Topologle structurale 21 Structural Topology 1995

Haresh Lalvani School of Architecture Pratt Institute Brooklyn, New York 1 1205 U.SA.

French translation: Traduction franyise : Jean-Luc Raymond

I I Families of Multi-directional

amilies of multidirectional space labyrinths having regular faces are presented. The technique uses placing polyhedra on the vertices of space grids in paired

combinations and interconnecting with open-ended connectors. The labyrinths are restricted to those derived from three kaleidoscopic fundamental regions of space-fillings, and are organized within higherdimensional frameworks as an extension of our earlier work. The technique generates 51 new labyrinths composed of two types of vertices in addition to the known labyrinths with one type of vertex. Ex- tensions to n-dimensional labyrinths in Euclidean and non- Euclidean spaces are mentioned along with other possibili- ties which include non-periodic labyrinths and labyrinths with curved faces.

This work is a continuation of our previous thesis ‘Struc- tures on Hyper-Structures’ [ 1-31 whereby higherdimensional space structures, notably hyper-cubes, are used as

Topologle structurale * 21 - Structural Topology * 1995

- Familles de labyrinthes spatiaux multidirectionnels et periodiques

R6surn6

ous presentons ici les familles de labyrinthes spatiaux multidirectionnels possedant des faces regulieres. La technique repose sur la localisation de

polyedres sur les sommets de grilles spatiales ; les polyedres sont places en combinaisons appariees et sont relies par des elements de jonction a bouts ouverts. Les labyrinthes se limitent a ceux derives de trois regions fondamentales kaleidoscopiques de remplissages de l’espace et sont organises a l’interieur de structures de dimensions elevees selon nos travaux anterieurs. La technique permet dengendrer 51 nouveaux labyrinthes composes de deux types de sommets en plus des labyrinthes connus a un seul type de sommets. On evoque des generalisations aux labyrinthes de dimension n s’inscrivant dans des espaces euclidiens ou non euclidiens, ainsi que dautres possibilites tels les labyrinthes non periodiques et les labyrinthes a faces courbes. B

Cet article s’inscrit en continuite avec notre these Struc- tures on Hyper-Structures [l-31, ou on utilisait des structures

47

41

Families of Multi-d irectional Periodic Space Labyrinths

Familles de labyrinthes spatiaux rnultidirectionnels et pdriodiques

Figure 1 Three kaleidoscopic fundamental regions of space- fillings based on tetrahedral and octahedral symmetries. Trois regions fon damentales kaleidompiques de remplissages de I’espace bases sur des symktrie s tetrakdres et octaedres.

conceptual spaces to organize and generate space structures and their transformations. Earlier we had shown the application of this concept to polyhedra, close-packings and mis- cellaneous space structures. Here, we extend this technique to the systematic generation of space labyrinths and present interim results.

Space labyrinths are a class of space structures which are characterized by a continuous infinite surface which divides space into two parts, inside and outside, without self-inter- sections. In our search for orderly structures, we restrict our study to polyhedral plane-faced space labyrinths composed of regular polygons and having one or two types of vertices. Surfaces with one type of vertex, termed 1 -verteu labyrinths here, are known from the prior work of Petrie and Coxeter [4], and Burt et a1 [5] where they were alternatively described as infinite poZyhedra. Our technique for labyrinth- generation permits the derivation of the known 1 -vertex labyrinths and a new class of 2-vevtex labyrinth.

Our method of labyrinth-generation is termed the polyhedra placement method wherein any space grid, symmetric or asymmetric, provides a starting point. In symmetric space grids, polyhedra corresponding to the symmetry or sub- symmetry of the vertices (nodes) of the chosen grid are placed at the vertices of the grid and interconnected by connector pieces. The faces common to the polyhedra and the connectors are “open” or “removed to produce a continuous space that links the interior of adjacent nodal spaces

332

432 332 332

la l b 1c

spatiales de dimensions elevees, en particulier des hyper- cubes, comme espaces conceptuels pour organiser et engendrer des structures spatiales et leurs transformations. Nous avons deja montre l’application de ce concept aux polyedres, aux remplissages par juxtapositions et a diverses structures spatiales. Nous etendons ici cette technique a la generation systematique de labyrinthes spatiam, et nous presentons quelques resultats intermediaires.

Les labyrinthes spatiaux constituent un ensemble de structures spatiales caracterisees par une surface continue infinie qui divise l’espace en deux parties, l’interieur et l‘exterieur, sans auto-intersection. Dans notre recherche de structures ordonnees, nous avons limit6 notre etude aux labyrinthes spatiaux polyedriques de faces planes composes de polygones reguliers et possedant un ou deux types de sommets. On connait les surfaces a un seul type de sommets, nommees ici lubyrinthes a 1 sommet, depuis les travaux anterieurs de Petrie et Coxeter [4] et Burt et al. [5], ou on les decrivait comme des poly2dres infinis. Notre technique de generation de labyrinthes permet la derivation des labyrinthes a 1 sommet deja connus et dune nouvelle classe de labyrinthes a 2 sommets.

