Snub Disphenoid

Total Page:16

File Type:pdf, Size:1020Kb

Snub Disphenoid� E COLE NORMALE SUPERIEURE ________________________ Lists of faceregular p olyhedra Gunnar BRINKMANN Michel DEZA LIENS ________________________ Département de Mathématiques et Informatique CNRS URA 1327 Lists of faceregular p olyhedra Gunnar BRINKMANN Michel DEZA LIENS November Lab oratoire dInformatique de lEcole Normale Superieure rue dUlm PARIS Cedex Tel Adresse electronique dezadmiensfr Fakultat f ur Mathematik Universitat Bielefeld D Bielefeld Adresse electronique gunnarmathematikunibielefeldde Lists of Faceregular Polyhedra Gunnar Brinkmann Michel Deza Fakultat f ur Mathematik CNRS and LIENSDMI Universitat Bielefeld Ecole Normale Superieure y D Bielefeld Paris France Abstract We introduce a new notion that connects the combinatorial concept of regularity with the geometrical notion of facetransitivity This new notion implies niteness results in case of b ounded maximal face size We give lists of structures for some classes and investigate p olyhedra with constant vertex degree and faces of only two sizes Introduction A planar nite or innite graph is called facetransitive if the automorphism group acts transitively on the set of faces For nite p olyhedra see Ma as well as for innite graphs in the plane with nite faces and nite vertex degree that is tilings see BaDe it is well known that the graph can b e realized with its full com binatorial automorphism group as its group of geometrical symmetries Restricting the attention to p olyhedra with constant vertex degree up to combinatorial equiva lence only the Platonic solids have an automorphism group acting transitively on their faces In the remaining text we will restrict our attention to p olyhedra with constant vertex degree A natural generalisation of this concept let us call it weakly facetransitive is to require that only faces of the same size are equivalent under the automorphism group If we dene the th corona of a face to b e the face itself and the nth corona to b e the set of all those faces that are contained in the nth corona or share an edge with it we can further relax this concept and only require some coronas of xed size to b e isomorphic by an isomorphism mapping the central faces onto each other A p olyhedron with all ncoronas of faces of the same size isomorphic is called weakly ntransitive Obviously all p olyhedra are weakly transitive and if a p olyhedron is weakly ntransitive it is also weakly ntransitive So the rst interesting case to study is the case of weakly transitive p olyhedra Still relaxing this condition by not requiring the rst coronas to b e isomorphic but just to b e isomorphic as multisets that is every face of a given size i must have the same number of neighbours of size i for every i still gives a very restrictive condition email gunnarmathematikunibielefeldde y email dezadmiensfr and as we will see it already implies niteness in case the maximal size of a face is b ounded We call this condition strong faceregularity So the class of all face regular p olyhedra contains all weakly nfacetransitive p olyhedra for any n and therefore also the weakly facetransitive or even facetransitive ones The same concept can b e reached by strengthening the notion of a mono chro matic regular dual Let p denote the number of igons in a given p olyhedron We use i the notation p p p p p for the facevector or pvector of a p olyhedron i b b is the maximal number for which a face with size b exists A less restrictive denition of faceregularity but only for bifaced p olyhedra was considered in DGrc Namely if only p and p are nonzeros and a b a b then the number f of ifaces edgeadjacent to any given iface was required to b e indep endent of the choice of the iface for i either a or b For a k valent p olyhedron we write aR or bR if this partial or weak faceregularity holds for agonal or f f resp ectively bgonal faces All such simple p olyhedra with b as well as all valent ones with b except the cases R for k a b and aR aR for k a b f g were found in DGrc For example all resp p olyhedra bR for all ve p ossible cases k k b and f k b a f g are listed there The graphs of all R ful lerenes f ie k a b are given in list b elow In these cases resp p olyhedra are also aR ie faceregular in the sense of the present pap er f The faceregularity which we consider is a purely combinatorial prop erty of the skeleton of a p olyhedron It is dierent from the ane notion of regularfaced ie all faces b eing regular p olygons p olyhedra We use the abbreviation frp for faceregular polyhedron An frp in one of the lists b elow is describ ed by i where j is the number of the List and i is its number in j List j We also use the notation i for i We call two