EP0701850A2 - Game comprising pieces carrying the elements of at least one pattern to be assembled with the aim of reproducing said pattern entirely - Google Patents

Abstract

The cube game comprises simple solid cubes (C) which are assembled together to form a solid composite cube (1). The simple cubes have on their side faces elements of different patterns. When the patterns are reconstituted by assembling the simple cubes they present different patterns on the different faces of the composite cube, and thus extend in three dimensions. The faces of the simple cube pieces carry parts of the pattern belonging to at least two different patterns. The pieces allow several assembly configurations enabling reconstituting a number of patterns greater than the number of external faces of the composite solid, by modifying the assembly mode.

Description

Games as defined above are known per se. However, in these known games, the decorative motif, once reconstituted, is two-dimensional, in the sense that it extends in one and the same plane.

This is the case, for example, of games constituted by cubic studs, each face of which carries a pattern element and which allow the reconstruction of six different patterns corresponding to the six faces of each of said cubic studs.

The object of the present invention is to provide a game of this kind which allows the reconstitution of patterns extending in three dimensions, in at least two planes intersecting each other, which, of course, increases the difficulty and interest of such games.

This object is achieved by the means defined in claim 1.

• La fig. 1 représente, vu en perspective, un solide élémentaire cubique dont les faces sont destinées à recevoir des éléments de motifs à reconstituer et dont l'assemblage de plusieurs d'entre eux permet la réalisation de solides composés dont les faces présentent soit un seul et même motif, soit plusieurs motifs, à trois dimensions.
• La fig. 2 représente, vu en perspective, un solide composé de forme cubique réalisé à l'aide de plusieurs solides élémentaires cubiques tels que celui de la fig. 1.
• La fig. 3 représente, vu en perspective, une pièce de base munie d'alvéoles destinés à recevoir et à maintenir en position des solides élémentaires tels que celui de la fig. 1 placés "sur pointe", c'est-à-dire dont un des sommets est tourné vers le bas.
• La fig. 4 représente, vu en perspective, un solide composé réalisé à l'aide de solides élémentaires tels que celui de la fig. 1 disposés "sur pointe".
• La fig. 5 représente une partie d'un flan de papier fort (bristol) ou de carton sur lequel sont imprimés les motifs élémentaires des six faces de cubes tels que celui de la fig. 1 et dans lequel ces faces sont découpées tout en demeurant attenantes, ce qui permet, par pliage, de réaliser des cubes élémentaires.
• La fig. 6 est une vue en perspective d'un solide élémentaire en forme de pyramide à base carrée.
• La fig. 7 est une vue en perspective d'un tétraèdre régulier, c'est-à-dire dont les côtés sont constitués par des triangles équilatéraux.
• La fig. 8 est une vue en perspective d'une pyramide composée à base carrée réalisée à l'aide de pyramides et de tétraèdres élémentaires tels que ceux des figs. 6 et 7, respectivement.
• La fig. 9 est une vue en perspective d'un tétraèdre composé réalisé à l'aide de pyramides et de tétraèdres élémentaires tels que ceux des figs. 6 et 7, respectivement.
• La fig. l0 est une vue en plan d'une partie d'un flan à partir duquel peuvent être réalisés des tétraèdres tels que celui de la fig. 7.
• La fig. 11 est une vue en perspective d'un solide élémentaire en forme d'octaèdre régulier.
• La fig. l2 est une vue en perspective d'un tétraèdre régulier identique à celui de la fig. 7.
• La fig. l3 est une vue en perspective d'une pyramide composée à base carrée réalisée à l'aide d'octaèdres et de tétraèdres tels ceux des figs. 11 et l2, respectivement.
• La fig. l4 est une vue en perspective d'un tétraèdre composé également réalisé à l'aide d'octaèdres et de tétraèdres tels que ceux des figs. ll et l2, respectivement.
• La fig. l5 est une vue en plan d'une partie d'un flan à partir duquel peuvent être réalisés des octaèdres tels que celui de la fig. ll.
• La fig. l6 est une vue en perspective d'un solide élémentaire en forme de dodécaèdre rhombique.
• La fig. l7 est une vue en perspective d'un solide composé réalisé à l'aide de dodécaèdres rhombiques tels que celui de la fig. l6.
• La fig. l8 est une vue en perspective d'un autre solide composé réalisé à l'aide de dodécaèdres rhombiques tels que celui de la fig. l6.
• La fig. l9 représente le dodécaèdre rhombique de la fig. l6 développé à plat, et
• La fig. 20 est une vue en perspective, éclatée, d'un dodécaèdre rhombique tel que celui de la fig. l6 formé de quatre hexaèdres rhombiques.

The drawing represents, by way of example, several embodiments of the subject of the invention.

