WO1990012626A1

WO1990012626A1 - Dice

Info

Publication number: WO1990012626A1
Application number: PCT/GB1990/000601
Authority: WO
Inventors: Alfred Albert Charles Eardley; Darryl Eardley; Blake Eardley
Original assignee: ALPHADICE Ltd
Current assignee: ALPHADICE Ltd
Priority date: 1989-04-21
Filing date: 1990-04-19
Publication date: 1990-11-01
Anticipated expiration: 1991-10-21
Also published as: GB2230464A; GB2230464B; GB9008854D0; GB8909082D0; EP0469028A1; AU5443590A

Abstract

Dice are disclosed which have more than six faces arranged in opposed pairs, the faces of each pair of opposed faces being substantially identical and with their centres substantially lying on an axis passing substantially through dice centre, the faces being formed as flats on a generally spherical body. In passing substantially through the dice centre the axes may be substantially symmetrically spaced one from the other or be arranged such that their angular spacing in both bearing and elevation is maximised. One embodiment provides that all the faces are similar and non-contiguous - preferably that all the faces are substantially identical and circular in plan. Dice are disclosed having fourteen and twenty six circular faces arranged in seven and thirteen opposed pairs with the axes joining opposed faces substantially symmetrically spaced one from the other. A die is disclosed having twenty six circular faces arranged in thirteen opposed pairs with the angular spacing in both bearing and elevation of the axes joining opposed faces maximised. Dice are disclosed the faces of which are contiguous and the shape of each in plan being selected from one of a number of differing substantially regular shapes and so configured that no two adjacent faces have the same substantially regular shape. One form of this die has six substantially identical square faces, eight substantially identical equisided triangular faces, and twelve substantially identical rectangular faces. Dice embodying the invention may be formed by moulding a plastics material, or by machining the faces thereof from a sphere of moulded plastics or other material; and the faces be provided with any desired indicia - symbols, letters, numerals or the like by cutting, etching or engraving.

Description

DICE

DESCRIPTION

The invention relates to dice.

In this specification the term die relates to any object having a number of faces some or all of which carry indicia and which may be cast or thrown to select at random between ones of those indicia. BACKGROUND ART

Dice have been known for many years to comprise a regular cuboid shape are made from a material of constant density and each of the six faces of which carries one of a number (between one and six) spots. The regular shape of such dice means that they are unbiased and one or other of the six faces will be uppermost when the die comes to rest after having been thrown or cast by a user, the particular face being selected at random by the fall of the die.

Dice having four faces have been proposed (each face being an equilateral triangle) and made and these have proved to be, when made properly, effective at selecting a number at random - they are unbiased.

Although attempts have been made to produce dice with more than six faces which are unbiased, these attempts to our knowledge have not until now been successful.

Objects of the invention include the provision of dice having more than six faces and which are substantially unbiased.

SUMMARY OF THE INVENTION In one aspect the invention provides a die having more than six faces arranged in opposed pairs on opposite sides of the die, the faces of each pair of opposed faces being substantially identical and with their centres substantially lying on an axis passing substantially through the centre of .the die and wherein the faces are formed as flats on a generally spherical body.

In passing substantially through the centre of the die the axes may be substantially symmetrically spaced one from the other or alternatively their angular spacing in both bearing and elevation may maximised. A second aspect of the invention provides a die having more than six faces arranged in opposed pairs on opposite sides of the die, the faces of each pair of opposed faces being substantially identical and with their centres substantially lying on an axis passing substantially through the centre of the die and wherein the said axes are substantially symmetrically spaced one from the other in passing substantially through the centre of the die.

A third aspect of the invention provides a die having more than six faces arranged in opposed pairs on opposite sides of the die, the faces of each pair of opposed faces being substantially identical and with their centres substantially lying on an axis passing substantially through the centre of the die and wherein the angular spacing of the various axes one from the other in passing substantially through the centre of the die in both bearing and elevation is maximised.

