim 72, wherein some of the pixels include more male edges than female edges and the remainder of the pixels include more female edges than male edges.

75. The set of picture elements of claim 69, wherein the pixels have tessellating shapes or, if the shapes of one or more of the pixels is non-tessellating, then there exists at least one plurality of said pixels which is tessellating.

76. The set of picture elements of claim 69, wherein, if a pixel of said set is opaque, then the coloring scheme on at least one face thereof is a solid coloring scheme in which resolution of the color is sufficiently high to allow no visually discernible patterns of any sort and if a pixel of said set is transparent or light-emitting, then the coloring scheme thereof is a solid coloring scheme in which resolution of the color is sufficiently high to allow no visually discernible patterns of any sort.

77. The set of picture elements of claim 69, wherein, if a pixel that is a member of said set is opaque, the coloring scheme on at least one face of the picture element is a pattemed coloring scheme in which resolution of the color is sufficiently low to permit clearly visible patterns, and, if a pixel that is a member of said set is transparent or light-emitting, the coloring scheme thereof is a pattemed coloring scheme in which resolution of the color is sufficiently low to permit clearly visible patterns.

78. The set of picture elements of claim 69, wherein one or more of the pixels is a convertible picture element having convertible means for selectively providing a male or female edge, said convertible means comprising a removable double peninsula which when removed from the picture element forms a bay in the picture element.

79. The set of picture elements of claim 69, wherein at least one of the pixels, but not all of them, is a framing pixel having at least one interlocking edge and having at least one non- interlocking edge.

80. The set of picture elements of claim 69, wherein the non-coupling edge is straight.

81. The set of picture elements of claim 69, wherein one of the pixels is a comer pixel having two adjacent finished edges which are not male or female edges.

82. The set of picture elements of claim 69, wherein a first one of the picture elements has a basic shape of an equilateral triangle with two female edges and one male edge, a second one of the picture elements having a basic shape of an equilateral triangle with two male edges and one female edge.

83. The set of picture elements of claim 69, wherein each of the pixels is substantially flat and normalized edges thereof form a basic shape, the edges of said pixel including deviations from the normalized shape thereof, wherein said deviations affect visual perception of color contours of a visual image produced by juxtapositioning a plurality of pixels having the same or different basic shape thereof.

84. The set of picture elements of claim 69, wherein the first picture element has a basic shape of a square and the second and third picture elements having the basic shape of an equilateral triangle, the side of the triangle being equal in length to the side of the square.

85. The set of picture elements of claim 69, the first and second picture elements having a basic shape of a right isosceles triangle and the third and fourth picture elements having the basic shape of an equilateral triangle, the side of the triangle being equal in length to the side of the right isosceles triangle.

86. The set of picture elements of claim 69, at least one of the picture elements having one interlocking edge of a first species and a second coupling edge of a second species, the two interlocking species being different.

87. The set of picture elements of claim 86, wherein the first coupling species is capable of forming well defined color contours and the second interlocking species is capable of forming displaced color contours.

88. The set of picture elements of claim 86, wherein the first interlocking species is capable of forming well defined color contours and the second interlocking species is capable of forming ambiguous color contours.

89. The set of picture elements of claim 69, wherein a first and a second picture elements interlock with each other along one edge and along the two edges that are perpendicular to their coupling edge said pixels together form a single bay which can couple with a single peninsula of a single pixel.

90. The set of picture elements of claim 69, wherein at least a few of the picture elements are shaped such that they maybe assembled together to form recognizable shapes of common animated or inanimate objects independently of or as part of a work of art.

91. A sheet of transparent or light-emitting or opaque pixels for simultaneously constmcting a support stmcture while creating a work of art, the sheet of pixels including at least one group of pixels, each of which has a color scheme on opposite faces of opaque pixels or a color scheme through transparent pixels or a color scheme radiated by light-emitting pixels and including connecting means for connecting each of the pixels to all adjacent pixels having the same or different coloring schemes.

92. The sheet of claim 91, wherein the sheet includes a frame having at least one plain edge, the frame including connecting means connecting the frame with the pixels.

93. The sheet of claim 92, wherein the sheet includes more than one group of pixels and a frame, including connecting means interlocking the pixels with the frame, the frame extends between the groups of pixels, separating each group from the adjacent groups, and the area of the frame into which pixels are interlocked has the same coloring scheme as said interlocked pixels.

94. The sheet of claim 91, wherein the sheet includes at least one second group of pixels, each of the second group of pixels having a shape which is different from a shape of each of the first group of pixels, each of the second group of pixels having different coloring schemes on opposite faces thereof and including connecting means for connecting one of the pixels to another one of the pixels having the same or different coloring schemes on opposite faces thereof.