frp frisomers if they have the same parameters as frp ie v the pvector and the numbers f a b ie the number of bfaces edgeadjacent to each aface for any a b coincide All frisomers in List are bifaced They are v v v and faced v All frisomers in List are for v for v for v for v for v for v for v for v All frisomers in Lists and are with v Considering the p olyhedra of Lists and with resp ect to collapsing of all triangular faces to p oints ie the inverse to vertextruncation we see that in List any such collapsing gives a member of List But in List there are p olyhedra such that this collapsing do es not give an frp The smallest one is Examples of sequences of frp such that each of them comes from the previous one by edge truncation are and Some innite families of valent frp Bifaced Prism and Barrel ie two ngons separated by two layers of gons n n faced Prism Barrel truncated on all n vertices of b oth ngons n n Prism edgetruncated on n disjoint edges of only one ngon n Prism edgetruncated on n edges separated by at least edges of only one n ngon faced Prism vertex truncated on all vertices of only one ngon n faced Barrel truncated on all vertices of only one ngon n In fact many of the frp in the lists are some partial truncations of Prism and n Barrel For example there are exactly frp which are partial truncations of the n Cub e There are resp p ossibilities for truncations on resp vertices Remarks i Among the chiral p olyhedra in the lists are for example Nrs in List Nr in List and esp ecially Nrs in List and in List with symmetry T O I and O resp ectively ii None of the p olyhedra in any of our Lists has a trivial symmetry group The Finiteness of Classes with Bounded Face Size Theorem For every n N there is only a nite number of faceregular polyhedra with constant vertex degree and face sizes not exceeding n Pro of We will assume that the p olyhedra in question all contain an ngon The total number can b e obtained by summing up over all m n Remind that for i j N the number f i j denotes the number of neighbouring jgons of an igon So f i j p f j ip is the number of edges b etween igons and i j f ij jgons and we can express p as p p in case igonal and jgonal faces share j j i f ji at least one edge Lo ok at the fgraph G with vertex set V fijp g and edge set E i ffi j gjf i j g This graph is connected since the dual of the underlying p oly hedron is connected We can express every other value p by a formula of the kind i f i i f i i f i i 1 2 1 b k p g ip n n f ii f i i f i i 1 1 2 k b if i i i n is a eg shortest path from i to n in G k Since for xed n all the f i j as well as the length of the path are b ounded and since the number of graphs on n vertices is also nite we have only a nite number of p ossible sets of equations p g ip i n i n P n As a well known consequence of Eulers formula we get ip in the i i P P n n ip ip for valent p olyhedra and valent case i i i i for valent p olyhedra Substituting p by g ip in this formula every set of equations gives exactly one i n solution for p and therefore also for each p So for every set of equations there is a n i well determined number of faces and therefore there is a maximum number of faces that is p ossible 2 Corollary If in the cubic case the number of nonhexagons is bounded or in the quartic case the number of nonsquares is bounded then there is only a nite number of faceregular polyhedra Pro of The fact that the number of faces smaller than resp is b ounded gives an upp er b ound on the maximum face size implying the result by the previous theorem 2 Statistics In this section we will give some statistics ab out the number of faceregular p olyhedra compared to the number of all p olyhedra for some classes vertices p olyhedra faceregular p olyhedra p olyhedra faceregular p olyhedra vertices Table Cubic p olyhedra Table Cubic p olyhedra without triangles p olyhedra faceregular p olyhedra vertices Table Cubic p olyhedra without faces larger than a hexagon For all vertex numbers not mentioned no faceregular p olyhedra exist List all faceregular simple p olyhedra with b 6 Among the p olyhedra of the List the rst three are regular then there are bifaced ones six with b four for n nine for n n n and fullerenes which are F D F T n d d F D F C F T F T F T F I F T F I F D h v d h d
Recommended publications
  • Icosahedral Polyhedra from 6 Lattice and Danzer's ABCK Tiling
