WO2016096132A1

WO2016096132A1 - Basic bricks

Info

Publication number: WO2016096132A1
Application number: PCT/EP2015/002535
Authority: WO
Inventors: Pierre Moreau
Original assignee: Moreau, Olivier
Priority date: 2014-12-18
Filing date: 2015-12-16
Publication date: 2016-06-23
Also published as: FR3030855A1; FR3030855B1

Abstract

Set of solid or hollow bricks, which are truncated (9) in an identical manner at each pole (O-O') at truncation angles that originate from the breakdown of the five regular polyhedrons and the two rhombohedral polyhedrons having twelve and thirty lozenges into pyramidal sections. They differ by way of their truncation angles, of which there are seven. They are joined together, in groups, by their truncated faces, or by their decapitated poles and the latter are linked to other groups by sharing one or more common bricks, thereby creating isotropic media. In the case of hollow bricks, the ducts thus created allow the circulation of fluids by virtue of the interconnections which allow good distribution. The bricks originating from truncated antiprisms can serve as models for jewellery cutting or for the creation of moulds, and the various assemblies resemble crystal lattices. These bricks can be of use for heat exchangers, the field of research into energy storage and/or molecular biology. The fact that identical parts are used for each assembly demonstrates its simplicity.

Description

BASIC BRICKS.

In the patent FR 2 977 967, we presented new polyhedral forms under the title:

Atoms and Molecules for Constructions.

From this study, our reflection has evolved towards the framework of these polyhedra and this has allowed us to develop a series of truncated bricks, suitable for the realization of the assemblies described in the patent mentioned, but also in another form as well as the creation of other montages.

For the understanding, we take again the description of this patent of origin, very slightly modified in the form but keeping the same appellations and notations, to present you the innovations brought.

Description Priorities.

It is a family of polyhedra where each of them is composed of a right antiprism (APn) forming the nucleus, surrounded by a toric ring of tetrahedral appearance (Tn).

The assembly of these two pieces, antiprism and torus, forms the polyhedron to which we give the name of atom (An). The index (n) represents the number of sides of the regular polygons constituting the two poles of the right antiprisms at the origin of these polyhedra and determines its serial number.

The length of the sides of the polar polygons of the antiprisms is taken as unit, equal to 1.

The crown, of toric appearance, is formed of 6n equilateral triangles of edges equal to 1. This ring has, at its outer periphery 2n vertices (5), arranged in sawtooth, fig10, symbolizing the equatorial belt and dividing the atom in two symmetrical hemispheres.

Each vertex of the toric crown is associated with five triangles of the torus constructed from the connection to the three vertices of one of the 2n triangles of the antiprism and to the two contiguous vertices on the crown.

This way of constructing this atom determines, in a unique way:

the height of the antiprism, the different angles of the faces between them and with the poles of the antiprism, the radius of the cylinder circumscribed to the toric crown.

In this atom, each of the faces of one hemisphere finds, on the other hemisphere, a face which is parallel to it.

The icosahedron is the only known polyhedron corresponding to this description because it can be decomposed into an antiprism of order 3 (AP3), fig 1, and a ring crown to six vertices (T3), fig.2.

We thus have a new description of the icosahedron, and it is the first polyhedron (A3), fig.3, of this series, going to infinity.

In addition to the atom (A3), which is the icosahedron, only the atom (A), fIg.7, is convex. All the following ones to infinity are concave.

This family of atoms meets common criteria:

the equatorial radius, denoted R, distance of the points (5) to the axis of symmetry, which is also the radius of the cylinder circumscribing the crown ring, satisfies the equation of the ^4th degree:

4 sin ⁴ (^ ⁴ -4 sin ³ QR ³ + (4 cosg) - 7) sin ² ^) R ² + 2 (l + cos) sin (^) R - 3 cos ² Q + 2 = 0

The height h of the right antiprism (APn) satisfies the equation:) W ^l - ( ^J, -H- :)) ^{if 3} - ⁹

The product of h: 1a height of antiprism,

¾ H = R tan (- ^)

by Η: 1a height of the crown is: 2 -n

A special case occurs for order 9: the vertices of the nonahedron antiprism and the vertices of its star-shaped branches forming the toric crown are on the same plane: h = H.