• Fig. 1 represents, seen in perspective, an elementary cubic solid whose faces are intended to receive elements of patterns to be reconstituted and whose assembly of several of them allows the production of solid compounds whose faces have either one and the same pattern, or several patterns, in three dimensions.
• Fig. 2 represents, seen in perspective, a solid composed of cubic form produced using several elementary cubic solids such as that of FIG. 1.
• Fig. 3 shows, seen in perspective, a basic part provided with cells intended to receive and hold in position elementary solids such as that of FIG. 1 placed "on point", that is to say one of the vertices facing down.
• Fig. 4 shows, seen in perspective, a solid compound made using elementary solids such as that of FIG. 1 arranged "on point".
• Fig. 5 shows part of a blank of strong paper (cardboard) or cardboard on which are printed the elementary patterns of the six faces of cubes such as that of FIG. 1 and in which these faces are cut while remaining adjoining, which allows, by folding, to produce elementary cubes.
• Fig. 6 is a perspective view of an elementary solid in the shape of a square-based pyramid.
• Fig. 7 is a perspective view of a regular tetrahedron, that is to say the sides of which are constituted by equilateral triangles.
• Fig. 8 is a perspective view of a composite pyramid with a square base produced using pyramids and elementary tetrahedra such as those of FIGS. 6 and 7, respectively.
• Fig. 9 is a perspective view of a compound tetrahedron produced using pyramids and elementary tetrahedra such as those of FIGS. 6 and 7, respectively.
• Fig. 10 is a plan view of part of a blank from which tetrahedra such as that of FIG. 7.
• Fig. 11 is a perspective view of an elementary solid in the form of a regular octahedron.
• Fig. 12 is a perspective view of a regular tetrahedron identical to that of FIG. 7.
• Fig. l3 is a perspective view of a square pyramid composed using octahedra and tetrahedra such as those in figs. 11 and 12, respectively.
• Fig. 14 is a perspective view of a compound tetrahedron also produced using octahedra and tetrahedrons such as those of FIGS. ll and l2, respectively.
• Fig. 15 is a plan view of part of a blank from which octahedra such as that of FIG. ll.
• Fig. 16 is a perspective view of an elementary solid in the form of a rhombic dodecahedron.
• Fig. 17 is a perspective view of a solid compound produced using rhombic dodecahedrons such as that of FIG. l6.
• Fig. 18 is a perspective view of another solid compound made using rhombic dodecahedrons such as that of FIG. l6.
• Fig. l9 represents the rhombic dodecahedron of fig. l6 developed flat, and
• Fig. 20 is an exploded perspective view of a rhombic dodecahedron such as that of FIG. l6 formed by four rhombic hexahedra.

In the description which follows, and this is valid for all the embodiments, the elementary patterns can consist either of plain surfaces, of different colors, the difference of the colors distinguishing from each other the reconstituted patterns, or by symbols, oriented or not, either by fractions of images, drawings, engravings, photographs or others, the assembly of which makes it possible to reconstruct the image as a whole. These elementary patterns have not been shown in the drawings.

Before entering into the actual description of the particular embodiments of the invention, it may not be unnecessary to engage in some general considerations:
It is admitted that the compound polyhedra obtained by the assembly of elementary polyhedra can not only have planar faces, but also embossed faces having hollow parts and projecting parts.

The solid compounds may be "multiple", that is to say have several vertices facing in the same direction. In this case, they can be broken down into several solid compounds of simpler shape such as pyramids, tetrahedra, cubes or others.

An important notion of the invention is the notion of configuration S. S expresses the number of possible arrangements of elementary solids leading to a compound polyhedron, most generally of the same shape and the same dimension, the appearance of which on the surface is different from the fact that the facets of the elementary solids which are apparent are not the same for these different compound solids. In in other words, the number S of the configurations represents the number of reconstitutable patterns, which is greater than the number of external faces of the compound solids.

It should be noted that the different arrangements of elementary solids can also lead to compound polyhedra differing from each other, or even of different dimensions.

The compound solids can be formed from identical elementary solids or from groups of complementary elementary solids, it being understood that, in a group, the elementary solids have edges of the same length.

All the elementary solids (monolithic or themselves composed) are regular polyhedra with F side faces or identical facets.

The number of possible configurations S in a stack of elementary solids is determined as follows:
The stack has successive layers of elementary solids containing K (i) elementary solids each, i being the layer index varying from 1 to N, and N being the total number of layers of the stack.