    Icosahedral Polyhedra from 퐷6 lattice and Danzer’s ABCK tiling Abeer Al-Siyabi, a Nazife Ozdes Koca, a* and Mehmet Kocab aDepartment of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khoud, 123 Muscat, Sultanate of Oman, bDepartment of Physics, Cukurova University, Adana, Turkey, retired, *Correspondence e-mail: [email protected] ABSTRACT It is well known that the point group of the root lattice 퐷6 admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group 퐻3 , its roots and weights are determined in terms of those of 퐷6 . Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of 퐷6 determined by a pair of integers (푚1, 푚2 ) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of 퐻3 and it is shown that the inflation of the tiles can be obtained as projections of the lattice vectors characterized by the pair of integers which are linear combinations of the integers (푚1, 푚2 ) with coefficients from Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the < 퐴퐵퐶퐾 > octahedral tilings in 3D space with icosahedral symmetry 퐻3 and those related transformations in 6D space with 퐷6 symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in 퐷6 .The tetrahedron K constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of “K-polyhedron”, “B-polyhedron” and “C-polyhedron” generated by the icosahedral group have been discussed.
    [Show full text]
  • Bilinski Dodecahedron, and Assorted Parallelohedra, Zonohedra, Monohedra, Isozonohedra and Otherhedra
    The Bilinski dodecahedron, and assorted parallelohedra, zonohedra, monohedra, isozonohedra and otherhedra. Branko Grünbaum Fifty years ago Stanko Bilinski showed that Fedorov's enumeration of convex polyhedra having congruent rhombi as faces is incomplete, although it had been accepted as valid for the previous 75 years. The dodecahedron he discovered will be used here to document errors by several mathematical luminaries. It also prompted an examination of the largely unexplored topic of analogous non-convex polyhedra, which led to unexpected connections and problems. Background. In 1885 Evgraf Stepanovich Fedorov published the results of several years of research under the title "Elements of the Theory of Figures" [9] in which he defined and studied a variety of concepts that are relevant to our story. This book-long work is considered by many to be one of the mile- stones of mathematical crystallography. For a long time this was, essen- tially, inaccessible and unknown to Western researchers except for a sum- mary [10] in German.1 Several mathematically interesting concepts were introduced in [9]. We shall formulate them in terms that are customarily used today, even though Fedorov's original definitions were not exactly the same. First, a parallelohedron is a polyhedron in 3-space that admits a tiling of the space by translated copies of itself. Obvious examples of parallelohedra are the cube and the Archimedean six-sided prism. The analogous 2-dimensional objects are called parallelogons; it is not hard to show that the only polygons that are parallelogons are the centrally symmetric quadrangles and hexagons. It is clear that any prism with a parallelogonal basis is a parallelohedron, but we shall encounter many parallelohedra that are more complicated.
    [Show full text]
  • Rhombohedra Everywhere
    Rhombohedra Everywhere Jen‐chung Chuan [email protected] National Tsing Hua University, Taiwan Abstract A rhombohedron is a 6-face polyhedron in which all faces are rhombi. The cube is the best-known example of the rhombohedron. We intend to show that other less-known rhombohedra are also abundant. We are to show how the rhombohedra appear in the algorithm of rhombic polyhedral dissections, in designing 3D linkages and in supplying concrete examples in mathematics amusement. A tongue-twisting jargon: in case all six rhombic faces are congruent, the rhombohedron is known as a trigonal trapezohedron. 1. Minimum Covering for the Deltoidal Icositetrahedron Why choosing deltoidal icositetrahedron? It is known that every convex polyhedron is the intersection of the half spaces defined by the supporting planes. It is not straightforward to find one such 6n-face polyhedron so the supporting planes enclose n rhombohedra exactly. It turns out that the deltoidal icositetrahedron, having 24 congruent “kites” as its faces, fits such requirement. The coloring scheme below shows the possibility of painting the faces with four colors so each edge-adjacent faces shall receive distinct colors: Figures 1(a) and (b) It turns out that each fixed color, say the blue, occupies exactly three pairs of parallel faces. The six supporting planes enclosed the red-edge rhombohedron. This together with green-, yellow- red- edge rhombohedra cover all faces. The rich symmetries possessed by the deltoidal icositetrahedron enables us to visualize the interplay between the dynamic geometry and combinatorial algorithm. Figure 2 2. Minimum Covering for the Rhombic Icosahedron Unlike the deltoidal icositetrahedron, no choice for any collection of six faces of the rhombic icosahedron can have non-overlapping vertices.