The angle between the plane of the polar polygon and the plane of the star branch, for example: fig. l 1, for some remarkable cases:

order 3 → - 41 ° 81 order 4 → - 34 ° 55

order 5 → - 25 ° 17 order 6 → - 16 ° 03

order 8 → - 3 ° 66 order 9 → 0 °

order 10 → + 2 ° 69 order 12 → + 6 ° 33

order 25 → + 14 ° 02 infinite order + 19 ° 47

It should be noted: when the number of sides of polar polygons of antiprism tends

V3

towards infinity, the height of the antiprism tends towards - and the height of the toric crown tends

6

3 1

to - a ratio of -; in addition, the angle between the star branch and the polar polygon

2 2 ^"

associate tends to: arccos (- 19 ° 47 or 0.3398 radians.

These atoms (An) are grouped together with common parts to form coherent sets that we will call molecules (Mn).

One of the ways to associate these identical atoms is to take one of the hemispheres of each that we will call half-shell (Cn), fig.4.

These half-shells are recessed, their equatorial rim is sawtoothed, the angles between the facets remain identical to those of the initial atom.

These half-shells (Cn) are joined together by the tops of their starred branches (5) and this three by three or four by four.

By this method of assembly, it thus leads to cavities (Vn), fig.8, whose hexagonal or octagonal base is non-coplanar and whose six or eight faces are not contiguous.

Thus the molecule of order 3 (M3), fig.4, consists of four half-shells (C3) symbolizing four atoms. It has the appearance of a triskele or tetrapod.

It is composed of forty faces, namely the four starry triangles and the four cavities whose hexagonal base is non-coplanar and their six faces not joined because too long.

The molecule (M'3) changes its notation and now becomes the molecule (M 8), it consists of eight half-shells (C3), fig.8, symbolizing eight atoms and is cubic in appearance.

The junction of the half-hulls is then four by four. It is composed of eighty faces, namely eight starry triangles and six cavities (V'3) whose octagonal base is non-coplanar and whose eight faces are non-contiguous because too short.

The molecule of order 4, (M4), fig.12, results from the quadratic assembly of six half-shells

(C4) symbolizing six atoms.

It is composed of seventy-eight faces: the six star squares and eight cavities (V4) whose hexagonal base is non-coplanar and their six faces not joined because too short.

The molecule of order 5 (M5), fig.13, results from the assembly of twelve half-shells, symbolizing twelve atoms, joined together in a dodecahedral manner.

It is composed of one hundred and ninety-two faces, namely twelve pentagons

stars whose axes of symmetry form an angle of: arccos (-i)

V 5

It also has twenty cavities (V5) whose base

hexagonal is non-coplanar and the six faces are not joined because they are too short.

These different molecules M3, M4, M5 and M8 allow assemblies, of the same type of molecules, to develop tree-style montages.

To this end, the molecules must be adapted to fit into each other and lead to stable assemblies with the help of assembly means, such as permanent magnets, mechanical internal fastening system or the method of molecular assembly.

Thus, for the molecules M3, M4, M5 and M8, a cavity (Yn) takes the place of a half-shell, FIG. This is designed to receive and contain another adjusted half-shell belonging to another identical molecule and fusing with each other.

It can be said for then that two neighboring molecules have a common atom between them and that each of the bonds between molecules is made by means of a common atom.

The cohesion of the molecules is ensured by the use of permanent magnets which are fixed on the inner walls of the faces called to be in contact two by two.

By means of six identical molecules (M5) it is possible to produce a ring of toric appearance, fig.17.

Each molecule is connected to two others thanks to two solid atoms. This ring of six molecules is on two different levels alternately.

On a molecule (M5), it is possible to graft a maximum of three other identical molecules (M5), fig.18, and this in three different directions from four possible locations on the starting central molecule.

The latter thus has three atoms in common with the other three molecules.

From this principle, we can realize a tree structure and fill the space to infinity and this in a regular way.

Thanks to this binding method, it is possible to make sets of twenty molecules (M5) arranged in a dodecahedral manner and linked together by common atoms (A5) at each bond.

Eight molecules have a triple bond while the other twelve have a double bond.

This assembly of twenty molecules defines the macromolecule (MM5) and constitutes the level 1 in the construction.

In this module (MM5) the twenty molecules have a point of contact to an internal fictional sphere and the module fits and is contained in an icosahedron.

To go to the next step in the construction, one duplicates the level 1 (MM5) twelve times by placing them on the 12 vertices of the icosahedron associated with the level 1 module.

At each of the twelve vertices of this icosahedron, a link is made to another modulus (MM5) via a free bond of the molecule which allows to have a common bonding atom.

This group of twelve macromolecules is composed of two hundred and sixty molecules, it is part of an icosahedron and constitutes level 2 of the construction.