The faces of a die in accordance with any of the above noted aspects and features may all be substantially similar and non-contiguous, preferably, all the faces are substantially identical and are circular in plan.

One die according to this embodiment has fourteen substantially identical circular faces arranged in seven substantially diametrically opposed pairs, the centres of the opposed pairs of faces substantially lying on substantially equilength axes which are substantially symmetrically spaced one from the other in passing substantially through the centre of the die and which have bearing and elevation relative to a reference axis in substance as follows:- AXIS B BEEAARRIINNGG ((°°)) ELEVATION (°) 1 * * 90.00 2 0 0.. ,0000 0.00

3 9 900.. ,0000 0.00

4 4455.. . 0000 Sin^-1(l/3⁵)

5 113355.. , 0000 Sin^"1(l/3^I)

6 222255.. . 0000 Sin^"1(l/3 ) 7 331155.. . 0000 Sin^"1(l/3 )

(* the bearing of this axis may be assigned any value between 0° and 360°).

Another die in accordance with this embodiment has twentysix substantially identical circular faces arranged in thirteen substantially diametrically opposed pairs, the centres of the opposed pairs of faces substantially lying on substantially equilength axes which are substantially symmetrically spaced one from the other in passing substantially through the centre of the die and which have bearing and elevation relative to a reference axis in substance as follows:-

AXIS BEARING (°) ELEVATION (°)

1 * 90.00

2 0.00 0.00 3 90.00 0.00

4 45.00 Sin^-1(l/3*)

5 35.00 Sin~¹(l/3^)

6 25.00 Sin^-1(l/3 )

7 15.00 Sin^_1(l/3*) 8 0.00 45.00

9 90.00 45.00 10 180.00 45.00

11 170.00 45.00

12 145.00 0.00

13 135.00 0.00 A third die in accordance with this embodiment of the invention has twentysix substantially identical circular faces arranged in thirteen diametrically opposed pairs, the centres of the opposed pairs of faces substantially lying on substantially -equilength axes the angular spacing of which one from the other in passing substantially through the centre of the die in both bearing and elevation is maximised and which have .bearing and elevation relative to one another in substance as follows:-

AXIS BEARING ( °) ELEVATION (°)

1 * 90.00

2 0.00 51.04

3 90.00 51.04

4 180.00 51.04

5 270.00 51.04

6 45.00 30.49

7 135.00 30.49

8 225.00 30.49

9 315.00 30.48

10 19.48 0.00

11 70.52 0.00

12 109.48 0.00

13 160.52 0.00

(* the bearing of this axis may be assigned any

0° and 360°). Another die embodying the invention has faces which are contiguous and the shape of each face of the die in plan is selected from one of a number of differing substantially regular shapes and the die is so configured that no two adjacent faces have the same substantially regular shape. A fourth aspect of the invention provides a die having more than six faces arranged in opposed pairs on opposite sides of the die, the faces of each pair of oppose faces being substantially identical and with their centre substantially lying on an axis passing substantially throug the centre of the die and wherein the faces are contiguou and the shape of each face of the die in plan is selecte from one of a number of differing substantially regula shapes and the die is so configured that no two adjacen faces have the same substantially regular shape.

A die embodying this aspect of the invention ha six substantially identical square faces eight substantiall identical equisided triangular faces arranged in pairs o four axes, and twelve substantially identical rectangula faces arranged in pairs on six axes, arranged such that n faces of the same substantially regular shape are adjacen one to another.