A(i,V) étant le nombre de solides présentant V facettes visibles dans la couche i, V étant évidemment inférieur à F.The K (i) elementary solids of each layer can be grouped as follows: $$\text{K (i) = A (i, 0) + A (i, 1) + A (i, 2) + A (i, 3) + A (i, 4) + ... + A (i, V)}$$

A (i, V) being the number of solids having V visible facets in the layer i, V being obviously less than F.

et que le nombre de facettes total à disposition est de $${\text{FΣ}}^{\text{N}}{}_{\text{i=1}}{\text{Σ}}^{\text{V}}{}_{\text{v=1}}\text{A(i,V)}$$

   D'une façon générale, le nombre de configurations S s'écrit comme suit : $${\text{SΣ}}^{\text{N}}{}_{\text{i=1}}{\text{Σ}}^{\text{V}}{}_{\text{v}}{\text{A(i,V)V≦FΣ}}^{\text{N}}{}_{\text{i=1}}{\text{Σ}}^{\text{V}}{}_{\text{v=1}}{\text{A(i,V)=FΣ}}^{\text{N}}{}_{\text{i=1}}\text{K(i)}$$

d'où $${\text{S≦FΣ}}^{\text{N}}{}_{\text{i=1}}{\text{K(i)/Σ}}^{\text{N}}{}_{\text{i=1}}{\text{Σ}}^{\text{V}}{}_{\text{v=1}}\text{A(i,V)V}$$

   Il est à remarquer que les F facettes de l'ensemble des solides élémentaires constituant un solide composé ne sont pas toutes obligatoirement utilisables. C'est ainsi que dans le cas d'une pyramide à base carrée formée de dodécaèdres rhombiques, le dodécaèdre rhombique du sommet, qui présente A(i=1,V=8)=8 facettes apparentes, n'est pas utilisable comme dodécaèdre rhombique à trois facettes à sommet obtus (à l09,47°), puisque les quatre facettes cachées forment une pyramide aiguë à 70,53°.As a result, the total number of visible facets is $${\text{Σ}}^{\text{NOT}}{}_{\text{i = 1}}{\text{Σ}}^{\text{V}}{}_{\text{v = 1}}\text{A (i, V) V}$$

and that the total number of facets available is $${\text{FΣ}}^{\text{NOT}}{}_{\text{i = 1}}{\text{Σ}}^{\text{V}}{}_{\text{v = 1}}\text{A (i, V)}$$

In general, the number of configurations S is written as follows: $${\text{SΣ}}^{\text{NOT}}{}_{\text{i = 1}}{\text{Σ}}^{\text{V}}{}_{\text{v}}{\text{A (i, V) V ≦ FΣ}}^{\text{NOT}}{}_{\text{i = 1}}{\text{Σ}}^{\text{V}}{}_{\text{v = 1}}{\text{A (i, V) = FΣ}}^{\text{NOT}}{}_{\text{i = 1}}\text{K (i)}$$

from where $${\text{S ≦ FΣ}}^{\text{NOT}}{}_{\text{i = 1}}{\text{K (i) / Σ}}^{\text{NOT}}{}_{\text{i = 1}}{\text{Σ}}^{\text{V}}{}_{\text{v = 1}}\text{A (i, V) V}$$

It should be noted that the F facets of the set of elementary solids constituting a compound solid are not necessarily usable. Thus in the case of a pyramid with a square base formed by rhombic dodecahedra, the rhombic dodecahedron at the top, which has A (i = 1, V = 8) = 8 apparent facets, cannot be used as a dodecahedron rhombic with three facets with obtuse apex (at l09.47 °), since the four hidden facets form an acute pyramid at 70.53 °.

The object set of the first embodiment of the invention comprises elementary solids C, in the form of cubes, one of which has been shown in FIG. 1.

Each side face or facet of these cubes carries elements of patterns to be reconstructed. All or part of all the cubes in the game may have the same set of elementary patterns or even elementary patterns differing from one cube to another.

These cubes C allow the production of compound solids, for example the cubic compound solid 1 shown in FIG. 2, formed by the superposition of four layers of cubes C, each layer being itself formed by the juxtaposition of four strips of cubes each comprising four cubes C.

Thus, the compound cube 1 is formed of sixty-four cubes C, each of its faces being itself formed of sixteen square facets formed by the sides of the elementary cubes C.

In general, the number of facets available is 6N³ while the number of facets per face of the compound cube is N², N expressing the number of layers of elementary cubes in the compound cube.

If we consider that five faces of the compound cube are visible, the sixth being in contact with the surface on which the said compound cube rests, we will have 5N² facets which will be used by configuration S, hence the relation 5N²S ≦ 6N³, or again S ≦ 6N / 5.

Thus, for example, still in the case of the cube, for N = 3, the compound cube being formed of twenty-seven elementary cubes, there will be three possible configurations of five compound images appearing simultaneously and, for N = 5 (one hundred and twenty -five elementary cubes), there will be six possible configurations of five composite images appearing simultaneously.

Fig. 3 shows a triangular support 2, for example made of plastic, having four rows of cells 3, of trihedral shape, intended to each receive a cube such as the cube C in FIG. 1 placed "on point", that is to say having one of its vertices facing downwards.

It is thus possible, using such a support, to produce a stack of cubes such as the stack 4 of FIG. 4 where all the cubes C are placed "on point".

In such a stack, the number of elementary cubes will be N (N + 1) (N + 2) / 6.

The number of available facets is N (N + 1) (N + 2).