At this stage of the assembly, the twelve branches of the macromolecules (MM5) of level 2 are interconnected by twenty molecules (M5, X5), this by sharing an atom common to each pair of branches.

Thanks to these interconnections, the possible communications between the elements

will continue to occur if there is a break in one or more branches of the tree, and this to the image of a cellular tissue.

This type of construction by (MM5) has the lowest density of material compared to the following two from the macromolecules MM4 and MM3.

By means of four identical molecules (M4), it is possible to produce an O-ring, each of the molecules being connected to the other two contiguous thanks to two common atoms made integral with the aid of permanent magnets or another binding method.

This ring of four molecules is presented on two levels alternately.

The macromolecule of order 4 (MM4) consists of a central molecule on which six other molecules (M4) are grafted by sharing at each bond a common atom, figl 5-16.

This construction can be continued indefinitely and it is this assembly that has the least empty space between the molecules, and therefore a high density of material.

Six molecules (M3) allow the realization of a crown of appearance toric on two levels.

By this same method of connection, we can come to crown this crown by a set of four molecules grouped in the manner of a triskele or a tripod, and this on both sides of the crown.

This new assembly constitutes the third order macromolecule (MM3) composed of fourteen molecules (M3).

This configuration can also be continued indefinitely. All the empty spaces communicate with each other and can act as a heat exchanger.

From this last assembly based on (MM3) all forms of constructions are allowed.

In all models falling within this description, each of the macromolecules (MMn) can be taken as the starting point of the set because the basic elements and their provisions are homogeneous.

These constructs from macromolecules (MMn) are suitable for creating isotropic media.

Description What's new.

We start from the molecules M3, M4, M5 and M8, plate 1, whose half-shells of atoms will be reduced to the right antiprisms (APn) forming the nuclei of these atoms to which

we give here the name of brick (ABn), boards 4-5-6.

However, the antiprisms of the different molecules M3, M4, M5 and M8 are prolonged for the polygonal medium (I) of the outer poles, up to the circumscribed sphere that each of these four molecules possesses and this gives us the point (Ο ') on the sphere.

Our views do not represent these circumscribed spheres, and the antiprisms stop at the polygonal middle (I) of the outer pole.

For these bricks, originally in the form of antiprism, we can give them the shape of a straight cylinder whose diameter will be that of the circle inscribed in the polar polygon of the starting antiprisme and therefore this cylinder will be tangent to each faces of this antiprism.

We can also give them the form of a right prism of the same section and the same number of faces than the polar polygon.

For bricks, in the form of a straight prism or antiprism, we can also choose multiples with this number of faces, provided that their polygonal sections share the same inscribed circle, ie they have in common a part of each of the faces of the starting antiprism. By this construction, our molecules become: MB3, MB4, MB5 and MB8.

For the molecule of order 5, (M5) which has twenty cavities with six sides we can, from the six coplanar media of their six sides, construct prisms, antiprisms going to the center (O) of this molecule.

The length of the edges of these last hexagons is in this case: 0, 860 for three of them and 0.865 for the other three; this gives us an irregular hexagonal base which, therefore, is not suitable for our construction.

For this reason in this case, we will opt for cylinders inscribed in these cavities whose diameter will be worth: 0 = 1, 725, plate 7.

Another possibility is to take the alternate vertices, low or high, polygons of these cavities, which are coplanar, to build the prisms, antiprisms and cylinders inscribed from the polygons thus created. These prisms, antiprisms and cylinders are extended to the center (O) of this molecule. The diameter of these latter cylinders will be worth: 0 = 0.993.

This construction gives the ABC5 bricks and the MBC5 molecule.

The same goes for the molecule (M4). It has eight six-sided cavities.

From three coplanar alternating vertices we can construct prisms, antiprisms and cylinders, at the origin of the ABC4 bricks and the MBC4 molecule.

The four cavities of the molecule (M3) give, by the same method, ABC3 bricks and the MBC3 molecule.

The six cavities of the M8 molecule have eight vertices, from four alternating coplanar vertices we can obtain the ABC8 bricks and the MBC8 molecule.

These molecules, coming from the bricks coming from the cavities, take the name:

MBC3, MBC4, MBC5 and MBC8 for the last one.

These molecules MBn and BCn, are composed of bricks ABn and ABCn and the latter will be solid or hollow according to the desired need. The thickness of their walls will depend on the use.