    [Show full text]
  • Polytopes and Projection Method : an Approach to Complex Structures R
    POLYTOPES AND PROJECTION METHOD : AN APPROACH TO COMPLEX STRUCTURES R. Mosseri, J. Sadoc To cite this version: R. Mosseri, J. Sadoc. POLYTOPES AND PROJECTION METHOD : AN APPROACH TO COMPLEX STRUCTURES. Journal de Physique Colloques, 1986, 47 (C3), pp.C3-281-C3-297. 10.1051/jphyscol:1986330. jpa-00225742 HAL Id: jpa-00225742 https://hal.archives-ouvertes.fr/jpa-00225742 Submitted on 1 Jan 1986 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. JOURNAL DE PHYSIQUE Colloque C3, supplement au no 7, Tome 47, juillet 1986 POLYTOPES AND PROJECTION METHOD : AN APPROACH TO COMPLEX STRUCTURES R. MOSSERI' and J.P. SADOC**(~) "~aboratoirede Physique des Solides, 1, Place A. Briand, F-92195 Meudon, France i * Imperial College, Physics Department, GB-London SW7 282, Great-Britain ~ksurn6: La mise en place de motifs atomlques dans les espaces de grandes dimensions est envisagde dans des cas simples en rapport avec la thgorie des espaces courbes pour les structures n~n-~&riodiques. Abstract: Decoration with atomic motifs in high dimension space is considered in simple cases, and related to the theory of curved spaces in non-periodic structures. I - INTRODUCTION The discovery of AlMn /1/ quasicrystals has raised the urgency for a geometrical description of orientationally ordered 3D structures without periodicity.
    [Show full text]
  • Icosahedral Polyhedra from D6 Lattice and Danzer's ABCK Tiling
    S S symmetry Article Icosahedral Polyhedra from D6 Lattice and Danzer’s ABCK Tiling 1 1, 2, Abeer Al-Siyabi , Nazife Ozdes Koca * and Mehmet Koca y 1 Department of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khoud, Muscat 123, Oman; [email protected] 2 Department of Physics, Cukurova University, Adana 1380, Turkey; [email protected] * Correspondence: [email protected] Retired. y Received: 10 November 2020; Accepted: 26 November 2020; Published: 30 November 2020 Abstract: It is well known that the point group of the root lattice D6 admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H3, its roots, and weights are determined in terms of those of D6. Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of D6 determined by a pair of integers (m1, m2) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of H3, and it is shown that the inflation of the tiles can be obtained as projections of the lattice vectors characterized by the pair of integers, which are linear combinations of the integers (m1, m2) with coefficients from the Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the <ABCK> octahedral tilings in 3D space with icosahedral symmetry H3, and those related transformations in 6D space with D6 symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in D6.