On nomme mkthode de localisation de polytkires notre methode de generation de labyrinthes. Selon celleci, toute grille spatiale, symetrique ou asymetrique, fournit un point de depart. Dans les grilles spatiales symetriques, les polyedres correspondant a la symetrie ou a la sous-symetrie des sommets (noeuds) de la gnlle spatiale choisie sont places aux sommets de la grille, et sont relies par des elements de jonction. Les faces communes aux polyedres et aux elements de jonction sont (( ouvertes )) ou (( supprimees )) pour produire un espace continu qui lie l‘interieur des espaces nodaux adjacents par l’interieur de l’element de jonction. Lorsqu’on applique ce proc6de a tous les sommets et toutes les arktes (etais) de la gnlle sous-jacente, on obtient un espace continu en forme de labyrinthe. Les aretes de la gdle spatiale correspondent aux axes des polyedres de jonction et definissent les joints entre les centres de symetrie de l’espace interieur du labyrinthe. Les joints entre les centres de symetrie de l’espace exterieur definissent une grille spatiale complementaire, et la surface du labyrinthe separe la paire de gnlles complementaires. A l’aide de notre methode, une seule grille suffit comme point de depart, con- trairement a la methode de Burt qui necessite deux gnlles

Figure 2 Fundamental regions of eight space grids with equal edges, one or two types of vertices. and derived from the three fundamental regions of Fig.1. Regions fondamentales de huit grilles spatiales dar&tes bales, ayant un ou deux types de sommets, et d&v&s des trois @ions fondamentales de la figure 1.

Topologia atructumle - 21 - Structural Topology 1995

2d 432 432 28 2b 2c

432

432

2e 332 2h 332 I f 332

through the interior of the connector. As this process is a p plied over every vertex and every edge (strut) of the under- lying grid, a continuous labyrinth-like space is obtained. The edges of the space grid correspond to the axes of the connector polyhedra and define the connections between the centers of symmetry of the interior space of the labyrinth. The connections between the centers of symmetry of the exterior space defines a complimentary space grid, and the surface of the labyrinth separates the pair of complementary grids. By our method, only one grid provides a suffi- cient starting point in contrast with Burt's method where two complementary grids are the starting point. This en- larges the number of possible space labyrinths.

Though the polyhedra placement method is a general one, we report the interim results of labyrinths based on periodic space grids derived from three kaleidoscopic space- fillings. These space-fillings and their kaleidoscopes are known from prior literature (see, for example, Ref [6]). The corresponding fundamental regions are shown in Fig. 1; (a) is the 1/48th segment of a cube, (b) is 1 /24th segment of a cube or 1/48th segment of a rhombic dodecahedron, and (c) is 1/24th segment of a rhombic dodecahedron. The sym-

complementaires. Ceci accroit le nombre de labyrinthes spatiaux possibles.

Bien que la methode de localisation de polyedres soit une methode generale, nous signalons ici les resultats intermediaires de labyrinthes fondes sur des gnlles spatiales periodiques derivees de trois remplissages kaleidoscopiques de l'espace. On connaissait deja ces remplissages de l'espace ainsi que leurs kaleidoscopes (voir par exemple, [6]). La figum 1 illustre les regions fondamentales correspondantes ; (a) illustre la 48e partie d'un cube, (b) la 24" partie dun cube ou la 48" partie dun rhombidodecaedre, et (c) la 24" partie dun rhombidodecaedre. La symetrie des sommets est indiquee dans chaque cas, et on doit repeter les regions fondamentales par des operations de symetrie pour remplir l'espace. Ces trois regions engendrent 8 grilles spatiales distinctes composees dune longueur d'arbte et dun ou deux types de sommets. On derive les grilles spatiales en considerant chaque arbte individuelle de la region fondamentale, et toutes les combinaisons daretes individuelles, comme generateurs. La figure 2a-h illustre les regions fondamentales correspondantes de ces 8 grilles spatiales. Ici, (a) engendre le treillis cubique simple, @) engendre le

I 49

Families of Multi-directional Periodlc Space Labyrinths

Familles de lab rinthes spatiaux multidrectionnels el phriodiques

Figure 3 Eight polyhedra of octahedral symmetry based on Ref [1.2]. Huit polyedres de symbtrie octaedre bases sur [l, 21.

metry of the vertices are marked in each case, and the fundamental regions must be repeated by symmetry operations to fill space. These three regions generate 8 distinct space grids composed of one edge-length and one or two types of vertices. The space grids are derived by consider- ing each individual edge of the fundamental region, and all combinations of individual edges, as generators. The corresponding fundamental regions of these 8 space grids are shown in Fig. 2a-h, where (a) generates the simple cubic lattice, (b) generates the body-centred cubic lattice, (e) generates the close-packing of rhombic dodecahedra, (9 generates the octet truss, (g) the diamond lattice and (h) the close-packing of Miraldi rhombohedra.