The number of facets per plane face is N (N + 1) / 2.

facettes utilisées par configuration, d'où la relation $\text{S3N(N+l)/2≦N(N+l)(N+2}$

), ou encore $\text{S≦2(N+2)/3}$

.If we consider three visible faces of such a "pyramidal" stacking of cubes, we therefore have $\text{3N (N + l) / 2}$

facets used by configuration, hence the relationship $\text{S3N (N + l) / 2 ≦ N (N + l) (N + 2}$

), or $\text{S ≦ 2 (N + 2) / 3}$

.

Thus, for example, for N = 4, the "pyramidal" stacking being formed of twenty elementary cubes, there will be four possible configurations of three composite images appearing simultaneously and, for N = 7 (eighty-four elementary cubes), there there will be six possible configurations of three composite images appearing simultaneously.

Incidentally, in the case of cubes posed on point, described here, we can hide the half-squares of the visible faces of the cubes of the base, which will give an image without niche on its base, that is to say at base straight. This will have the consequence, in the aforementioned case where N = 4, leading to a tetrahedral pyramid of twenty cubes, that we can achieve five configurations instead of four by hiding the four basic half-squares of each composite image.

Still in the case of "pyramidal" stacking of the kind of that of FIG. 4, the top layer, called layer l, contains a cube, the next layer, called layer 2, contains three cubes, layer 3 contains six cubes, layer 4 contains ten cubes, etc.

cubes.Layer N contains Σn from l to $\text{N = N (N + l) / 2}$

cubes.

de l à N de $\text{n(n+l)/2 = N(N+l)(N+2)/6}$

puisque la Σn² de l à N vaut N(N+l)(2N+l)/6.As a result, the total number of cubes is $\text{ΣΣn = Σ}$

from l to N from $\text{n (n + l) / 2 = N (N + l) (N + 2) / 6}$

since the Σn² from l to N is worth N (N + l) (2N + l) / 6.

.The total number of facets (square) per flat face of the "pyramidal" stack is l + 2 + 3 + 4 + ... + N = Σn from l to $\text{N = N (N + l) / 2}$

.

It should be noted that, in the case shown in figs. 3 and 4, the different layers can be either parallel to the planes of the outer faces of the completed tetrahedral pyramid, or horizontal.

The elementary solids may be solid, made of wood or plastic for example, or even be hollow, being produced from a folded sheet material. The latter solution has the advantage that the elementary patterns can be printed on the sheet material.

This is how fig. 5 represents a blank 5 of strong paper (cardboard) or cardboard, on which a series of drawings 6 has been printed, each representing a cube C in the developed state, flat. The square has 6 parts of these drawings show the faces of the cube. The projecting parts 6b are folded inwardly cubes once they mounted; as for the protruding parts 6 c , they constitute assembly tongues. The elementary patterns are printed on the blank 5, each drawing 6 receiving the desired elementary patterns, identical for some of these drawings, different for others. The drawings 6 are then detached, by striking with a cutter, from the blank 5.

It should be noted that the game can be delivered with the cubes being assembled, as also the cubes in the developed state, their assembly being then ensured by the users.

Fig. 6 represents an elementary pyramid with a square base P, and FIG. 7 a regular elementary tetrahedron T. Using solids of these two types, we can make either a square pyramid compound 7 (fig. 8), or a regular compound tetrahedron 8 (fig. 9).

In the case of the compound pyramid 7, the first layer of elementary solids, in this case the top layer, is constituted by a pyramid P. All the elementary angle solids are also constituted by pyramids P. The elementary tetrahedra T inserted between the elementary pyramids P are all placed "on edge".

In general, in the case of the composite pyramid produced using elementary pyramids P on side a and elementary tetrahedra T on side a , the length of the side of the composite pyramid will be Na.

The number of elementary pyramids P is N (2N² + 1) / 3, and the number of elementary tetrahedrons T is 2N (N²-1) / 3.

The number of the triangular facets of the set of elementary pyramids is 4N (2N² + 1) / 3, while the number of the triangular facets of the set of elementary tetrahedrons is 8N (N²-1) / 3.

The number of triangular facets of elementary pyramids visible on one face of the composite pyramid is N (N + 1) / 2, while the number of triangular facets of elementary tetrahedrons visible on one face of the composite pyramid is N (N -1) / 2.

, d'où il résulte que ${\text{Sp≦2(2N}}^{\text{2}}\text{+1)/3(N+1)}$

.The number of possible configurations counted on the elementary pyramids, designated by Sp, is ${\text{2N (N + 1) Sp ≦ 4N (<figure-callout id="2" label="2N" filenames="imgf0001.png" state="{{state}}">2N</figure-callout>}}^{\text{2}}\text{+1) / 3}$

, from which it follows that ${\text{Sp ≦ 2 (<figure-callout id="2" label="2N" filenames="imgf0001.png" state="{{state}}">2N</figure-callout>}}^{\text{2}}\text{+1) / 3 (N + 1)}$

.

