LLáászlszlóó VVöörröössUniversity of Pécs, M. Pollack Faculty of Engineering, Institute of Architecture

POINTPOINT--NET STRUCTURES OF SPACENET STRUCTURES OF SPACE--FILLINGFILLING PLATONIC AND ARCHIMEDEAN SOLIDSPLATONIC AND ARCHIMEDEAN SOLIDS

MATHEMATICS IN ARCHITECTURE AND CIVIL ENGINEERING DESIGN AND EDUCATION, 27-28 May 2011

The The mmoreore--dimensional dimensional ccubes and their ubes and their sshadowshadows, multiplication of the edges, multiplication of the edges’’ lengthlength

Tesellations in tessellationsTesellations in tessellations

The length of the edges of the 3-dimensional cube (3-cube) can be doubled by 23 and tripled by 33 cubes, and so on, joining each other by faces. In this way, the cubes produce a space-filling mosaic, their faces, being on common planes, create plane-tiling. The projected tiling elements can have 3 different shapes. Those belonging to one group are congruent, in case of parallel projection. The shadows of the faces belonging to different planes overlap one another.However it is possible to select groups of these shadows which compose planar tessellations of the projections plane. The selection may be based on combinatorial rules: how many different shapes may have the adjacent elements in what order? However we can choose cubes from the spatial mosaic whose projection results in plane-tiling.

The 3The 3--dimensional model (3dimensional model (3--model) of themodel) of the kk--dimensional dimensional cubecube ((kk--cubecube))

These cubes compose the space-filling mosaic of the k-dimensional space. We gain the interpenetrating spatial tessellation of the three-dimensional space if we construct this structure with the 3-model of the k-cube. This figure shows the doubling of the 4-cube.

We need We need m m kk kk--cube joined by faces to cube joined by faces to m m -- tuplicate the length of the tuplicate the length of the kk--cubecube’’s edges.s edges.

The edges of the Platonic 3The edges of the Platonic 3--cube have cube have 33 different spatial stands. Our first set [group 3] different spatial stands. Our first set [group 3] has only 1 element. has only 1 element. The edges of members of thThe edges of members of the nexte next set of Platonic and Archimedean set of Platonic and Archimedean solidssolids have have 66 different spatial stands. We call this set as [group 6].different spatial stands. We call this set as [group 6].

The former edges are parallel with those of a The former edges are parallel with those of a 33--model of the 6model of the 6--cubecube..The hull of this is the truncated octahedron, member of the [groThe hull of this is the truncated octahedron, member of the [group 6].up 6].

The truncated octahedron is the common part of a cube and an octThe truncated octahedron is the common part of a cube and an octahedron and ahedron and the inner vertices of the 6the inner vertices of the 6--cube's 3cube's 3--model join the vertices of two solids similar to model join the vertices of two solids similar to the above ones. The solid model self can fill the space.the above ones. The solid model self can fill the space.

The inner vertices of the 6The inner vertices of the 6--cube's 3cube's 3--model compose shifted parallel layers of model compose shifted parallel layers of detached squares. This net structure will be multiplied by the sdetached squares. This net structure will be multiplied by the spacepace--filling models filling models and also by the 3and also by the 3--dimensional shadow of 6dimensional shadow of 6--cubes filling the 6cubes filling the 6--space, that means space, that means by the multiplication of the edge length of the 6by the multiplication of the edge length of the 6--cube's 3cube's 3--model.model.

Based on some examples, we can see, that the elements of solids,Based on some examples, we can see, that the elements of solids,members of our [group 6], join the parts of the 6members of our [group 6], join the parts of the 6--cubecube’’s 3s 3--model.model.

Based on the former interrelations, we can state that the tesselBased on the former interrelations, we can state that the tessellations composed lations composed from the solids of the from the solids of the [[group 6group 6]] fit into nets of squaresfit into nets of squares..

The probably most frequently used lattice girderThe probably most frequently used lattice girderis based on mosaic of tetrahedra and octahedra.is based on mosaic of tetrahedra and octahedra.

The next example isThe next example is the periodical tessellation of tetrahedra and truncatedthe periodical tessellation of tetrahedra and truncated tetrahedratetrahedra..

The former spatial mosaics may have fractal structure because thThe former spatial mosaics may have fractal structure because their elements can be composed eir elements can be composed from tetrahedra and octahedra. The vertices of these mosaics alsfrom tetrahedra and octahedra. The vertices of these mosaics also join the above nets of squares.o join the above nets of squares.