For each space grid, polyhedra corresponding to the symmetry of the vertices of the fundamental region, and termed noda7 poZyhedra, are placed on adjacent pairs of vertices and inter-connected with connector polyhedra. The nodal polyhedra for the three space-fillings correspond to four symmetries, namely 432 (octahedral), 332 (tetrahedral), 422 (square prismatic) and 222 (rectangular prismatic) as indicated in Fig. 1. Restricting to regular-faced structures with mirror-symmetry, each symmetry comprises a family of eight structures which are placed at the vertices of a reference cube as described earlier [l, 21. Each symmetry has at least one degenerate structure, the point. In addition, symmetry 422 has only three regular-faced structures and symmetry 222 has only one regular-faced structure. The remaining structures in these two symmetries are degenerate polyhedra.

- 4 Six examples of pairs of polyhedra with octahedral symmetry connected with six different types of connectors. and based on the fundamental region shown in Fig. 2b. Six exemples de paires de polyedres de symetrie octaMre relic par six differents types de joints, et bas& sur la region fondamentale illustr6e ii la figure 2b.

treillis cubique centre, (e) engendre la juxtaposition de rhombidodecaedres, ( f ) engendre le reseau octet, (9) le treillis de diamant et (h) la juxtaposition de rhomboedres de Miraldi.

Pour chaque gdle spatiale, des polyedres correspondant a la symetrie des sommets de la region fondamentale, nom- mes polykdres nodaux, sont places sur des paires adjacentes de sommets, et relies par des polyedres de jonction. Les polyedres nodaux pour les trois remplissages de l’espace correspondent aux quatre symetries, nommement 432 (octaedre), 332 (tetraedre), 422 (prismatique carree) et 222 (prismatique rectangulaire) (figurn 1). En se restrei- gnant aux structures a faces regulieres avec symetrie miroir, chaque symetrie comprend une famille de huit structures qu’on place aux sommets dun cube de reference comme on l’a deja decrit [I, 21. Chaque symetrie possede au moins une structure degeneree, le point. De plus, la symetrie 422 ne possede que trois structures a faces regulieres et la symetrie 222 ne possede qu’une structure a faces regulieres. Les autres structures de ces deux symetries sont des polyedres degeneres.

A titre dexemple, la figurn 3 montre les huit polyedres de la symetrie octaedre; ils sont identifies par p (octaedre), q (cube), r (cuboctaedre), ij (cube tronque), 4 (octaedre tronque), F (rhombicuboctaedre), b (cuboctaedre tronque) et 0 (point). Comme on l’a montre dam un article anterieur, les structures peuvent se transformer de faGon continue vers dautres structures a l’interieur de l’espace cubique. Ces huit polyedres sont places aux sommets des grilles spatiales

Topologie structurale - 21 - Structural Topology * 1995

As an example, the eight polyhedra of the octahedral symmetry are shown in Fig. 3 and are notated as p (octahedron), q (cube), r (cuboctahedron), 6 (truncated cube), 4 (truncated octahedron), 7 (rhombicuboctahedron), 0 (truncated cuboctahedron) and 0 (point). As shown in earlier work, the structures can transform to others continuously within the cubic space. These eight are placed at the vertices of the space grids in paired combinations and interconnected to generate the labyrinths. The continuous transformations of the nodal polyhedra lead to corresponding transformations of the labyrinths from one to another.

The pairing-and-interconnecting technique is shown with six examples in Fig. 4 where the pair is notated ac- cording to the polyhedra being connected.

r - 5 joins a cuboctahedron with a truncated cube, and so on. In the examples shown, the connector polyhedra comprise a hexagonal prism (a), a triangular prism (b), a triangular antiprism (c), a special truncation of a pyramid, simi- lar to a semi-cuboctahedron (d), a hexagonal pyramid (e) and a triangular pyramid (e). These six examples cover the basic classes of connectors. Clearly, the examples with py- ramidal connectors are “degenerate” labyrinths, though the configurations are a part of the same family of structures. The connectors transform to other connectors as the nodal polyhedra transform to other nodal polyhedra.

All combinations obtained by this pairing method generate 64 configurations each for seven space grids obtained from Fig. 2a-g, and 8 configurations for the remaining

For example, 0 - 0 joins a pair of truncated cuboctahedra,

en combinaisons appariees, et relies pour engendrer les labyrinthes. Les transformations continues des polyedres nodaux menent aux transformations correspondantes des labyrinthes vers dautres labyrinthes.