The axes of this die may have bearing and elevatio of each axis and the shape of the faces associated therewit is substantially as follows:-

AXIS BEARING (°: ) ELEVATION (°) FACES

Al * 90.00 1, 2 SQUARE

A2 0.00 0.00 3, 4 SQUARE

A3 90.00 0.00 5, 6 SQUARE

A4 45.00 0.00 7, 8 RECTANGLE

A5 135.00 0.00 9, 10 RECTANGLE

A6 0.00 45.00 I¹' 12 RECTANGLE

A7 90.00 45.00 13, 14 RECTANGLE

A8 180.00 45.00 15, 16 TRIANGLE

A9 270.00 45.00 17, 18 TRIANGLE

A10 45.00 45.00 19, 20 TRIANGLE

All 135.00 45.00 21, 22 TRIANGLE

A12 225.00 45.00 23, 24 TRIANGLE

A13 315.00 45.00 25, 26 TRIANGLE

(* the bearing of this axis may be assigned any value betwee 0° and 360°). In one preferred arrangement of this die th nominal dimensions of the sides of the faces have i substance the following relationship:- squares 152x152 triangles 192x192x192, rectangles 164x130; - the corner where three faces meet being rounded or bevelled.

Dice embodying the invention may be formed b moulding a plastics material, or by machining the faces fro a sphere of moulded plastics or other material.

The faces of dice embodying the invention ar preferably provided with indicia in the form of letter symbols, or numerals whilst being moulded or by cutting etching, or engraving after the faces have been forme thereon.

EMBODIMENTS OF THE INVENTION

The above and other aspects, features an advantages of the invention will become apparent from th following description of various embodiments of the inventio now made with reference to the accompanying drawings, i whic :-

Figure 1 diagra atically illustrates the thre cartesian coordinates, X, Y and Z and the angular bearing an elevation conventions used in this specification,

Figure 2 shows at A a fourteen faced die embodyin the "invention, and at B diagramatically illustrates th arrangement of the axes joining opposed pairs of faces o this die (viewed looking along the Z axis referred to i Figure 1) ,

Figure 3 shows in a manner similar to Figure 2 at a first twentysix faced die embodying the invention, and at the arrangement of the axes joining opposed pairs of faces o this die, Figure 4 shows in a manner similar to Figure 2 at a second twentysix faced die embodying the invention, and a B the arrangement of the axes joining opposed pairs of face of this die,

Figure 5 shows in a manner similar to Figure 2 at a second twentysix faced die embodying the invention, and a

B the arrangement of the axes joining opposed pairs of face -7-

of this die ,

Figure 1 illustrates the three axes mutuall orthogonal X, Y, and Z extending in positive and negativ directions from an origin 10 and from which the position o 5 any point may be defined by using the cartesian coordinate of the point - the distances of the point from the origi along the X, Y and Z axes.

The Figure also shows the convention used fo defining the the bearing (B) and elevation (E) angles used i

10 this specification. As can be seen the bearing angle (B) i measured from the X (positive) axis and increases in movin counterclockwise around the Z axis in the X-Y plane, and th elevation angle (E) is measured from the X (positive) axi and increases in moving clockwise around the Y axis in the X

15 Z plane. The position of any point may be defined by usin the elevation (E) and bearing (B) and of the point fro reference axes and the distance of that point from th origin.

Figure 2 shows at A plan view of a fourteen face

20 die embodying the invention, and at B is a diagrammatic vie (looking along the Z axis in Figure 1 toward the X-Y plane illustrating the relative positions of the seven axes Al, A2 A3, A4, A5, A6 and A7 joining the seven opposed pairs o faces making up the fourteen faces of the die. The axes ar

25 substantially symetrically arranged in space in passin through the origin.

In Figure 2B only one face in indicated (in dotte outline at 12), however, the centre points of each of th fourteen faces are indicated at Fl, F2, F3, F4, F13

30 F14.

Each face is substantially identical in size, the all lie at substantially the same distance G from the centr of the die and each is disposed in a plane substantiall normal to the axis on which it lies.

35 The centre points of opposed pairs of faces li substantially on the axis joining them. The bearing and elevation of each axis is given in substance in Table 1A - which also indicates which Faces of the die lie on each axis.

TABLE 1A

AXIS BEARING (°) ELEVATION (°) FACES

Al * 90.00 1, 2

A2 0.00 0.00 3, 4

A3 90.00 0.00 5, 6

A4 45.00 Sin^-1(l/3²) 7, 8

A5 135.00 Sin^"1(1/3 ) 9, 10

A6 225.00 Sin^"1(l/3 ) 11, 12

A7 315.00 Sin^"1(1/3 ) 13, 14

(* the bearing of this axis may be assigned any value between 0° and 360°).