In these molecules, the bricks virtually merge in the center and, for the assembly is possible, these bricks are truncated in the central part so that each can find its place without encroaching on another, plate 8.

The virtual junction lines, on the axis (VW) of these bricks, towards the center (O) of the polyhedra, and the main axis (Oy) of the bricks define the cutting plane of the truncations operated. between two neighboring bricks towards the center, fig.37, 43.

This maneuver is repeated for each line of junction; the truncations of the poles of these bricks generate oblique facets (9) concurrently.

These truncations are reproduced identically on the opposite poles of the bricks, from the junction point (Ο ') of the central axis of the bricks and the sphere circumscribed to their molecule MB3, MB4, MB5 and MB 8, boards 9 to 12.

On each of these boards the 2 bricks on the left are from a right prism, the 2 from the center are from a right antiprism and those from the right are based on a cylinder.

However, our diagrams, Fig.19 to 36, show bricks slightly shorter because they are stopped at the center (I) of the outer polar polygons, except for Figures 37 to 50, where the given lengths of the bricks (ΟΟ ') are exact:

this changes nothing, either in reasoning or in the appearance of graphic representations.

The truncated portions of these solid bricks may be topped, generating a plane (10). In the case of hollow bricks, in the form of a cylinder, prism or antiprism, the poles will be topped according to the section of the internal conduit thus arranged.

Each of the poles of the bricks can be connected to one or more other bricks, fig.59-60.

This makes it possible to create meshes or structures, fig.61 -69-72, in which the circulation of fluids and / or the communication of information between all the bricks are feasible and this in all the desired directions.

These interconnections would allow a better fluidity in the distribution.

The peculiarity of these bricks lies in the angle of truncation HÔ, namely the angle that makes the central axis (Ο-Ο ') of the brick with the median line (OK.) Where the point () is the middle of one of the edges of the pyramidal base located on the axis (VW), fig.37-40-43-46., and (H), located on the axis (00 ') is the center of gravity of the pyramidal base, so that (OH) is the height of the pyramid.

This angle is different for each antiprism according to its starting order number, but this angle remains the same for the bricks in which are cylinder-shaped, prism or antiprism if they meet the conditions described above.

Here is a table giving the value of the truncation angles measured for the different bricks according to their order number and we can notice the complementarity existing with the bevel angles used in the mechanical construction of the edges for the five regular polyhedrons, called: "the solids of Plato ".

In polyhedra, bevel angles are the angles obtained between their faces and the projection of their edges towards the center (O) of these polyhedra. Angle angle arc-tangent angle brick for polyhedron solid truncation of Plato

M3 MB3 AB3 54 ° 735 35 ° 264 tetrahedron

M3 MBC3 ABC3 54 ° 735

35 ° 264 tetrahedron

M4 MB4 AB4 45 ° 45 ° hexahedron

M4 MBC4 ABC4 35 ° 264 54 ° 735 octahedron

M5 - MB 5 AB5 31 ° 717 27-9 V5 58 ° 282 dodecahedron

10

3- V5

M5-- MBC5 ABC5 20 ° 905 69 ° 094 icosahedron

M8 MB8 AB8 35 ° 264 54 ° 735 octahedron

M8 MBC8 ABC8 45 ° 45 ° hexahedron

This way of proceeding, for the construction of the eight truncated bricks but before a possible heading, MB3, MBC3, M4, MBC4, M5, MBC5, M8 and MBC8 from the four molecules M3, M4, M5, and M8, determines the length (ΟΟ ') of these bricks, as being the radius of the spheres circumscribed to each of these molecules at the same index.

The length of the bricks AB3 and ABC3 = radius of the sphere circumscribed to M3 = 13.659

AB4 and ABC4 = "" "M4 = 18.453

AB5 and ABC5 = "" "M5 = 28.786

"AB8 and ABC8 =" "" M8 = 19.843

The M3 molecule contains the roots of two tetrahedra.

"M4" "of the cube and the octahedron.

"M5" "of the dodecahedron and the icosahedron.

M8 "" of an octahedron and a cube.

The present study reveals the potentialities of the M3, M4, M8 and M5 molecules described in the reference patent already cited. The number of their cavities, their axes of symmetry and the distribution of these latter show their interest.

For these cavities, we take as a base a vertex out of two, because they are coplanar and from there, we build the prisms, antiprisms and cylinders.

The cavities V3 of M3 generate another regular tetrahedron, the principle of self-duality. "" V4 of M4 "" a regular octahedron; we can say the duality of M4.