To construct spatial grid structures, we need to form the initiaTo construct spatial grid structures, we need to form the initial ray group of the 6 l ray group of the 6 basic edges in a centrally symmetric shape and we gain 12 rays. basic edges in a centrally symmetric shape and we gain 12 rays. We can We can determine these basic struts by the treated polyhedra but from pdetermine these basic struts by the treated polyhedra but from practical point of ractical point of view, the better usable node element could be a rhombic dodecaheview, the better usable node element could be a rhombic dodecahedron whose dron whose faces are perpendicular to the 12 basic rays.faces are perpendicular to the 12 basic rays.

The edges of the next polyhedra The edges of the next polyhedra -- [group 9] [group 9] -- have have 99 different spatial stands: Archidifferent spatial stands: Archi--medean truncated cube, (small) rhombicuboctahedron and truncatedmedean truncated cube, (small) rhombicuboctahedron and truncated cuboctahedron.cuboctahedron.

Gathering the 9 starting edges, we can construct a 3Gathering the 9 starting edges, we can construct a 3--dimensional model of thedimensional model of the99--dimensional cube. The hull of this model is the truncated cuboctdimensional cube. The hull of this model is the truncated cuboctahedron. ahedron.

This solid is the common part of a cube, an octahedron and a rhoThis solid is the common part of a cube, an octahedron and a rhombdodecahedron mbdodecahedron The last solid is our node element of the group 6.The last solid is our node element of the group 6.

The inner vertices of the 9The inner vertices of the 9--cube's 3cube's 3--model joinmodel join the vertices of cubes and octahedrons the vertices of cubes and octahedrons as wellas well. T. The elements of the 6he elements of the 6--cube's 3cube's 3--model join those of the 9model join those of the 9--cube. Thus the cube. Thus the members of the group 6 fit into the 9members of the group 6 fit into the 9--cube's 3cube's 3--model. The set of the group 9 holds model. The set of the group 9 holds the sets of the groups 3 and 6: the sets of the groups 3 and 6: [[[[group 3group 3]] [[group 6group 6]] group 9group 9]]..

The The elements of Archimedean solidelements of Archimedean solid--members of themembers of thegroup 9group 9 also join the parts of the 9also join the parts of the 9--cube's 3cube's 3--model.model.

We can construct nets of attached squares inside of the above We can construct nets of attached squares inside of the above aa and and bb nets. Thus nets. Thus the vertices of mosaics created from members of the vertices of mosaics created from members of [[group 6group 6]] can join nodes of our can join nodes of our new structure of nets.new structure of nets.

The doubling of the edge length of our 9The doubling of the edge length of our 9--cube's 3cube's 3--model generates a new pattern model generates a new pattern of vertices. We can describe the whole net of nodes in the folloof vertices. We can describe the whole net of nodes in the following way as well: wing way as well: the above congruent the above congruent aa and and bb nets are parallel with three planes perpendicular to nets are parallel with three planes perpendicular to each other. These nets must be copied in their own planes in direach other. These nets must be copied in their own planes in directions of three ections of three vectors determined by nodes of the basic net. vectors determined by nodes of the basic net.

We can see on the left site the new, shifted We can see on the left site the new, shifted aa and and bb nets together which are signed with red and nets together which are signed with red and green points, respectively. green points, respectively. Based on all interrelations described up till now, we may say thBased on all interrelations described up till now, we may say that all at all the possible 11 spacethe possible 11 space--filling mosaics of Platonic and Archimedean solids fit into thisfilling mosaics of Platonic and Archimedean solids fit into this structure. structure. The last right description of all convex uniform honeycombs was The last right description of all convex uniform honeycombs was given by Branko Grgiven by Branko Grüünbaum: nbaum: „„Uniform tilings of 3Uniform tilings of 3--spacespace”” Geombinatorics 4 (1994).Geombinatorics 4 (1994).

In two cases of tilings (described later), we may erase layers oIn two cases of tilings (described later), we may erase layers of nodes. The thinned planar pattern f nodes. The thinned planar pattern is showed on the right site. We can construct nets of attached sis showed on the right site. We can construct nets of attached squares onto the layers and these quares onto the layers and these show that the vertices of tilings created with cubes as well as show that the vertices of tilings created with cubes as well as with solids of the [group 6] can join with solids of the [group 6] can join the points of this structure as well.the points of this structure as well.

The former four figures together show the deriving of our nets fThe former four figures together show the deriving of our nets from each rom each other.other.