The cartesian coordinates of the centre points of the faces of this die are substantially as shown in Table IB.

TABLE IB

FACE COORDINATES (X,Y,Z)

0 0 1

2 0 0 -1 3 1 0 0

4 -1 0 0

5 0 1 0

6 0 -1 0

1 1 1

7 1/3² 1/3²

1 1/3²

1 1

8 -1/3 -1/3²

1 -1/3²

1 1

9 -1/3² 1/3²

1 1/3²

1 1

10 1/3² -1/3²

1 -1/3²

1 1

11 -1/3² -1/3² 1/3²

1 1 1

12 1/3² 1/3² -1/3²

1 1 1

13 1/3² -1/3² 1/3²

1 1 1

14 -1/3² 1/3² -1/3² The surface of the die between the faces is rounded - that is to say the faces are formed as flats on a sphere.

The dice may be made by moulding any suitable plastics material (e.g. perspex) in a mould of the desired shape or in a mould producing a sphere and thereafter cutting or grinding the sphere to produce the faces.

Figure 3 shows at A plan view of a first twentysix faced die embodying the invention, and at B a diagrammatic view (looking along the Z axis in Figure 1 toward the X-Y plane) illustrating the relative positions of the thirteen axes Al, A2, A3, A13 joining the thirteen opposed pairs of faces making up the twentysix faces of the die. The axes are substantially symetrically arranged in space in passing through the origin.

In Figure 3B only one face in indicated (in dotted outline at 16), however, the centre points of each of the twentysix faces are indicated at Fl, F2, F3, F4,

F25, F26. The faces are substantially identical in size, they all lie at substantially the same distance H from the centre of the die and each is disposed in a plane substantially normal to the axis on which it lies.

The centre points of opposed pairs of faces substantially lie on the axis joining them. As with the die of Figure 2, the surface of this die between the faces is rounded - that is to say the faces are formed as flats on a sphere. Again, this die may be made by moulding a plastics material in either a pre-for ed mould of the desired shape or in a mould producing a sphere and thereafter cutting or grinding the sphere to produce the faces.

The bearing and elevation of each axis is given in substance in Table 2A - which also indicates which Faces of the die lie on each axis. TABLE 2A

AXIS BEARING {") ELEVATION (^u) FACES

Al * 90.00 1, 2

A2 0.00 0.00 3, 4

A3 90.00 0.00 5, 6

A4 45.00 Sin^"1 (1/3*) 7, 8

A5 135.00 Sin^"1 (1/3*) 9, 10

A6 225.00 Sin^_1ι (1/3*) 11, 12

A7 315.00 Sin^"1' (1/3*) 13, 14

A8 0.00 45.00 15, 16

A9 90.00 45.00 17, 18

A10 180.00 45.00 19, 20

All 270.00 45.00 21, 22

A12 45.00 0.00 23, 24

A13 135.00 0.00 25, 26

(* the bearing of this axis may be assigned any value between 0° and 360°). The cartesian coordinates of the centre points of the faces of this die are substantially as shown in Table 2B. TABLE 2B

FACE COORDINATES (X,Y,Z)

1 0 0 1

2 0 0 -1

3 1 0 0

4 -1 0 0

5 0 1 0

6 0 -1 0

7 1/3* 1/3* 1/3

8 -1/3* -1/3* -1/3

9 -1/3* 1/3* ^{1 3'}

10 1/3* -1/3* -1/3^

11 -1/3* -1/3* 1/3 -

12 1/3 1/3² -1/3- TABLE 2B (cont'd)

Figure 4 shows at A plan view of a second twentysix faced die embodying the invention, and at B a diagrammatic view (looking along the Z axis in Figure 1 toward the X-Y plane) illustrating the relative positions of the thirteen axes Al, A2, A3, A13 joining the thirteen opposed pairs of faces making up the twentysix faces of the die. The axes are arranged in space in passing through the origin to provide substantially for their maximum separation.