, d'où $\text{St≦4(N+1)/3}$

.As for the number of possible configurations counted on the elementary tetrahedra, designated by St, it is ${\text{2N (N-1) St ≦ 8N (N}}^{\text{2}}\text{-1) / 3}$

, from where $\text{St ≦ 4 (N + 1) / 3}$

.

We see that Sp <St for all N. Consequently, the number of configurations is determined by the number of elementary pyramids if we consider a planar pattern on each face of the composite pyramid.

For example, in the case of a three-layer composite pyramid (fig. 8), it is easy to verify that three configurations are possible. For N layers, Sp ≧ N.

In the case of the compound tetrahedron 8 (fig. 9), the first layer of elementary solids, in this case the top layer, is constituted by a tetrahedron T. All the elementary angle solids are also constituted by tetrahedra T. The elementary pyramids P interposed between the elementary tetrahedra T are all placed on one of their triangular faces.

In general, in the case of compound tetrahedra produced using elementary pyramids P on side a and elementary tetrahedra T on side a , the length of the side of the compound tetrahedra is equal to Na.

The number of elementary pyramids is N (N²-1) / 3, and the number of elementary tetrahedrons is N (N² + 2) / 3.

The number of triangular facets of the set of elementary pyramids is 4N (N²-1) / 3, while the number of triangular facets of the set of elementary tetrahedrons is 4N (N² + 2) / 3.

The number of the triangular facets of the elementary pyramids visible on one face of the compound tetrahedron is N (N-1) / 2, while the number of the triangular facets of the elementary tetrahedrons visible on one face of the compound tetrahedron is N (N + 1 ) / 2.

, d'où il résulte que $\text{Sp≦8(N+1)/9}$

.The number of possible configurations counted on the elementary pyramids, designated by Sp, is ${\text{Sp3N (N-1) / 2 ≦ 4N (N}}^{\text{2}}\text{-1) / 3}$

, from which it follows that $\text{Sp ≦ 8 (N + 1) / 9}$

.

, d'où ${\text{St≦8(N}}^{\text{2}}\text{+2)/9(N+1)}$

.As for the number of possible configurations counted on the elementary tetrahedra, designated by St, it is ${\text{St3N (N + 1) / 2 ≦ 4N (<figure-callout id="2" label="N" filenames="imgf0001.png" state="{{state}}">N</figure-callout>}}^{\text{2}}\text{+2) / 3}$

, from where ${\text{St ≦ 8 (<figure-callout id="2" label="N" filenames="imgf0001.png" state="{{state}}">N</figure-callout>}}^{\text{2}}\text{+2) / 9 (N + 1)}$

.

We see that St <Sp for all N. Consequently, the number of configurations is determined by the number of tetrahedra if we consider a planar pattern on each face of the compound tetrahedron.

For example, in the case of a three-layer compound tetrahedron (fig. 9), it is easy to verify that two configurations are possible. For N layers, St <N.

Fig. 10 represents a blank 9 on which has been printed a series of drawings 10 each representing a tetrahedron T in the developed state, flat. The triangular parts 10 a of these drawings represent the faces of the tetrahedron. The overflowing parts 10 b constitute assembly tabs. The elementary patterns are printed on the blank 9, each drawing 10 receiving the desired elementary patterns, identical for some of these drawings, different for others. The drawings 10 are then detached, by knocking with the cutter, from the blank 9.

Fig. 11 represents a regular elementary octahedron 0 and FIG. l2 a regular elementary tetrahedron T identical to that of fig. 7. Using solids of these two types, you can make either a square-based composite pyramid 11 (fig. 13), or a regular compound tetrahedron 12. The solid 11 of FIG. 13 differs from the solid 7 of FIG. 8 by the fact that its base is not flat but has an embossed appearance. It is therefore necessary, to produce this solid 11, to have a cell support similar to the support 2 in FIG. 3, but whose alveoli will have the shape of a half-octahedron 0.

As for the solid 12 of FIG. l4, it is identical to the solid 8 of FIG. 9, the pyramids composing it being all placed square base against square base, two by two, thus forming octahedra. The base of the tetrahedron l2 is plane.

In the case of the composite pyramid 11 (fig. 13), the first layer of elementary solids, in this case the top layer, is constituted by an octahedron 0. All the elementary solids of angle are also constituted by octahedra 0 The elementary tetrahedra T are interspersed between the elementary octahedra.

In general, in the case of composite pyramids produced using regular elementary octahedra 0 on side a and elementary tetrahedra T on side a , the length of the side of the compound pyramids is Na.

The number of elementary octahedra 0 is N (N + 1) (2N + 1) / 6 and the number of elementary tetrahedrons T is 2N (N²-1) / 3.

The number of the triangular facets of the set of elementary octahedra is 4N (N + 1) (2N + 1) / 3, while the number of the triangular facets of the set of elementary tetrahedra is 8N (N²-1) / 3.

The number of the triangular facets of the elementary octahedra visible on one face of the composite pyramid is N (N + 1) / 2, while the number of the triangular facets of the elementary tetrahedrons visible on one face of the composite pyramid is N (N -1) / 2.