This This table shows the fitting of the possible tilings, built from Plattable shows the fitting of the possible tilings, built from Platonic and Archimedean onic and Archimedean solids, into the net structure derived from the multiplied 9solids, into the net structure derived from the multiplied 9--cube's 3cube's 3--model.model.

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We may see the tiling 8 with points of the net structure in fronWe may see the tiling 8 with points of the net structure in front and top t and top view as well as in isometric orthogonal axonometric projection, view as well as in isometric orthogonal axonometric projection, as as an an example.example.

We may see the tiling 11 with points of the net structure in froWe may see the tiling 11 with points of the net structure in front and top view as nt and top view as well as in isometric orthogonal axonometric projection, as a furwell as in isometric orthogonal axonometric projection, as a further example.ther example.

To construct spatial grid structures, we need to form the initiaTo construct spatial grid structures, we need to form the initial ray group of the l ray group of the edges into a centrally symmetric shape and we gain 18 rays. Fromedges into a centrally symmetric shape and we gain 18 rays. From practical point practical point of view, the better node element could be a small rhombicuboctahof view, the better node element could be a small rhombicuboctahedron whose edron whose square faces are perpendicular to the 18 basic rays.square faces are perpendicular to the 18 basic rays.

This solid is also the common part ofThis solid is also the common part ofa cube an octahedron and a rhombdodecahedrona cube an octahedron and a rhombdodecahedron..

The edges of the next polyhedra have The edges of the next polyhedra have 1515 different spatial stands: Platonic dodecahedron (5.5.5) different spatial stands: Platonic dodecahedron (5.5.5) and icosahedron (3.3.3.3.3) as well as the Archimedean icosidodeand icosahedron (3.3.3.3.3) as well as the Archimedean icosidodecahedron (3.5.3.5), truncated cahedron (3.5.3.5), truncated dodecahedron (3.10.10), truncated icosahedron (5.6.6), rhombicododecahedron (3.10.10), truncated icosahedron (5.6.6), rhombicosidodecahedron (3.4.5.4) and sidodecahedron (3.4.5.4) and truncated icosidodecahedron (4.6.10)truncated icosidodecahedron (4.6.10). These are the member of the . These are the member of the [group 15].[group 15].

Gathering 15 starting edges, parallel with the differently orienGathering 15 starting edges, parallel with the differently oriented edges of these solids and ted edges of these solids and joining a common point, we can construct a 3joining a common point, we can construct a 3--dimensional model of the 15dimensional model of the 15--dimensional cube. dimensional cube.

The hull of the gained 15D cubeThe hull of the gained 15D cube’’s 3D model is the Archimedean truncated icosidodecahedron.s 3D model is the Archimedean truncated icosidodecahedron.

This solid is the common part ofThis solid is the common part ofa dodecahedron an icosahedron and a rhombic triakontahedrona dodecahedron an icosahedron and a rhombic triakontahedron..

The statistic from the parts of the 15D cubeThe statistic from the parts of the 15D cube’’s 3D model. 32768 vertices, 245760 edges, s 3D model. 32768 vertices, 245760 edges, ……

Some layers of the vertices of the 15D cubeSome layers of the vertices of the 15D cube’’s 3D models 3D model

Vertices of the 15D cube’s 3D model– fragment of the top view

The centrally symmetric arrange of the basic edges of the 15D cube’s 3D model.

These are perpendicular to the square faces of the rhombicosidodThese are perpendicular to the square faces of the rhombicosidodecahedron (3.4.5.4).ecahedron (3.4.5.4).This solid is also the common part ofThis solid is also the common part of

a dodecahedron an icosahedron and a rhombic triakontahedron.a dodecahedron an icosahedron and a rhombic triakontahedron.

Fit of the truncated icosahedron or buckyball into the 15D cube’s 3D model.

Fit of the truncated icosahedron or buckyball into the 15D cubeFit of the truncated icosahedron or buckyball into the 15D cube’’s 3D model.s 3D model.

It is known that five cubes can be constructed inside of a dodecIt is known that five cubes can be constructed inside of a dodecahedron. Thus we can start the ahedron. Thus we can start the basic edges of the group 15 and ((group3) (group 6) group 9) frobasic edges of the group 15 and ((group3) (group 6) group 9) from a common point in a way m a common point in a way that we gain 3 common edges. We have 21 differently oriented raythat we gain 3 common edges. We have 21 differently oriented rays parallel with the edges of s parallel with the edges of the solids which are members of our former 4 groups. Thus the abthe solids which are members of our former 4 groups. Thus the above solids can create the ove solids can create the group 21 in this form: group 21 in this form: [[[[[[group 3group 3]] [[group 6group 6]] group 9group 9]] [[[[group 3group 3]] group 15group 15]] group 21group 21].].