In Figure 4B only one face in indicated (in dotted outline at 18), however, the centre points of each of the twentysix faces are indicated at Fl, F2, F25, F26.

The faces are substantially, identical in size, they all lie at substantially the same distance I from the centre of the die and each is disposed in a plane substantially normal to the axis on which it lies. The centre points of opposed pairs of faces substantially lie on the axis joining them.

As with the dice of Figures 2 and 3, the surface of this die between the faces is rounded - that is to say the faces are formed as flats on a sphere. Again, this die may be made by moulding a plastics material in either a pre¬ formed mould of the desired shape or in a mould producing a sphere and thereafter cutting or grinding the sphere to produce the faces.

The bearing and elevation of each axis is given in in Table 3 - which also indicates which Faces of the die lie on each axis. TABLE 3

AXIS BEARING (°) ELEVATION (°) FACES

Al * 90.00 1, 2

A2 0.00 51.04 3, 4

A3 90.00 51.04 5, 6

A4 180.00 51.04 7, 8

A5 270.00 51.04 9, 10

A6 45.00 30.49 11, 12

A7 135.00 30.49 13, 14

A8 225.00 30.49 15, 16

A9 315.00 30.48 17, 18

A10 19.48 0.00 19, 20

All 70.52 0.00 21, 22

A12 109.48 0.00 23, 24

A13 160.52 0.00 25, 26

(* the bearing of this axis may be assigned any value between 0° and 360°).

Figure 5A is a plan view of a die having twenty¬ six faces, the shape of each face of the die being either substantially a square (such as shown at 20), substantially an equilateral triangle (such as shown at 22), or substantially a rectangle (such as shown at 24).

Figure 5B is a diagrammatic view (looking along the Z axis in Figure 1 toward the X-Y plane) illustrating the relative positions of the thirteen axes Al, A2, A3, A13 joining the thirteen opposed pairs of faces making up the twentysix faces of the die.

The centre points of each of the twentysix faces are indicated at Fl, F2, F3, F4, F25, F26. The centre point of each square and triangular face lies at substantially the same distance J from the centre of the die, whilst the centre points of the rectangular faces lie substantially at a distance 0.986J from that centre point. Each face is disposed substantially in a plane normal to the axis on which it lies.

The bearing and elevation of each axis is given in substance in Table 4A - which also indicates which Faces of the die lie on each axis.

TABLE 4A

AXIS BEARING (° ) ELEVATION (°) FACES

Al * 90.00 1, 2 SQUARE

A2 0.00 0.00 3, 4 SQUARE

A3 90.00 0.00 5, 6 SQUARE

A4 45.00 0.00 7, 8 RECTANGLE

A5 135.00 0.00 9, 10 RECTANGLE

A6 0.00 45.00 11, 12 RECTANGLE

A7 90.00 45.00 13, 14 RECTANGLE

A8 180.00 45.00 15, 16 TRIANGLE

A9 270.00 45.00 17, 18 TRIANGLE

A10 45.00 45.00 19, 20 TRIANGLE

All 135.00 45.00 21, 22 TRIANGLE

A12 225.00 45.00 23, 24 TRIANGLE

A13 315.00 45.00 25, 26 TRIANGLE

(* the bearing of this axis may be assigned any value between 0° and 360°). The cartesian coordinates of the centre points of the faces of this die are substantially as shown in Table 4B. TABLE 4B

FACE COORDINATES (X,Y,Z)

0 0 1

2 0 0 -1

3 1 0 0

4 -1 0 0

5 0 1 0

The faces of the die formed as flats in a generally spherical body and as a result the corners where three faces meet are rounded or bevelled. The nominal dimensions of the sides Of the faces in this embodiment (that is to say the dimensions the sides would have if they were not truncated by the rounding) may have in substance the following relationship:- squares 152x152, triangles 192x192x192 rectangles 164x130. Clearly these nominal dimensions may b varied (and will probably vary in manufacture ) relative t one another without in substance adversely affecting th unbiased characteristics of the die.