    [Show full text]
  • Cube” – a Hidden Relationship of a Topological Nature Between the Cube and Truncated Octahedron
    International Journal of Scientific Engineering and Science Volume 2, Issue 1, pp. 1-4, 2018. ISSN (Online): 2456-7361 A Generalized Formulation of the Concept of the “Cube” – A Hidden Relationship of a Topological Nature between the Cube and Truncated Octahedron Nikos Kourniatis Assistant Professor, Department of Civil Engineering, Piraeus University of Applied Sciences Abstract— This first objective of this paper is to search for higher-dimensional cube structures and explore their relationship with certain Archimedean polyhedra. In particular, the research focuses on the relationship between the six-dimensional cube and the truncated octahedron. The problem is approached through the study of higher-dimensional hypercube projections in three-dimensional space and, subsequently, on a plane, and the finding that each n-hypercube can result from a composition of lower-dimensional cubes. This procedure basically leads to the generalization of the concept of the cube through its successive projections in different dimensions, and finally reveals a relationship of a topological nature between the cube and Archimedean solids, such as the truncated octahedron. Keywords—Cube, polyhedra topology, truncated octahedron. I. THE CONCEPT OF THE CUBE IN THE DIFFERENT II. THE EXAMPLE OF N-HYPERCUBES DIMENSIONS – SUCCESSIVE PROJECTIONS A three-dimensional cube can be projected parallel to any vector in a certain plane, thus providing an axonometric image of it. A lower-dimensional shape. According to Pohlke's theorem, if we map any axonometric image of a cube on a plane, then there is always a vector (d) parallel to which the cube was projected, thus providing this particular image. This means that a cube has infinite axonometric images on the plane, some of which provide a high level of monitoring of the cube, while others do not.
    [Show full text]
  • SUSY Equation Topology, Zonohedra, and the Search For
    x**University of Maryland * Center for String and Particle Theory* Physics Department***University of Maryland *Center for String and Particle Theory** x x x x x x x x x x x x x x x x x x x x Dec 2011§ ¤ UMDEPP 011-020 x x x x x x ¦ ¥ x x hep-th/1201.0307 x x x x x x x x x x x x x x x x x x SUSY Equation Topology, x x x x x x x x Zonohedra, and the Search for x x x x x x x x Alternate Off-Shell Adinkras x x x x x x x x x x x x x x x x x x 1 2 x x Keith Burghardt , and S. James Gates, Jr. x x x x x x x x x x Center for String and Particle Theory x x x x x x Department of Physics, University of Maryland x x x x x x College Park, MD 20742-4111 USA x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x ABSTRACT x x x x x x x x x x x x Results are given from a search to form adinkra-like equations based x x x x x x on topologies that are not hypercubes.
    [Show full text]
  • Cambridge University Press 978-1-107-41160-9 - New Geometries for New Materials Eric a Lord, Alan L Mackay and S Ranganathan Index More Information
    Cambridge University Press 978-1-107-41160-9 - New Geometries for New Materials Eric A Lord, Alan L Mackay and S Ranganathan Index More information Index Page numbers in italics refer to illustrations ␣-Al–Mn–Si, 57 Aston, M. W., 67 ␣-boron, 89 asymmetric unit, 11, 50, 172 ␣-helix, 207 Atiyah, M. & Sutcliffe, P., 109 ␣-manganese, 100, 101, 153, 192, 203 Atlas of Zeolite Framework Types, 126, 136, Abe, E. et al., 200 139 AcrobatReader, 210 Audier, M. et al., 94, 202 Acta Crystallographica, 133, 203 Audier, M. & Duneau, M., 76 adjoint surfaces, 163 Audier, M. & Guyot, P., 85, 86, 102, 106 Al–Co, 104 Algorithmic Beauty of Plants, The,78 ␤-brass 194 alloys, 193 ␤-crystobalite, 41, 137, 139 Almagest, 4 ␤-manganese, 114, 115, 207 Almgren, F. J., 162 ␤-quartz, 111, 138, 139, 144 Al–Mn, 102, 105 ␤-tungsten (␤-W), 44, 45, 145 Al–Ni–Co, 21, 104, 105, 201 Baer, S., 107 aluminosilicate, 136 Baerlocher, Ch. et al., 136, 137, 139 Amelinckx et al., 121 balance surface, 14, 149, 168 Ammann, R., 18, 55, 199 Barlow, W., 4 Ammann bars, 16 BASIC, 209 Ammann tilings, 85, 86 Baumé, A., 3 Anaxagoras, 1 Bausch, A. R. et al., 71 Andersson, S., 161 Belin, C. H. E. & Belin, R. C. H., 98 Andersson, S. & Hyde, S. T., 182, 186 Bergman, G. et al., 92, 94, 202 Andreini, A., 36 Bergman cluster, 91, 92, 95, 203 anti-Kythera mechanism, 1 Bernard, C., 6 aperiodic tiling, 16, 55, 84, 85 Bernal, J. D., 66, 198, 199, 204 Applebaum, J. & Weiss, Y., 69 Bernal, J.