The centrally symmetrical shape of the basic ray group holds 90 The centrally symmetrical shape of the basic ray group holds 90 rays. rays. This fThis figure shows the igure shows the solid whose 60 rhombic and 30 symmetrical octagonal faces are pesolid whose 60 rhombic and 30 symmetrical octagonal faces are perpendicular to these rays rpendicular to these rays and their center points join them. This is no more an Archimedeaand their center points join them. This is no more an Archimedean solid but the hull of a 16n solid but the hull of a 16--cube's 3cube's 3--modelmodel. . We gain with a possible correction 20 new faces (that means 20 nWe gain with a possible correction 20 new faces (that means 20 new initial ew initial rays) which are parallel with the faces of an icosahedronrays) which are parallel with the faces of an icosahedron . .

If we fit a rhombicuboctahedron (our node element in the group 9If we fit a rhombicuboctahedron (our node element in the group 9) into a cube and they are ) into a cube and they are five time fitted in a dodecahedron, we can also gain a solid whofive time fitted in a dodecahedron, we can also gain a solid whose faces are parallel with se faces are parallel with those of the last onethose of the last one. . Thus both above solids can be constructed from the common part oThus both above solids can be constructed from the common part of f 55 rhombicuboctahedron by parallel and centrally symmetric moving rhombicuboctahedron by parallel and centrally symmetric moving of the resulted faces.of the resulted faces.

The next topic could be the construction of lattice girders and The next topic could be the construction of lattice girders and spatial tessellations if wespatial tessellations if weuse the members of the former soliduse the members of the former solid--groups and lowergroups and lower--dimensional parts of the describeddimensional parts of the described33--models of moremodels of more--dimensional cubes. dimensional cubes.

LLáászlszlóó VVöörröössUniversity of Pécs, M. Pollack Faculty of Engineering, Institute of Architecture

POINTPOINT--NET STRUCTURES OF SPACENET STRUCTURES OF SPACE--FILLINGFILLING PLATONIC AND ARCHIMEDEAN SOLIDSPLATONIC AND ARCHIMEDEAN SOLIDS

References:[1] H. S. M. Coxeter, Regular Polytopes (3rd ed. Dover, 1973)[2] Grünbaum, Branko "Uniform tilings of 3-space" Geombinatorics 4 (1994): 49 - 56.[3] Kabai, Sándor and Bérczi, Szaniszló "Space Stations Construction by Mathematica: Interactive Programs to Use the Double Role of the Golden Rhombohedra Modules (The Crystallography of a Space Station)" 37th Lunar and Planetary Science Conference, #1121, LPI, Houston, Texas, USA, 2006. CD-ROM.[4] S. Kabai and Sz. Bérczi, Rhombic Structures. Geometry and Modeling from Crystals to SpaceStations / Rombikus Szerkezetek. Geometria és modellezés a kristályoktól az űrállomásig (Püspökladány: Uniconstant, 2009, ISBN 978-963-87767-3-0)[5] Vörös, László "Reguläre Körper und mehrdimensionale Würfel" KoG 9 (2005): 21-27. http://master.grad.hr/hdgg/kog/[6] Vörös, László "Regular and Semi Regular Solids Related to the 3-dimensional Models of the Hypercube" 13th Scientific-Professional Colloquium on Geometry and Graphics, Poreč, Croatia, September 7-11, 2008.http://www.grad.hr/sgorjanc/porec/abstracts.pdf[7] Vörös, László "Planar tessellations based on shadows of more-dimensional cubes" Paper presented at the 25th National and 2nd International Scientific Conference, monGEometria, Belgrade, Serbia, June 24-27, 2010. Proceedings-CD, ISBN 978-86-7924-038-5, paper[8] Vörös, László "Sets of Space-filling Zonotopes with Connection to Art and Design" Paper presented at the 14th International Conference on Geometry and Graphics, ICGG, Kyoto, Japan, August 5-9, 2010. Proceedings-DVD, ISBN 978-4-9900967-1-7, extended abstract and full paper, 231[9]http://epitesz.pmmk.pte.hu/tervezesi_es_epiteszeti_ismeretek_tanszek/oktatok_tervezes/voros_laszlo/voros_laszlo_videok[10] http://en.wikipedia.org/wiki/Convex_uniform_honeycomb

The creation of the constructions and figures required for the lecture was aided by the AutoCAD program as well as AutoLisp routines developed by the lecturer.