The centre points of opposed pairs of faces li substantially on the axis joining them.

As with the dice already described the angle between the faces are rounded. This die may be made b moulding a plastics material in either a pre-formed mould o the desired shape or in a mould producing a regular cuboi and thereafter cutting or grinding the corners of the cuboi to produce the faces.

It will be seen that the faces are arranged suc that no faces of the same regular shape are adjacent one t another and that faces of the same regular shape are oppose to one another - substantially lie on the same axis passin through the die. With this disposition of the faces of thi die each square face is bounded by four rectangular faces each triangular face is bounded by three rectangular face and each rectangular face is bounded by two square faces (o its shorter sides) and two triangular faces (on its longe sides) .

INDUSTRIAL APPLICABILITY

The faces of dice made in accordance with th above described arrangements may be marked with any desire indicia whilst being moulded or by cutting, etching engraving or in any other suitable way after it has bee formed. For example the fourteen faced die may be marked t display a complete suit of cards (including a joker) if th die is to be used (perhaps with others marked with othe cards) in a "card" type game. Similarly two twentysix face dice may be marked with different values and suits of card to provide a complete "deck" of cards (excluding the jokers) Alternatively the twentysix faced dice may simply be marke with _.the numbers 1 to 26 or with the letters of the alphabet to etiable a user to select at random one of the twentysix letters or numbers thereon.

Claims

-17-CLAIMS

1. A die having more than six faces arranged opposed pairs on opposite sides of the die, the faces of e pair of opposed faces being substantially identical and w their centres substantially lying on an axis pass substantially through the centre of the die and wherein faces are formed as flats on a generally spherical body.

2. A die as claimed in claim 1, wherein the s axes are substantially symmetrically spaced one from other in passing substantially through the centre of the d

3. A die as claimed in claim 1, wherein angular spacing of the various axes one from the other passing substantially through the centre of the die in b bearing and elevation is maximised.

4. A die having more than six faces arranged opposed pairs on opposite sides of the die, the faces of e pair of opposed faces being substantially identical and w their centres substantially lying on an axis pass substantially through the centre of the die and wherein said axes are substantially symmetrically spaced one from other in passing substantially through the centre of the d

5. A die having more than six faces arranged opposed pairs on opposite sides of the die, the faces of e pair of opposed faces being substantially identical and w their centres substantially lying on an axis pass substantially through the centre of the die and wherein angular spacing of the various axes one from the other passing substantially through the centre of the die in b bearing and elevation is maximised.

6. A die as claimed in any one of claims 1 to wherein all the faces are substantially similar and are n contiguous.

7. A die as claimed in claim 6, wherein all faces are substantially identical and are circular in plan

8. A die as claimed in any one of claims 1, 2 4, having fourteen substantially identical circular fa arranged in seven substantially diametrically opposed pair the centres of the opposed pairs of faces substantially lyi on substantially equilength axes which have bearing a elevation relative to a reference axis in substance follows:-

AXIS BEARING (°) ELEVATION (°)

1 * 90.00

2 0.00 0.00

3 90.00 0.00

4 45.00 Sin^-1(l/3*)

5 135.00 Sin^-1(l/3²)

6 225.00 Sin~¹(l/3*)

7 315.00 Sin^~1(l/3*)

(* the bearing of this axis may be assigned any value betwe 0° and 360°).