"" V5 of M5 "" a regular icosahedron, and also, its duality is manifest.

"" V8 of M8 "" a regular hexahedron, the duality is proven.

This method, to shape truncated bricks and not head from right order antiprisms (n), also allows to obtain convex polyhedral bricks with only pentagonal faces, a total of four (n) facets, but of two different models, provided that the length (OO ') of these bricks is greater than twice the height (OH).

These last polyhedra, thanks to the topping of one of the poles authorizes the creation of a table that can serve as a model for the size of stones in jewelry, or for the realization of molds.

For the four truncated bricks, ABC3-ABC4-ABC5 and ABC8, resulting from cavities of molecules M3-M4-M5 and M8 we can, before a possible topping, give them different lengths: instead of taking the radius of the circumscribed sphere to these molecules, we can, from the center (O), choose the height (OE), whose point (E) is the center of gravity of the polar polygon resulting from the alternating and coplanar summits of the low level of the cavities of these molecules .

Thus the bricks, ABC3, ABC4 and ABC5 from antiprisms, take a new form: with the height (OE), the bricks are in the form of a trigonal trapezohedron whose diamond-shaped faces are identical and the edges equal.

ABC3, ABC4 and ABC5 have six identical diamond-shaped faces and their twelve edges are equal. ABC8 has eight identical faces, but not diamond-shaped because its sixteen edges have two different sizes.

From the molecules 1V13, 4, M5 and M8 we can also obtain bricks AB3,

AB4, AB5 and AB8 in the form of trigonal, tetragonal and pentagonal trapezohedrons: this greatly reduces the length of the antiprisms and thus bricks. In these configurations we end up with identical quadrilaterals for each one of them.

Molecule brick angle angles of rhombus length faci polyhedron truncation small large brick edge

M3 ABC3 54 ° 735 70 ° 528 109 ° 47l 6,6055 6,605 6

M4 ABC4 35 ° 264 90 ° 90 ° 14.008 8.088 6

M5 ABC5 20 ° 905 63 ° 435 116 ° 565 26, 155 10.971 6

M8 ABC8 45 ° 1 1.44 8

M3 AB3 54 ° 735 70 ° 528 109 ° 471 2,041 2,041 6

M4 AB4 45 ° 5,000 8

M5 AB5 3 Γ717 1 1, 135 10

M8 AB8 35 ° 264 90 ° 90 ° 4,082 2,357 6 We note, for the bricks AB3 and ABC3, the equality of their lengths and those of their edges.

For the ABC4 brick, the ratio of the length of the brick to that of its edges is: 3

As we noticed the complementarity that existed between the truncation angles of bricks AB3-AB4-AB5-ABC5 and AB8 on the one hand, and the bevel angles allowing the construction of the edges for Plato solids on the other hand, we can adopt another method to obtain the truncation angles of these same bricks:

For this, we start from the five regular polyhedra to which we can join the two rhombohedral polyhedra with twelve and thirty lozenges:

The common points of these seven polyhedra reside in the fact that for each model, their edges and their faces are identical on the one hand, and that each of them has a registered sphere, tangent to each of their faces of other go.

Moreover, the projection of their edges towards the center (O) divides these seven polyhedra into as many identical pyramids, for each of them, as these polyhedra have no faces.

We thus arrive at the creation of pyramids whose polygonal bases are 3-4 or 5 sides.

These pyramids, we can duplicate them and stick them back to back on each of the faces of these seven polyhedra, in an octahedral way, fig.49-50.

For these two polyhedra with 12 and 30 diamonds, we can also draw a table giving the angles of truncation and the radius of their sphere inscribed:

polyhedron brick angle of arc angle radius molecule inscribed

M30-- MB30 AB30 8 ° 72 ^e V25 + 10- V5

These M 12 and M 30 polyhedra, with twelve and thirty diamonds, surmounted by pyramids give the MB 12 and MB30 molecules, fig 75-78.

For the bricks from these seven polyhedra, all the summits of the pyramids will be found on the same sphere whose radius will be twice that of the sphere inscribed in the polyhedron of origin.

For these seven polyhedra, the bricks can be solid or hollow and the polar pyramidal spikes can be beheaded.

The truncated surfaces of the poles (9) or the poles (10) themselves when they are topped serve as junction zones for these bricks together, either by a mechanical internal fastening system or by a magnet system. either by chemical bonding, or again by a process atomic or molecular bond.