, d'où il résulte que $\text{So≦2(2N+1)/3}$

.The number of possible configurations counted on the elementary octahedra, designated by So, is $\text{2N (N + 1) So ≦ 8N (N + 1) (2N + 1) / 6}$

, from which it follows that $\text{So ≦ 2 (2N + 1) / 3}$

.

, d'où $\text{St≦4(N+1)/3}$

.As for the number of possible configurations counted on the elementary tetrahedra, designated by St, it is ${\text{2N (N-1) St ≦ 8N (N}}^{\text{2}}\text{-1) / 3}$

, from where $\text{St ≦ 4 (N + 1) / 3}$

.

We see that St> So for all N. Consequently, the number of configurations is determined by the number of elementary octahedra if we consider a planar pattern on each face of the composite pyramid.

For example, in the case of a three-layer composite pyramid (fig. 13), it is easy to verify that four configurations are possible. For N layers, moreover St ≧ N.

In the case of the compound tetrahedron 12 (fig. 14), the first layer of elementary solids, in this case the top layer, is constituted by a tetrahedron T. All the elementary angle solids are also constituted by tetrahedrons T. The elementary octahedra P are interspersed between the elementary tetrahedra T.

In general, in the case of compound tetrahedra produced using regular elementary octahedra 0 on side a and elementary tetrahedra T on side a , the side length of the compound tetrahedra is equal to Na.

The number of elementary octahedra is N (N²-1) / 6, and the number of elementary tetrahedrons is N (N² + 2) / 3.

The number of the triangular facets of the set of elementary octahedra is 8N (N²-1) / 6, that is 4N (N²-1) / 3, while the number of the triangular facets of the set of elementary tetrahedrons is 4N (N² + 2) / 3.

The number of the triangular facets of the elementary octahedra visible on one face of the compound tetrahedron is N (N-1) / 2, while the number of the triangular facets of the elementary tetrahedrons visible on one face of the compound tetrahedron is N (N + 1 ) / 2.

, d'où il résulte que $\text{So≦8(N+1)/9}$

.The number of possible configurations counted on the elementary octahedra, designated by So, is ${\text{So3N (N-1) / 2 ≦ 4N (N}}^{\text{2}}\text{-1) / 3}$

, from which it follows that $\text{So ≦ 8 (N + 1) / 9}$

.

, d'où ${\text{St≦8(N}}^{\text{2}}\text{+2)/9(N+1)}$

.As for the number of possible configurations counted on the elementary tetrahedra, designated by St, it is ${\text{St3N (N + 1) / 2 ≦ 4N (<figure-callout id="2" label="N" filenames="imgf0001.png" state="{{state}}">N</figure-callout>}}^{\text{2}}\text{+2) / 3}$

, from where ${\text{St ≦ 8 (<figure-callout id="2" label="N" filenames="imgf0001.png" state="{{state}}">N</figure-callout>}}^{\text{2}}\text{+2) / 9 (N + 1)}$

.

We see that St <So for all N. Consequently, the number of configurations is determined by the number of tetrahedra if we consider a planar pattern on each face of the compound tetrahedron.

For example, in the case of a three-layer compound tetrahedron (fig. 14), it is easy to verify that two configurations are possible. For N layers, St <N.

Fig. 15 represents a blank 13 on which a series of drawings 14 has been printed, each representing an octahedron 0 in the developed state, flat. The triangular portions l4 has these drawings depict the faces of the octahedron. The overhanging portions 14 b will be folded inwards of the octahedra once they have been mounted; as for the protruding parts 14 c , they constitute assembly tongues. The elementary patterns are printed on the blank 13, each drawing 14 receiving the desired elementary patterns, identical for some of these drawings, different for others. The drawings 14 are then detached, by striking with a cutter, from the blank 13.

Fig. l6 represents an elementary rhombic dodecahedron DR whose lateral faces or facets are formed of diamonds. The acute angles of these diamonds are 70.53 °, and the obtuse angles of 109.47 °.

Fig. l7 represents a solid compound l5, of general tetrahedral shape, formed by the superposition of three layers of the rhombic dodecahedrons DR. In this solid l5, the rhombic dodecahedra DR have one of their obtuse vertices turned downwards.

As for the solid composed of fig. l8, designated by l6, it is also formed by the superposition of three layers of rhombic dodecahedrons DR. However, in the case of this solid compound l6, the rhombic dodecahedrons DR have one of their acute vertices facing downwards.

It should be noted that the use of a cell support of the kind of support 2 of FIG. 3 is necessary for the production of the solid compounds 15 and 16, different, by the shape of its cells, for each of them.

For the determination of the number of apparent facets of the solid compound l5 (fig. L7), it must be considered that the regular dodecahedron DR summit has nine apparent facets.

The three "edges" of the solid body 15 comprise a total of 3 (N-1) rhombic dodecahedrons with six apparent facets each.