9. A die as claimed in any one of claims 1, and 4, having twentysix substantially identical circul faces arranged in thirteen substantially diametrical opposed pairs, the centres of the opposed pairs of fac substantially lying on substantially equilength axes whi have bearing and elevation relative to a reference axis substance as follows:-

AXIS BEARING (°) ELEVATION (°)

1 * 90.00

2 0.00 0.00

3 90.00 0.00

4 45.00 Sin^"1(1/3*)

5 35.00 Sin^"1(1/3*)

6 25.00 Sin^~1(l/3*)

7 15.00 Sin^"1(1/3²)

8 0.00 45.00

9 90.00 45.00

10 180.00 45.00

11 170.00 45.00

12 145.00 0.00

13 135.00 0.00 -19-

10. A die as claimed in any one of claims 1, 3, and 5, having twentysix substantially identical circular faces arranged in thirteen diametrically opposed pairs, the centres of the opposed pairs of faces substantially lying on substantially equilength axes which have bearing and elevation relative to one another in substance as follows:-

AXIS BEARING (° ) ELEVATION <°>

1 * 90.00

2 0.00 51.04

3 90.00 51.04

4 180.00 51.04

5 270.00 51.04

6 45.00 30.49

7 135.00 30.49

8 225.00 30.49

9 315.00 30.48 10 19.48 0.00 11 70.52 0.00 12 109.48. 0.00 13 160.52 0.00

(* the bearing of this axis may be assigned any value between 0° and 360°).

11. A die as claimed in claim 2, wherein the faces are contiguous and the shape of each face of the die in plan is selected from one of a number of differing substantially regular shapes and the die. is so configured that no two adjacent faces have the same substantially regular shape.

12. A die having more than six faces arranged in opposed pairs on opposite sides of the die, the faces of each pair of opposed faces being substantially identical and with their centres substantially lying on an axis passing substantially through the centre of the die and wherein the faces are contiguous and the shape of each face of the die in plan is selected from one of a number of differing substantially regular shapes and the die is so configured -20-

that no two adjacent faces have the same substantially regular shape.

13. A die as claimed in claim 11 or claim 12, having six substantially identical square faces eight substantially identical equisided triangular faces arranged in pairs on four axes, and twelve substantially identical rectangular faces arranged in pairs on six axes, arranged such that no faces of the same substantially regular shape are adjacent one to another.

14. A die as claimed in claim 13, wherein the bearing and elevation of each axis and the shape of the faces associated therewith is substantially as follows:-

AXIS BEARING ( °) ELEVATION (°) FACES

Al * 90.00 1, 2 SQUARE

A2 0.00 0.00 3, 4 SQUARE

A3 90.00 0.00 5, 6 SQUARE

A4 45.00 0.00 7, 8 RECTANGLE

A5 135.00 0.00 9, 10 RECTANGLE

A6 0.00 45.00 11, 12 RECTANGLE

A7 90.00 45.00 13, 14 RECTANGLE

A8 180.00 45.00 15, 16 TRIANGLE

A9 270.00 45.00 17, 18 TRIANGLE

A10 45.00 45.00 19, 20 TRIANGLE

All 135.00 45.00 21, 22 TRIANGLE

A12 225.00 45.00 23, 24 TRIANGLE

A13 315.00 45.00 25, 26 TRIANGLE

(* the bearing, of this axis may be asεsigned any value between

0 and 360⁽ ^D).

15. A die as claimed in claim 13 or claim 14, wherein the nominal dimensions of the sides of the faces have in substance the following relationship:- squares 152x152, triangles 192x192x192, rectangles 164x130; - the corners where three faces meet being rounded or bevelled.

16. A die as claimed in any one of the preceeding claims when formed by moulding a plastics material.

17. A die as claimed in any one of claims 1 to 15, when formed by machining the faces from a sphere of moulded plastics or other material.

18. A die as claimed in any one of the preceeding claims, wherein the faces of the die are provided with indicia in the form of letters, symbols, or numerals whilst being moulded or by cutting, etching or engraving after the faces have been formed thereon.

19. A die as claimed in claim 1 or claim 4 and substantially as herein described with reference to Figure 2, or 4 of the accompanying drawings.

20. A die as claimed in claim 1 or claim 5 and substantially as herein described with reference to Figure 3 of the accompanying drawings.

21. A die as claimed in claim 1 or claim 12 and substantially as herein described with reference to Figure 5 of the accompanying drawings.