La figum 4 illustre la methode d’appairage et de jonction a l’aide de six exemples dans lesquels on identifie les paires selon les polyedres joints. Par exemple, 0 - 0 indique la jonction entre une paire de cuboctaedres tronques, r - 5 indique la jonction entre un cuboctaedre et un cube tronque, et ainsi de suite. Dans les exemples illustres, les polyedres de jonction comprennent un prisme hexagonal (a), un prisme triangulaire @), un antiprisme triangulaire (c), une tronca- tion particuliere dune pyramide, similaire a un semi-cuboctaedre (d), une pyramide hexagonale (e) et une pyramide triangulaire (e). Ces six exemples couvrent les classes fondamentales de polyedres de jonction. Manifestement, les exemples qui comportent des polyedres de jonctions pyra- midaux constituent des labyrinthes (( degeneres )), mCme si les configurations font partie de la m&me famille de structures. Les polyedres de jonction se transforment en dautres polyedres de jonction en m&me temps que les polyedres nodaux se transforment en d‘autres polyedres nodaux.

mutes les combinaisons obtenues par cette methode dappairage engendrent 64 configurations pour chacune des sept gnlles spatiales obtenues selon les figurns 2a-g, et 8 configurations pour la huitieme grille obtenue selon la figum 2h. On associe les 64 configurations aux 64 sommets dun cube a six dimensions, une pour chaque sommet, comme on l’a montre dans nos travaux precedents pour

4f

I ii

Families of Multi-directional Periodic Space Labyrinths

Familles de lab rinthes spatiaux multid!rectionnels el pdriodiques

eighth grid obtained from Fig. 2h. The 64 configurations are mapped on the 64 vertices of a Gdimensional cube, one for each vertex, as shown for other space structures in our earlier work. Each set of 64 configurations comprises a family of inter-related and inter-transforming configurations. This is shown in an alternative table in Fig. 5 which shows all the 64 combinations. The method of placing polyhedra in pairs can be applied to all grids composed of even-sided polygons.

grids are shown in Figs. 6-9 and are briefly described. In the photographs, the models show a portion of the infinite labyrinth for each paired combination. The models are placed on the vertices of a 2dimensional projection of a 6dimensional cube. In this projection only 49 vertices of the Gcube are visible and the remaining 15 vertices overlap with others. In three of the four examples shown, only regular-faced labyrinths having regular prismatic and anti-prismatic connectors are shown. In the fourth example, the “degenerate” labyrinths are included as part of the same family of configurations.

ing prismatic and anti-prismatic connectors based on the simple cubic lattice obtained from the fundamental region in Fig. 2a. This grid is composed of Gconnected vertices. In the photograph, only 20 vertices of the Gcube are occupied, leaving 44 vertices of the Gcube unoccupied. The unoccupied vertices can be filled if other types of connector polyhedra are permitted. Of the 20 shown, 6 are the known

Four families of configurations based on four of the eight

Fig. 6 shows a family of 20 regular-faced labyrinths hav-

~ ~ ~ ~~ ~~ ~ ~ ~~~ ~

Figure 5 0-0 p - 0 q - 0 r -0 i - 0 9-0 5-0 0 - 0 A table showing a family of 64 configurations

Un tableau d’une famille de 64 configurabons

- - based on all pairs of eight polyhedra. 0 - f i p - p 4-6 r-? r -p 4-6 6-6 0 - 5 bades sur toutes les paires de huit polyedres. 0-4 p - 9 4-4 r - 9 r -q 4-4 5-4 0 - 9

- -

- - 0 - F p - i q-F r-F r - r 4 - F 5-f 0-F

0- r p - r q-r r - r F-r 9- r p - r 0- r

0 - q p -q q - q r -q i - q 4 - 4 p - q 0 - q

0 - p p - p q - p r - p i - p 9 - p p - p 0 - p

0-0 p - o q - o r -0 F-o 4-0 p-o 0-0

dautres structures spatiales. Chaque ensemble de 64 configurations comprend une famille de configurations reliees et inter-transformees. Cela est illustre par un autre tableau a la figurn 5 montrant l’ensemble des 64 combinaisons. La methode qui consiste a apparier les polyedres peut &re appli- quee a toutes les gnlles composees de polygones ayant un nombre pair de cBtes.

Les figuxes 6-9 illustrent quatre familles de configurations fondees sur quatre des huit grilles. Elles sont brieve- ment decrites. Dans les photographies, les modeles mon- trent une portion du labyrinthe infini pour chaque combi- naison appariee. On a place les modeles sur les sommets d’une projection bidimensionnelle dun cube a six dimensions. Dans cette projection, seuls 49 sommets du cube a six dimensions sont visibles, les 15 sommets restants etant re- couverts par les autres. Dans trois des quatre exemples, on ne montre que les labyrinthes a faces regulieres possedant des polyedres de jonction prismatiques et antiprismatiques. Dans le quatrieme, les labyrinthes (( degeneres )) sont inclus comme une partie de la meme famille de configurations.