95. The sheet of opaque pixels of claim 91, wherein the coloring scheme of said pixels composes a first color scale on one face thereof and a second color scale on an opposite face thereof, each color scale may be of either discrete or continuous graduation and each color scale may be either monochromatic or spectral and the graduation of each color scale may progress along one or two axes of the plane of said sheet, thereby providing pixels for simultaneously creating an image on one side of the work of art and an image with different or same colors but of the same design on the opposite side of the work of art or for simultaneously creating two independent images, one on each surface of the artwork.

96. The sheet of claim 95, wherein a first color scale is provided on the first face and a second color scale is provided on the second face thereof, both having the same intensity value on the opposite ends of said sheet and the gradations of both scales progressing in the same steps of intensity toward the opposing ends of said sheet.

97. The sheet of transparent or light-emitting pixels of claim 91, wherein the coloring scheme of said pixels composes a color scale, the color scale may be of either discrete or continuous graduation and the color scale may be either monochromatic or spectral and the graduation of the color scale may progress along one or two axes of the plane of said sheet.

98. A method of picturing and sculpturing with opaque and/or transparent or light- emitting pixels, having different colors and/or patterns, having same or different normalized shapes, and having interlocking means, which are interconnected to form a two-dimensional or a three-dimensional self-supporting visual image, the method being carried out without the use of tools, devices and/or materials other than the pixels, and the method being intuitively understood by most people above the age of an infant including some mentally retarded persons, and the method being simple enough to be carried out by young children as well as handicapped persons who may lack eye-to-hand coordination or muscular coordination for using other imaging methods such as painting and drawing, comprising:

(a) Interlocking two or more of substantially flat opaque picture elements having the same or different color schemes on opposite faces thereof or transparent or light-emitting pixels such that a peninsula protmding outwardly from an edge of one of the pixels protmdes into a bay intmding inwardly from an edge of an adjacent one of the picture elements and the color scheme of the one picture element is the same or different from the color scheme on the adjacent picture element on faces thereof;

In order to couple two pixels, the two pixels must first be placed in two substantially parallel planes one above the other and the edges of the peninsula of one pixel and the edges of the bay of the second pixel must be aligned such that they are congruent; (it is not necessary to require that the complete male and female edges be congment for in some pixels have non- tessellating shapes, such as circular pixels. In the case of tessellating picture elements or tessellating sets of pixels, when the bay and the peninsula are congment, then the corresponding male and female edges are necessarily congment as well.) Then, the two pixels are pressed together by applying on them a force in the direction that is perpendicular to their parallel planes. (Figure 15a.)

(b) Removing zero or more pixels from the image, whether they are located along the current edge of the image or they are surrounded with other pixels. In order to disjoint two interlocked pixels apply a shear force on the joint between the two pixels in a direction that is almost perpendicular to their common plane. Due to the friction fit, some angular motion of one pixel with respect to the other is most likely required. (Figure 15c.)

(c) Repeating steps (a) and (b) until a desired visual image is produced.

99. The method of claim 98, wherein at least some of the picture elements include at least one male edge and at least some of the pixels include at least one female edge, the method further comprising a step of orienting male and female edges of pixels having a first color scheme with respect to male and female edges of other pixels having a second color scheme which is different from the first color scheme so as to create different perceptual effects.

100. The method of claim 98, wherein the method further comprising an initial step of selecting the type of pixels to be used according to what visual affect the bays and peninsulas thereof, being perceptual deviations from the normalized edges, have on the color contours.

101. The method of claim 98, further comprises another step whenever a pixel is to be placed in a work of art: selecting the desired color scheme available are any combination from the following determining perceptual factors - opaque or transparent or light-emitting, digital graduation or continuous graduation, spectral or monochromatic, and whether the graduation progresses along one axis or two axes of the plan of the pixel.

102. The method of claim 101, wherein the selected pixels affect the color contours such that the colors thereof appear to be blending or displaced, and the normalized shape of a pixel has several rotational symmetries and said symmetries are violated by the existence of either bay(s) and or peninsula(s), then the method further comprising another step prior to placing a pixel within the image: selecting a pixel that can interlock in the desired position and in an orientation that creates the desired perceptual affect.

103. The method of claim 101, wherein the normalized shapes of the pixels having a finite number of rotationally symmetric orientations, then the method further comprising another step prior to using the first pixel: determining how to orient the first pixels to be placed in the image as its orientation will necessarily dictate, with flexibility limited to the available rotational symmetries, the orientations of the rest of the picture elements in the work of art.

104. The method of claim 101, wherein the selected coloring scheme is discrete graduation, another initial step: determine the resolution of the artwork with respect to the overall size thereof.

105. The method of claim 98, wherein the peninsula of a first pixel is fitted in the bay of a second pixel by aligning a wider section of the peninsula with a length of the bay and an opening formed by the bay, inserting the peninsula through the bay, rotating the first pixel about a bilateral axis thereof until the wide section is aligned with a narrow section of the bay and the first pixel cannot be pulled away from the second pixel.