The bricks can be grouped together and form coherent assemblies that we call molecules and the latter are linked together by the pooling of one or more bricks and as in the patent mentioned above, these assemblages lead to the creation of macromolecules that can represent crystalline networks, fig.79-80.

For these 5 regular polyhedra and 2 rhombohedrons, these pyramids can be partly topped and, in this case, it results in a more or less significant cavity in the center of these seven polyhedra.

This cavity is then occupied by a polyhedron similar to that of origin, but in reduced model.

The pyramids, thus topped, can be considered as pyramidal slabs and take the denomination: DAn or DACn if they come from a cavity, fig. 1, 92 and 93.

Each of these seven polyhedra is thus broken down into another central polyhedron, similar but smaller, and this one is covered by as many slabs (DAn) that the polyhedron has no faces.

We duplicate these topped pyramids or slabs, and mount them head-to-tail on each of the faces of the seven polyhedra and this results in the creation of seven new bricks, in the form of slabs with double topped pyramids, contiguous to their common base.

Now, we can give these topped bricks, pole to pole lengths greater: on each of the faces of these seven polyhedra and along the axes of symmetry from their center (O) to the center of gravity (H) of each of their we have, coaxially, either a right prism or a right antiprism having at the poles a section identical to the faces of the polyhedra, or even a right cylinder whose diameter is that of the circle inscribed on these same faces, and this to the length desired.

The opposite pole of this prism, antiprism or cylinder will graft on one of the faces of another identical polyhedron, ie also composed of slabs and another central polyhedron.

This operation has the effect of creating bricks in the form of a prism, antiprism or straight cylinder whose poles have a pyramidal shape headless or not.

The number of faces, prisms and antiprisms, can be a multiple of the number of faces of origin of each of these polyhedra, provided to have the same circle inscribed.

In the case of hollow bricks, the poles are topped according to their inner ducts and, or, the dimensions of the central polyhedra.

For each of these seven polyhedra, removing one of the slabs creates a cavity that allows to receive a brick having the same truncations and topped according to the depth of this cavity. This operation can be repeated as many times as you want to create new branches and this allows you to connect the bricks together.

Another way of arranging the prisms, antiprisms and cylinders, serving to lengthen the bricks, exists for the five regular polyhedra: it consists in placing one of the vertices of these prisms, antiprisms or cylinders, not on one of the faces, but directly in the center (O) of these regular polyhedra.

Starting from each of the edges of these five polyhedra, we establish a plane of section towards the center of these polyhedra, which has the effect of truncating, in a regular way, these prisms, antiprisms and cylinders since the edges, or the line of tangency to the cylinder, to the center.

This results in the creation of the same number of facets as that of the polar polygons, to avoid any overlap of one element on another and dissociate each of the projected bricks. The shape of the bricks that results is similar to the first set of bricks derived from molecules M3, M4, M5 and M8.

The possible headings are made under the same conditions as those described above.

The MB30 molecule derived from the rhomboidal triacontahedron, a polyhedron with thirty diamonds, is very promising for the configuration of the constructions, since this polyhedron with thirty diamonds has in it the roots of the five regular polyhedra and has thirty-one axes of symmetry.

To summarize: we have just designed a set of bricks, full or hollow, truncated identically at each end to the image of the regular octahedron.

The seven truncation angles of all these bricks come from the five regular polyhedra and the two rhombohedrons with twelve and thirty lozenges, from which these angles are materialized by the cutting plane formed by the edge (VW) of the dihedron. of two contiguous faces and the center (O) of one of the seven polyhedra on the one hand and the height (OH) lowered from this same center (O) towards the center of gravity (H) of one of the two faces common to this edge on the other hand, and whose length (ΟΟ ') before their possible heading is equal to or greater than twice the height (OH) which is also the radius of the sphere inscribed with one of these seven polyhedra supra.

For bricks whose length is greater than twice the height (OH), their body is derived from a section of matter, in form, either of antiprism, or prism or straight cylinder.

The identical topping of solid or hollow pyramids leads to the obtaining of a central polyhedral cavity, similar to one of the seven original polyhedra, and this cavity is itself occupied by an identical polyhedron, but in reduction , full or hollow, covered by as many pyramidal headstones that the polyhedron has faces, fig.91, 92 and 93.