La figuw 6 illustre une famille de 20 labyrinthes a faces regulieres possedant des polyedres de jonction prismatiques et antiprismatiques bas& sur le treillis cubique simple ob- tenu de la region fondamentale dans la figurn 2a. Cette gnlle est composee de sommets dindice 6. Dans la photo- graphie, seuls 20 sommets du cube a six dimensions sont occupks, laissant 44 sommets libres. On peut remplir les sommets libres si on admet dautres types de polyedres de jonction. Parmi les 20 labyrinthes illustres, 6 sont des labyrinthes connus a 1 sommet composes dun polyedre nodal, et 14 sont des nouveaux labyrinthes a 2 sommets composes de deux polyedres nodaux differents. Ces derniers apparais- sent en paires reciproques de labyrinthes topologiquement identiques ; cela nous laisse donc un total de 7 nouveaux labyrinthes a 2 sommets.

La figuw 7 illustre une famille de 20 labyrinthes a faces regulieres ayant des polyedres de jonction prismatiques et antiprismatiques bases sur le treillis cubique centre correspondant a la region fondamentale de la figuw 2b. Cette gnlle est composee de sommets dindice 8. Comme dans la fkmille fondee sur le treillis cubique simple, les 20 labyrinthes comprennent 6 labyrinthes connus a 1 sommet composes dun polyedre nodal, et 7 differents labyrinthes a 2 sommets composes de deux polyedres nodaux distincts.

Topologie structiirale * 2 1 * Structural Topology * 1995

Figure 6 A fariiily of 20 space Iabyriiiths with regirlar faces t m e d on the cubic lattice and restricted to regular prisms arid anti-prisrns as connectors. The labyrinths occtipy selected positioris on the vertices of n 6-diniensional cube. (From the exhibit at the Cathedral of St.John the Divine, New Ywk. 1992). Une famille de 20 Inhyririthes spattaux a faces rkgtilieres basees sur le treillis cubique et limites a ceux dont les polyedres de jonction sont des prisnies regtiliers et des antiprismes. Les labyrinthes occupent des positions choisies sur les sonitnets d'un cube a six dimensions. (Extrait de I'exposition a la cathedrale St. John the Divine, New York, 1992.)

Families of Multi-directional Periodic Space Labyrinths

Familles de labyrinthes spatiaux multidirectionnels et periodiques

Fig. 8 shows the family of 20 regular-faced labyrinths with prismatic and anti-prismatic connectors based on the rhombic dodecahedra1 lattice obtained from the fundamental region in Fig. 2e. Here, since the space grid is composed of two different types of vertices, one set which is 8-connected and another set which is 6-connected, all 20 labyrinths are distinct 2-vertex labyrinths.

Fig. 9 shows the family of structures based on the diamond lattice derived from the fundamental region in Fig. 2g. This grid is composed of 4-connected vertices. The photograph shows legitimate as well as degenerate labyrinths. Of these, 20 labyrinths are regular-faced, as in the previous families. Of these 20, 6 are the known 1-vertex

Figure 7 A family of 20 space labyrinths with regular faces based on the body- centred cubic lattice, also restricted to those having regular prismatic and anti-prismatic connectors. The family is also arranged on the vertices of a 6- dimensional cube. (From the exhibit at the School of Architecture, Pratt Insti- tute, in 1991). Une famille de 20 labyrinthes spatiaux a faces regulieres bases sur le treillis cubique centre et aussi limites a ceux dont les polyedres de jonction sont des pr imes et des antiprismes. La famille est aussi disposee sur les sommets d’un cube a six dimensions. (Extrait de I’exposition a I’ecole d’architecture de I’lnstitut Pratt, 1991.)

labyrinthes, 20 sont a faces regulieres, comme dans les fa- milks precedentes. ces 20 labyrinthes comprennent 6 labyrinthes connus a 1 sommet et 14 labyrinthes a 2 sommets. Comme precedemment, les 14 labyrinthes a 2 sommets comportent des labyrinthes reciproques, ce qui nous laisse un total de 7 labyrinthes distincts a 2 sommets.

On peut appliquer de faqon systematique cette procedure aux quatre dernikres grilles parmi les 8 grilles de la figum 2. La grille fondee sur la region fondamentale de la figum 2c comporte des sommets dindice 12 et des sommets d’indice 4 ; elk engendre quatre differents labyrinthes a deux sommets et a faces rkgulieres composes de deux polykdres nodaux. La grille basee sur la region fonda-

Topologie structurale - 21 *Structural Topology - 1995

I 55

Figure 8 A family of 20 space labyrinths having regular faces and based on the rhonibic dodecahedra1 lattice. also arranged on selected vertices of a 6-dimensional cube. (From the exhibit at the Cathedral of St.John the Divine, New York. 1992). Une fariiille de 20 labyrinthes spatiaux A faces regulieres bases stir Ie treillis rhonibidodecaedre. Les lahyi-inthes sont disposes siii dcs sornnicts choisis d'tin cube 3 SIX diriicrisinns. (Extrait dc I'exposition 5 la c;itlicdrale St. Joli i i t t ic Divine, Ntrw York. 1992.)