The other apparent rhombic DR dodecahedrons are 3 (N-2) (N-1) / 2 and each have three visible facets.

As for the other completely hidden rhombic dodecahedra, constituting the solid l5, they are (N-3) (N-2) (N-1) / 6.

The total number of rhombic dodecahedra being N (N + 1) (N + 2) / 6, there are altogether 2N (N + 1) (N + 2) facets.

,
soit $\text{S≦4(N+2)/9}$

   Pour la détermination du nombre des facettes apparentes du solide composé l6, il faut considérer que le dodécaèdre régulier sommital présente huit faces apparentes. On remarque, incidemment, que ses quatre facettes cachées ne peuvent pas servir à former des configurations supplémentaires du fait que l'on aurait besoin, pour cela, de trois facettes disposées différemment.The number of visible facets being 9N (N + 1) / 2, we quickly deduce that
$\text{S ≦ 2N (N + 1) (N + 2) / (9N (N + 1) / 2)}$

,
is $\text{S ≦ 4 (N + 2) / 9}$

For the determination of the number of the apparent facets of the solid compound l6, it must be considered that the regular summit dodecahedron has eight apparent faces. We note, incidentally, that its four hidden facets cannot be used to form additional configurations because we would need, for this, three facets arranged differently.

The four edges of the solid body 16 comprise 4 (N-1) rhombic dodecahedrons with five apparent facets each. It should be noted, incidentally, that the seven hidden faces of these rhombic edge dodecahedrons can only be used in two ways, either by using five facets, or by using three other times only once.

The other rhombic dodecahedrons DR apparent from solid 16 are 2 (N-1) (N-2) each having three apparent facets.

As for the other rhombic dodecahedra DR constituting the solid 16, they are (N-2) (N-1) (2N-3) / 6, none of them having an apparent facet.

These dodecahedra which do not have visible facets, and which are therefore entirely hidden, must be in sufficient number to allow the various desired configurations, the number of which is designated by S.

2) S = 3 ou 4
   On a la condition $$\text{(N-2)(N-1)(2N-3)/6-4(N-1)(S-2)/2-S+1≧0,}$$

d'où $$\text{S≦}\frac{\text{(N-2)(N-1)(2N-3)/6 + 4N-3}}{\text{2N-1}}$$

3) S > 4
   On a la condition $$\begin{array}{c}\begin{array}{c}\text{(N-2)(N-1)(2N-3)/6-S+1-4(N-1)(S-2)/2-2(N-1)(N-2)(S-4)/4≧0}\\ \begin{array}{}\text{(3)}\text{d'où S≦(2N+1)/3}\end{array}\end{array}\end{array}$$

Exemples avec (1), (2) et (3) N S S max l ≦ 1 1 2 ≦ 1 1 3 ≦ 2 2 4 ≦ l8/7 2 5 ≦ 31/9 3 6 ≦ 51/11 4 7 5 5 8 ≦ l7/3 5 9 ≦ 19/3 6 10 ≦ 7 7 We must distinguish 3 cases:
1) S = 1 or 2
In this case, we have the condition $$\begin{array}{c}\begin{array}{c}\text{(N-2) (N-1) (2N-3) / 6-S + 1 ≧ 0,}\\ \begin{array}{}\text{(1)}{\text{hence S ≦ N (2N}}^{\text{2}}\text{-9N + 13) / 6.}\end{array}\end{array}\end{array}$$

2) S = 3 or 4
We have the condition $$\text{(N-2) (N-1) (2N-3) / 6-4 (N-1) (S-2) / 2-S + 1 ≧ 0,}$$

from where $$\text{S ≦}\frac{\text{(N-2) (N-1) (2N-3) / 6 + 4N-3}}{\text{2N-1}}$$

3) S> 4
We have the condition $$\begin{array}{c}\begin{array}{c}\text{(N-2) (N-1) (2N-3) / 6-S + 1-4 (N-1) (S-2) / 2-2 (N-1) (N-2) (S- 4) / 4 ≧ 0}\\ \begin{array}{}\text{(3)}\text{hence S ≦ (2N + 1) / 3}\end{array}\end{array}\end{array}$$

Examples with (1), (2) and (3) NOT S S max l ≦ 1 1 2 ≦ 1 1 3 ≦ 2 2 4 ≦ l8 / 7 2 5 ≦ 9/31 3 6 ≦ 51/11 4 7 5 5 8 ≦ l7 / 3 5 9 ≦ 19/3 6 10 ≦ 7 7

Fig. l9 represents a rhombic dodecahedron such as that of FIG. l6, in the developed state, flat, designated by l7. The design of this developed rhombic dodecahedron includes twelve diamonds l7 a corresponding to the twelve facets of the rhombic dodecahedron, parts projections 17 b intended to be folded inwards when the rhombic dodecahedron is mounted, and protruding parts 17 c constituting assembly tongues.