By giving the three truncated bricks (ABC3, ABC4, ABC5), resulting from antiprisms and the cavities of the molecules (M3, M4 and M5), a length (ΟΟ ') before topping, equal to the height (OE) lowered from the center (O) towards the center of gravity (E) of the face coming from the junction of the alternating and coplanar vertices of the low level of the polar cavities: these bricks take the form of a trigonal trapezohedron, with identical diamond-shaped faces and equal edges. DESCRIPTION

Figures from the old patent:

fig 1, 5 and 9 AP3, AP4 and AP5 antiprisms

fig 2, 6 and 10 belts T3, T4 and T5

fig 3, 7 and 11 atoms A3 (icosahedron), A4 and A5

fig 4, 8, 12 and 13 molecules M3, M8, M4 and 5

fig 14 molecule M4 with CA4 cavity

fig 15 M4 molecule surrounded by 4 other M4 molecules

fig 16 macromolecule MM4 formed of 7 molecules M4 (including 2 hidden) new figures:

fig 19 to 22 molecules MBn and its ABn bricks and in the form of antiprism in

their hulls Mn

fig 23 to 26 mol. MBn and its cylinder-shaped bricks ABn in their hulls Mn fig 27 to 30 mol.MBn and its bricks ABn in the form of right prisms in their hulls Mn fig 31, 32, 33 mol.MBC5 and its 20 bricks ABC5 in the three in their hulls M5 fig 34, 35, 36 partial views of the molecules MB3, MB8, MB4 and MB5 showing the

truncations on vertical bricks only.

Fig 37, 38, 39 bricks AB3, truncated at both poles, in three forms

fig 40, 41, 42 bricks AB8, truncated low and high, in the three forms

fig 43, 44, 45 bricks AB4, truncated low and high, in the three forms

fig 46, 47, 48 bricks AB5, truncated low and high, in the three forms

fig 49, 50 bricks AB12 and AB 30 truncated low and high

fig 51, 52 tetrapod MB3 and hexapod MB4 composed of 4 and 6 hollow bricks fig 53, octopod MB8 and its 8 bricks AB8 tops,

fig 54 dodecapod MB5 and its 12 hollow AB5 bricks

fig 55 hollow bricks AB3, the one on the right has a thickness

fig..56 AB4 hollow bricks, the right one has a thickness

fig 57, 58 hollow bricks AB8 and AB5, with thickness on the right

fig 59 molecule MB4 composed of 6 bricks in the form of hollow antiprisms in the right view and the same amputee of two bricks in the left fig 60 molecule MB5 composed of 12 bricks in the form of hollow antiprisms in the right view and amputated some bricks on the left fig 61 fig. interlocking of 10 hollow or solid octahedral MB8 molecules and

its bricks topped at the two poles AB8 fig 62 assembly of MB3 tetrapods, on 5 levels and links between neighboring molecules by the pooling of AB3 bricks, solid or hollow fig 63, 64 assembly of 20 hexagonal and hollow antiprisms (AB20) on a

icosahedra, and their truncations in the center, giving the polyhedron MB20 fig 65, 66 assemblages of hexagonal tetrapods MB3, hollow and topped on the right

and its full bricks on the left

fig 67, 68, 69 construction of a carpet composed of full or hollow tetrapods

fig 70, 71, 72 stack of 6 carpets of the previous model, composed of AB3 bricks

full or hollow

fig 73 rhombohedral polyhedron with 12 diamonds

fig 74 brick view AB12 going to the center (O) of Ml 2

and below, molecule MB 12 amputated some bricks

fig 75 molecule MB12 packed with 12 pyramids AB12

fig 76 rhombohedral polyhedron with 30 diamonds M30

fig 77 view of brick AB30 going to center (O) of M30

and below, molecule MB30 amputated some bricks

fig 78 molecule MB30 with 30 pyramids AB30

fig 79, 80 interlocking of molecules MB 12, thanks to the sharing of bricks AB12 fig. 81 brick AB3 trigonal lozenge, topped right fig. 82 brick AB8 trigonal cubic

fi. boy Wut. 83 brick AB4 tetragonal trapezohedron

Fig. 84 brick AB5 pentagonal trapezohedron

Fig. 85 brick ABC3 trigonal diamond

Fig. 86 brick ABC8 tetragonal trapezohedron

Fig. 87 cubic trigonal ABC4 brick

Fig. 88 brick ABC5 trigonal diamond

Fig. 89 brick AB12 length: three times the radius of the sphere inscribed

Fig. 90 brick AB30 length: three times the radius of the sphere inscribed

Fig. 91 dodecahedron equipped with all its slabs (DA5) pyramidal and topped fig. 92. dodecahedron deprived of three of its slabs and showing the small central dodecahedron fig. 93 same assembly as figure n ° 9l but equipped with twelve hollow slabs