Figure 9 A family of 64 configurations based on the diamond lattice and arranged on the vertices of a 6-dimensional cube. Only 49 configurations are shown. The family as shown includes legitimate space labyrinths and many degenerate cases. (From the exhibit at the School of Architecture, Pratt Institute. iii 1991) Une faniille de 64 Configurations hasees siir le ti-eillis en dinmarit et disposee stir les somrnets d'tin cube a six dinieiisions. On l ie iiiontre que 49 coiifigtrrations. La faniille illustr& iiicltit des Ihyrinthes spatiaux IC(1i- times et pliisieuis cas degdnCrds. (Extrait de I'expositinn 3 I'ecole d" ~. ,iic.hitccture de I'liistitut Pratt, 1991.)

Familles de lab rinthes spatiaux multidlrectionnels et p6riodiques

labyrinths and 14 are 2-vertex labyrinths. As before, the 14 2-vertex labyrinths comprise reciprocal labyrinths, leaving a total of 7 distinct 2-vertex labyrinths.

This procedure can be applied systematically to the remaining four of the 8 grids of Fig. 2. The grid based on the - fundamental region in Fig. 2c is composed of 12-connected and 4connected vertices and generates 4 distinct 2-vertex regular-faced labyrinths composed of 2 nodal polyhedra. The grid based on the fundamental region in Fig. 2d is composed of 4connected vertices and generates 4 regular- faced space labyrinths. Of these, 2 are the known 1-vertex labyrinths composed of prismatic nodal polyhedra, and the remaining two are a reciprocal pair of 2-vertex labyrinths. This leads to 1 topologically distinct labyrinth composed of two different nodal prisms obtained from this grid. The grid based on the fundamental region in Fig. 2f is the “octet truss” composed of 12connected vertices. It generates 2 new 1-vertex labyrinths, each composed of one nodal polyhedron. The grid based on the fundamental region in Fig. 2h is composed of 8-connected and 4connected vertices. It generates a total of 5 distinct labyrinths.

The total number of distinct “multidirectional” (Burt’s term) space labyrinths composed of regular-faced nodal polyhedra connected by regular prisms and anti-prisms and based on the eight grids equals 22 1-vertex labyrinths and 51 distinct 2-vertex space labyrinths. By varying the lengths of edges of the nodal and connector polyhedra in any labyrinth, an infinite number of labyrinths with varying sizes can be obtained. These are easily visualized as “intermedi- ates” in the continuous transformations between the labyrinths within their hypercubic organization. Additional 1-vertex labyrinths are obtained by shrinking the “height” of the regular prism connectors to zero. Such connector-less 1-vertex labyrinths are composed only of nodal polyhedra and can be derived in an alternative manner by removing appropriate faces from close-packings of regular and semi- regular polyhedra.

The work is presently in progress and has been restricted to multidirectional labyrinths derived from the tetrahedral and octahedral symmetries of polyhedra. 1 -vertex and 2-vertex labyrinths based on snub polyhedra have been constructed and remain to be included. At the time of this writing, the study has been extended to “multi- layered” labyrinths composed of one and two nodal polyhe-

mentale de la figum 2d comporte des sommets dindice 4 et engendre quatre labyrinthes spatiaux a faces regulieres. Parmi ceuxci, deux sont des labyrinthes connus a 1 sommet composes de polyedres nodaux prismatiques et les deux autres constituent une paire reciproque de labyrinthes a 2 sommets. De cette grille, nous obtenons donc un seul labyrinthe topologiquement distinct compose de deux prismes nodaux differents. La gnlle basee sur la region fondamentale de la figum 2f est le reseau octet compose de sommets d’indice 12. I1 engendre deux nouveaux labyrinthes a 1 sommet, chacun compose dun polyedre nodal. La gnlle basee sur la region fondamentale de la figurn 2h comporte des sommets dindice 8 et des sommets dindice 4. Elle engendre un total de cinq labyrinthes differents.

Le nombre total de labyrinthes spatiaux distincts (( multidirectionnels )) (le terme est de Burt) composes de polyedres nodaux a faces regulieres relies par des prismes reguliers et des antiprismes, et fondes sur les huit gdles, est de 22 labyrinthes a 1 sommet et 51 labyrinthes distincts a deux sommets. En faisant varier, dans un labyrinthe, les longueurs des arktes des polyedres nodaux et des polyedres de jonction, on peut obtenir un nombre infini de labyrinthes de tailles differentes. On peut facilement les visualiser comme des ((etapes intermediaires)) dans les transformations continues entre les labyrinthes a l’interieur de leur organisation hypercubique. On obtient des labyrinthes a 1 sommet addi- tionnels en reduisant la ((hauteur )) des prismes reguliers de jonction a zero. De tels labyrinthes a 1 sommet sans jonction ne sont composes que de polykdres nodaux, et on peut les obtenir dune autre maniere en supprimant les faces adequates de remplissages par juxtaposition de polyedres reguliers et semi-reguliers.