The developed rhombic dodecahedron l7 can be produced, as indicated for the previous embodiments, at the same time as several others from a blank which will be printed and then cut out.

The rhombic dodecahedron DR of fig. l6 may itself be composed and not be monolithic, as shown in fig. 20. In the example illustrated by this figure, where a rhombic dodecahedron DR is exploded, this rhombic dodecahedron is formed by four regular rhombic hexahedra HR. The rhombic dodecahedron DR compound will be used like the rhombic dodecahedron DR monolithic of fig. l6.

As a variant, provision may be made for this, especially when the elementary parts are cubes, but not exclusively, where said parts will present, on their faces, in addition to the elementary patterns serving for the reconstitution of patterns on the external faces of the compound solid body, additional elementary patterns which will appear as the solid body compound is reconstituted, after each layer of the elementary parts making it up.

These compound patterns will therefore appear momentarily inside the solid body being reconstituted and will be hidden as soon as the next layer of elementary parts has been put in place.

The additional patterns appearing on part of the faces of the elementary pieces can be determined in such a way that the successive interior compound patterns are in thematic relationship with one another. They can for example constitute successive illustrations of the different phases of a story.

Each layer of elementary parts may not form a compound pattern only, but, simultaneously, several different patterns. These will then be visible successively by observing the solid body composed in the process of reconstitution perpendicular to the direction of each of the sides of its base successively and by placing themselves in such a way that the successive rows of elementary pieces in the layer which has just to be implemented follow each other continuously. Thus, observed in this way, the elementary pieces will give the impression of a compound motif seen in perspective. Each layer of elementary parts can thus provide as many compound patterns as the base of the solid body will have sides, three for a triangular base (tetrahedron), four for a square base (pyramid), etc.

Claims (22)

Game comprising pieces carrying elements of at least one pattern, which are intended to be assembled for the purpose of reconstituting said pattern as a whole, characterized in that said pieces consist of elementary solids of polyhedral shape, the assembly of which leads to the production of composite solid bodies, also polyhedral, at least part of the faces of which presents the pattern to be reconstructed, the reconstituted pattern, which extends in three dimensions, being located in at least two different planes, intersecting the 'one another. Game according to claim 1, characterized in that at least part of the faces of said parts carry pattern elements belonging to at least two different patterns. Game according to claim 2, characterized in that said parts are arranged so as to allow several assembly configurations making it possible to reconstitute a number of patterns greater than the number of external faces of the solid compound, by modifying the method of assembly. Game according to claim 2, characterized in that the said pieces have, on a part of their faces, additional elementary patterns appearing, as and when the said solid compound is reconstituted, inside the latter and forming together at least one reconstructed motif. Game according to claim 4, characterized in that the said additional elementary patterns are arranged so that the succession in which they appear as the reconstitution of the solid compound constitutes successive reconstituted patterns which have a thematic relationship with each other others. Game according to Claim 4, characterized in that the said additional elementary patterns are arranged so as to allow the simultaneous production of several different reconstituted patterns, appearing one or the other depending on the incidence according to which the solid solid compound being reconstituted is observed. Game according to claim 1, characterized in that said elementary solids have identical square faces. Game according to claim 4, characterized in that said elementary solids are all constituted by cubes. Game according to claim 8, characterized in that a part of the square faces of said cubes are divided in two by one of the diagonals to allow a composite image inscribed in an isorectangle triangle whose base is rectilinear. Game according to Claim 1, characterized in that the said elementary solids have identical faces in the form of diamonds. Game according to Claim 10, characterized in that the said elementary solids are at least partly constituted by rhombic dodecahedrons. Game according to claim 10, characterized in that the said elementary solids are at least partly constituted by rhombic hexahedra. Game according to claims 11 and 12, characterized in that said elementary solids are of two different, complementary types, being made up of rhombic dodecahedra and rhombic hexahedra. Game according to claim 1, characterized in that said elementary solids have identical faces in the form of equilateral triangles. Game according to claim 14, characterized in that said elementary solids are of two different, complementary types, constituted by regular octahedra and regular tetrahedra, with the same edge length. Game according to claim 14, characterized in that said elementary solids are of two different, complementary types, constituted by regular pyramids with a square base and regular tetrahedrons, with the same edge length. Game according to claim l, characterized in that it comprises a basic part having cells intended to receive a part of the elementary solids, these cells being shaped so that the elementary solids which rest there have one of their vertices directed down. Game according to Claim 1, characterized in that the said elementary solids are in sufficient number to allow the production of several compound solids which differ from one another. Game according to claim 1, characterized in that at least part of the elementary solids are formed of plates assembled to each other along their edges, said solids thus being hollow. Game according to claim l9, characterized in that at least part of the elementary solids consist of an assembly of solid and hollow solids. Game according to claim l9, characterized in that the said plates are arranged so as to be all adjacent to one another by one of their edges, which makes it possible to develop them in a plane and to print the said on them pattern elements. Game according to any combination of claims 1 to 2l.

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