ABn brick, full or hollow, truncated from prism, antiprism or cylinder

AB20 brick of solid or hollow icosahedron, truncated from prism, antiprism or cylinder ABCn bricks from the cavities and its order number

An atonia and her order number

APn antiprism and its order number

Cn half-shell and its order number

DAn pyramidal slab topped and its order number

E center of gravity of the pole formed by the coplanar summits of the cavities

F center of gravity of the pole formed by the coplanar peaks of the cavities

H center of gravity of the base of the polar pyramids for the bricks

5 regular polyhedra and 2 polyhedra with 12 and 30 diamonds I polar polygon centers with shortened bricks

midpoint of one of the edges of the pyramidal bases

Mn molecule and its order number

M 12 polyhedron with 12 diamonds

M20 icosahedron

M30 polyhedron with rhombs

MBn polyhedric molecule, consisting of ABn bricks

MB 12 polyhedron with 12 lozenges and 12 bricks AB 12

MB20 polyhedral molecule consisting of 20 bricks AB20

MB30 polyhedral diamond with 30 bricks AB30

MBC5 molecule from the 20 cavities of MB5 and composed of 20 bricks A'B5

MBC4 molecule from the 8 cavities of MB4 and composed of 8 bricks A'B4

MBC3 molecule resulting from 4 cavities of MB3 and composed of 4 bricks

MMn macromolecule and its order number

O center of the molecules and vertices of the inner poles of the bricks

O 'center of peaks or outer poles of bricks

Pn polygon of the summit and its order number

n claim and its order number

Tn belt and its order number

A cavity with six faces and its order number

V W junction line of the dihedron constituted by 2 contiguous faces

Xn junction molecule between branches

5 peaks of the equatorial ring belt

9 truncation plane

10 polar plane from the top of the headed brick

Claims

CLAIMS (Rn).

R1) A set of bricks, either solid or hollow, characterized by the fact that they are truncated, identically at each end, according to one of the seven angles of truncation coming from the five regular polyhedra and from the two rhombohedrons to the twelve and thirty lozenges, from which these angles are materialized by the plane of section formed by the edge (VW) of the dihedron of two contiguous faces and the center (O) of one of the seven polyhedra on the one hand and the height (OH ) lowered from this same center towards the center of gravity (H) of one of the two faces common to this ridge

on the other hand, and whose length (ΟΟ ') before their possible topping is equal to or greater than twice the height (OH) which is the radius of the sphere inscribed with one of the seven aforementioned polyhedra.

R2) _ Set of bricks whose length is greater than twice the height (OH) according to

RI, characterized in that they come from a section of shaped material either antiprism, prism or cylinder straight, length (ΟΟ ').

R3) _A set of truncated bricks, whose length is greater than twice the height

(OH), formed from right antiprisms, of order (n), and from the five regular polyhedra according to RI, characterized in that this operation also allows the construction of convex bricks, composed only of four (n) pentagonal faces of two different models.

R4) Convergent set of solid or hollow bricks, derived from the five regular polyhedra and the two rhombohedrons with lozenges, truncated and topped according to RI, characterized by the fact that their identical topping results in obtaining a central cavity, of shape similar to the original polyhedron and that this cavity is itself occupied by an identical polyhedron, but in reduction, solid or hollow, covered by as many overhead pyramidal slabs that the polyhedron has faces.

R5) _Ensemble of eight bricks AB3, ABC3, AB4, ABC4, AB5, ABC5, AB8 and ABC8 truncated according to RI, and derived from molecules M3, M4, M5 and M8, characterized in that their length (ΟΟ '), before topping, is equal to the radius of the sphere circumscribed to each of these molecules at the same index.

R6) _Ensemble of three bricks ABC3, ABC4, and ABC5, resulting from antiprisms and cavities M1 molecules, M4, and M5 truncated according to RI, characterized in that their length (ΟΟ '), before tying, is equal to the height (OE) lowered from the center (O) towards the center of gravity (E) of the face coming from the junction of the alternating and coplanar vertices of the low level of the polar cavities.and that the bricks thus have the shape of a trigonal trapezohedron, with identical diamond-shaped faces and equal edges.

R7_ Set of bricks, according to RI, characterized by the fact that the truncated surfaces (9) of their poles, or the poles (10) themselves when they are topped, serve as junction zones for these bricks together, either by a mechanical internal fastening system, or by a magnet system, or by chemical bonding or else by an atomic or molecular bonding process.