La recherche se poursuit encore, et a ete limitee aux labyrinthes multidirectionnels derives des symetries tetra- bdres et octaedres de polyedres. I1 reste a inclure les labyrinthes a 1 sommet et a deux sommets bases sur des polyedres adoucis et, au moment d’ecrire ces lignes, la recherche s’etendait aux labyrinthes (( multicouches )) composes dun ou de deux polyedres nodaux et bases sur des remplissages periodiques de l’espace par des prismes reguliers. Nous les presenterons plus loin.

On fait une allusion aux labyrinthes spatiaux non periodiques dans [7] et un exemple a ete publie dans [8]. D’autres exemples developpes par l’auteur demeurent encore non

Topologie structurale * 21 * Structural Topology * 1995

dra and based on periodic space-fillings of regular prisms. These will be presented later.

Non-periodic space labyrinths were hinted at in [7], and one example published in [8]. Other examples developed by the author remain unpublished to date. We note that the families of 1- and 2-vertex labyrinths can be extended to 3- vertm, 4vertex , ......... n-vertex labyrinths. Some examples of multiple-vertex labyrinths composed of non-regular polygo- nal faces were illustrated in [9] and were based on the explosion-implosion of the fundamental region. The first example of a 3-vertex labyrinth with regular faces was discovered recently by Sharon Blmor in author’s morphology studio at Pratt. Such multiple-vertex labyrinths provide an area of hrther systematic study. One example of a collapsible labyrinth with cubic symmetry (Schlafli symbol 4 9 was developed by students Albert Quinones and Marc Ricketts, open- ing up the possibility of other collapsible labyrinths.

Other areas of further study include: the conversion of multiple-vertex labyrinths into close-packings of polyhedra, the derivation of curved-space labyrinths from plane-faced labyrinths, and so on. Some examples of periodic curved space labyrinths were described in [lo], and the concept extends to non-periodic labyrinths. ndimensional analogs of the space labyrinths descnied here are possible, and some additional hyper-space labyrinths have been worked out by the author (unpublished). In addition, space labyrinths in hyperbolic space in 2-, 3- and higher dimensions are also possible. The crystallographer Alan Mackay (per- sonal communication, 1987) mentions the concept of labyrinths that divide space into three or more parts !

Author‘s note and Acknowledgements From the early beginnings of the ideas in Refs. 1 and 2, this project has been carried out (since 1987) in author‘s Mor- phology Research Studio, a directed research studio in the School of Architecture, Pratt Institute. The first major exhi- bition of the work was held at Pratt in Fall 1991. A year later, the project was exhibited at the Cathedral of St. John the Divine, New York, in November 1992 along with the publication of a small fold-out brochure by Pratt [ll]. This brochure is available in limited edition from the School of Architecture, Pratt Institute, Brooklyn, New York 11205. The brochure shows detailed images of one family, but the total number of labyrinths was incorrectly reported there.

publies. On remarque que les familles de labyrinthes a 1 sommet et a 2 sommets peuvent 6tre etendues a des labyrinthes a 3 sommets, 4 sommets, . . . n sommets. On trouvera dans [9] des illustrations de certains exemples de labyrinthes a multiples sommets composes de faces polygonales non regulieres et bases sur l’explosion-implosion de la region fondamentale. Le premier exemple d’un labyrinthe a 3 sommets de faces regulieres a ete decouvert recemment par Sharon Tilmor a l’atelier de morphologie de l’auteur a Pratt. De tels labyrinthes a multiples sommets constituemnt le sujet dune nouvelle recherche systematique. Les etu- diants Albert Quinones et Marc Ricketts ont developpe un exemple dun labyrinthe pliant possedant une symetrie cubique (symbole de Schlafli 4‘7 ouvrant la voie vers la pos- sibilite dautres labyrinthes pliants.

dans les domaines suivants : la conversion des labyrinthes a multiples sommets en remplissages par juxtaposition de polyedres, la derivation de labyrinthes spatiaux courbes a partir de labyrinthes a faces planes, et ainsi de suite. Dans [lo], on decrit certains exemples de labyrinthes spatiaux periodiques courbes et le concept s’etend aux labyrinthes non periodiques. I1 est possible detablir des analogues de dimension n des labyrinthes spatiaux decrits ici, et l’auteur a m i s en evidence certains labyrinthes hyperspatiaux (travaux non publics). De plus, les labyrinthes spatiaux sont aussi possibles dans un espace hyperbolique de dimension deux, de dimension trois et de dimensions plus elevees. Le cristallographe Alan Mackay (communication personnelle, 1987) mentionne le concept de labyrinthes qui divisent l’espace en trois parties ou plus !