106. The method of claim 98, further comprising a step of mapping the visual image to be produced by determining a pattern of the color schemes of the picture elements which will correspond to a visual image of an original subject.

107. The method of claim 98, wherein a two-dimensional self-supporting image is formed.

108. The method of claim 98, wherein a three-dimensional self-supporting image is formed.

109. The method of claim 98, wherein the picture elements are interconnected with gaps therebetween.

110. The method of claim 98, wherein said picture elements are interconnected in a tessellating arrangement, without any unfilled gaps therebetween and with no pixel overlapping another pixel.

111. The method of claim 98, wherein steps (a) through (c) are repeated until the picture elements form a self-supporting stmcture which has an overall normalized shape in two- dimensions such as a triangle, or a square, or a parallelogram, or a quadrilateral, or a hexagon, or a circle, or an ellipsoid, or a regular or an irregular polygon.

112. The method of claim 98, wherein recognizing that the pixels within a work of art form a grid an initial step: determining the size of the grid, its position and its orientation with respect to the model of the artwork.

113. A method for designing substantially flat picture elements for picturing and sculpturing self supporting, solid and stable two- and/or three-dimensional works of arts:

(a) Select the type of pixel - light reflecting, light transmitting, or light emitting.

(b) Selecting the material(s) from which the pixels will be fabricated.

(c) Design the color schemes. For each color scheme, design digital or analog graduation scales, monochromatic or spectral color scales, single- or bi- dimensional color scale and solid or pattemed colors.

(d) Determine the most coarse resolution of the imaging system, that is, determine the size of the largest pixel.

(e) Design the basic shape of the pixel or, if set of pixels is to be used, design the shape of each pixel. The shape may be tessellating or non-tessellating. The shape may have bilateral axes of symmetries or not and it may have any number, from zero (e.g., scalene triangle) to infinity (e.g., a circle) axes of rotational symmetries. The shape may be bounded by straight or curved edges. (f) Design the coupling scheme or scheme or schemes. Symmetric or asymmetric bottleneck, or gooseneck or S-type interlocking schemes may be used. Consider the perceptual affect of the interlocking schemes with respect to well-defined, blending or displaced color contours. Also consider the capability to couple a pixel with its mirror-image pixel. Consider multi- layered and three-dimensional coupling.

(g) Arrange several pixels in a two dimensional array such that the edges of adjacent pixels butt together and at least one of the pixels is completely surrounded by other pixels. Along each edge that is formed by at least two pixels in this array, superimposed one or more of the designed interlockings from Step (f). The peninsula becomes part of the pixel(s) on one side of the shared edge and the bay becomes part of the pixel(s) on the other side of the shared edge. Repeat this step for all shared edges. Avoid having overlapping couplings or interlockings that are too close to each other within a single pixel for the fabricating material to support. During this process, consider symmetric versus asymmetric placement of the couplings along each of the edges. Also consider the capability to interlock a pixel with its mirror-image pixel. If a set of pixels is used, repeat this step until sufficient number of interlocking edges have been provided for, such that each pixel of the set can couple with every subset of pixels of said set.

114. A method for constmcting three-dimensional works of arts. Maximum stability is achieved when a stmcture is constmcted such that there exist an equilibrium of all of the forces that act on each and every picture element in all three dimensions.

(a) Orient two pixels such that, their planes intersect where the male edge of one meets the female edge of the other, and the peninsula is oriented with respect to the bay such that its wider section is aligned with the length of the bay and its opening;

(b) Insert the peninsula through the bay and rotate the pixel with the peninsula around its bilateral axis until the wide section is aligned with the narrower section of the bay such that one cannot simply be pulled away from the other and interlocking is achieved.

(c) Arrange the pixels such that there exists a force that continuously and without interruption presses the wide section of the peninsula of one pixel against the narrow section of the bay of the other pixel. (d) To disconnect any pair of pixels from each other within a three-dimensional construction, release the pressure enforced by step (c) above and rotate the pixel with the peninsula with respect to the bay into which it was inserted in step (b) until the two pieces can be separated by pulling them apart.

(e) Repeat steps (a) through (d) as many times as necessary to complete the work of art.

115. A method for producing motion pictures animation with picture elements which are substantially flat, having same or different color schemes and capable of coupling to form two- dimensional or three-dimensional self supporting visual images, the method being carried out without the use of tools, devices and/or materials other than the picture elements and the cinematographic equipment, the method comprising:

(a) placing zero or more assembled or unassembled picture elements in front of a camera;

(b) exposing one frame of film, photographing the image;

(c) disassembling and/or removing any number ~ from zero to all — of picture elements from the image;

(d) adding to the image any number of picture elements ~ from zero to more than the number of picture elements removed in Step (c); and

(e) repeating Step (b) through Step (d) as many times as necessary to complete the animation.