De nouvelles recherches peuvent aussi Ctre poursuivies

Notes de I’auteur et rememements Depuis le tout debut des idees emises en [l] et [2], on a rea- lise ce pmjet (depuis 1987) au Morphology Research Studio de l’auteur, un atelier de recherche dirigee a l’ecole darchitec- ture de 1’Institut Pratt. La premiere exposition majeure du travail s’est tenue a Pratt a l’automne 1991. Une annee plus tard, on a expose le projet a la cathedrale St. John the Di- vine, a New York, en novembre 1992 en m&me temps que la publication d’une petite brochure par 1’Institut Pratt [ll]. Cette brochure, dont l’edition est limitee, est disponible en s’adressant a School of Architecture, Pratt Institute, Broo- klyn, New York 11205. On y trouve des images detaillees

Families of Multi-directional Periodic Space Labyrlnths

Familles da labyrinthes spatiaux multidirectionnelr a1 pbriodiques

References / R6f6rences

All models have been executed by students during the period 1987-92. Credits for Fig. 6 (Fall 1991): E. Diga, A. Jachnik-Nass, R Hall, M. Macker, H. Sripadanna, S. lhlmor; Fig. 7 (Spring 1991): M. Campione, N. Caspi, J. Cormier, K. Crowley and E! Worthington; Fig. 8 (Spring 1992): G. Amaral, R Brooks, Y. Choi, G. Haddar, T. Nakajima, D. Nguyen, A. Oranchak, C. Rolfe, A. Yeh and R Zeismann; Fig. 9 (Fall 1988) : R. Hoppe, A. Keel, l? Ng, S. Schickendanz, L. Schnieder and M. Yaloz. Photo credit for Figs. 7 and 9: Bill Kontzias. The author wishes to thank the School of Archi- tecture, Pratt Institute, for its continued support.

[l] Lalvani, H. (1981). Multi-Dimensional PenorLC Arrangements of Ttansforrning Space Structures. Ph.D. Thesis, Univ. of Pennsylvania, 1981; published by University Microfilms, Ann Arbor, Michigan, 1982.

[2] Lalvani, H. (1982). SWu&m.s on Hyper- stncctures. Lalvani, New York.

[3] Lalvani, H. (1982). "Structures on Hyper- Structures." Structural 'hpology, 6, 13-16.

[4] Coxeter, H.S.M. (1968). ltvelue Geometric Essays. Southern University Press, Carbondale.

dune famille, mais le nombre total de labyrinthes y est in- correctement rapporte.

?bus les modeles ont ete rkalises par des Ctudiants pen- dant la periode 1987-1992. La figure 6 (automne 1991) est attriiuable a E. Diega, A. Jachnik-Nass, R Hall, M. Macker, H. Sripadanna et S. Tklmor; la figure 7 (printemps 1991) est le resultat des travaux de M. Campione, N. Caspi, J. Cor- mier, K. Crowley et l? Worthington ; la figure 8 (printemps 1992) : G. Amaral, R Brooks, Y. Choi, G. Haddar, T. Nakajima, D. Nguyen, A. Oranchak, C. Rolfe, A. Yeh et R Zeismann ; la figure 9 (automne 1988) : R Hoppe, A. Keel, €! Ng, S. Schickendanz, L. Schnieder et M. Yaloz. Les photos des figures 7 et 9 sont de Bill Kontzias. Eauteur tient a remercier l'ecole &architecture de l'Institut Pratt pour son soutien continu.

I [S] Burt, M., Kleinman, M., & Wachman, A.

(1 974). Infinite hlyhedra, Technion, Haifa, Israel.

[6] Shubnikov, A.V. & Koptsik, V.A. (1974). Symmehy in Science and Art. Plenum, New York.

[7] Lalvani, H. (1986-8T). "on-periodic Space Structures." Space Structures, Vol. 2 No.2, PP-PP.

[8] Lalvani, H. (1986). "Non-periodic Space- fillings of Golden Polyhedra." Pmceedings LSA86, First Internntional Sy?nposium on Lghtwight Structures in Architecture, Sydney.

[9] Lalvani, H. (1991). "Morphological Aspects of Space structures." In: Studies in Space Structures, ed. H. Nooshin, MultiScience Publ.,UIC

Morphology of Curved Space Structures!' Proceedings Vol. 1, htemational Symposium on Architectural Fabric Structures, IAFS, Glenview, Illinois.

[ll] Lalvani, H. (1992). Metarnorpholosy and Space Labyrinths, School of Architecture, Pratt Institute, New York.

[lo] Lahani, H. (1984). 'Generative

Documents

Topologle structurale 21 Structural Topology 1995