    [Show full text]
  • Golden Ratio Group” and Those of “Square-Root-Of-2 Ratio Group”
    Original Paper ________________________________________________________________________________________ Forma, 36, 1–14, 2021 Close Relationship between Ratios of “Golden Ratio Group” and Those of “Square-Root-of-2 Ratio Group” Hiroaki Kimpara Kawasaki Branch, Mathematical Association of Japan, 255 Ohkubo, Sakura-ku, Saitama 338-8570, Japan E-mail: [email protected] (Received February 3, 2020; Accepted September 22, 2020) While the golden ratio (1:φ) has been well known since the ancient Greece age and has already been studied in depth by many researchers, the square-root-of-2 ratio (1: 2 ) and other ratios of this group have gone largely unnoticed until now, despite the fact that the rhombic dodecahedron, one of the most important poly- hedra, is based on the square-root-of-2 ratio. According to a popular theory, these ratios are independent of one another. Against it, however, the author found out that there exists close relationship between the ratios of the golden ratio group and those of the square-root-of-2 ratio group through careful study on three kinds of rhombic polyhedron. Key words: Golden Ratio, Square-Root-of-2 Ratio, Rhombus, Rhombic Polyhedron, Dihedral 1. Introduction gon to a specific diagonal thereof. These ratios also rep- Until now, it has been the established fact that the golden resent the length ratios between two diagonals of spe- ratio is an independent ratio with no specific relationship cific rhombuses, and each of such rhombuses constitutes with other special ratios like the square-root-of-2 ratio. a specific group of rhombic polyhedron. These findings Against it, however, the author puts forth the following are explained in Sec.
    [Show full text]
  • Donald Golden Rhombohedra
    DONALD AND THE GOLDEN RHOMBOHEDRA Marjorie Senechal Models: RHOMBO, a geometric "toy" designed by Smith College Michael Longuet-Higgins . and much enjoyed by Donald Northampton, MA 1 Smith College April, 1984 The Five Golden Isozonohedra A rectangle having sides in the ratio :1 is a golden rectangle, a rhombus having diagonals in this ratio is a golden rhombus. A6 O6 HSM Coxeter, "The rhombic triacontahedron," Symmetry 2000, acute obtuse edited by I. Hargittai and T. C. Laurent, Portland Press Ltd., 2002. rhombus rhombus F20 Federov = (1 + √5)/2 rhombic icosahedron Golden rectangle Golden rhombus K30 Kepler triacontahedron Bilinski rhombic dodecahedron B12 The triacontahedron's face has diagonals in the "golden And a rhombohedron whose faces are six congruent golden section" ratio 1:. A model can be built up from twenty rhombi is a golden rhombohedron. Such a rhombohedron has rhombohedra, ten acute and ten obtuse, bounded by such an axis of trigonal symmetry and is said to be obtuse or acute rhombs. Regular Polytopes, 2nd edition, 1963 according to the nature of the three face angles at a vertex on this axis. Coxeter, 2002, contd. 2 By removing one of the p zones from a rhombic zonohedron and bringing together the two remaining pieces of the surface, one obtains a simpler zonohedron, with p-1 replacing p. In this manner the triacontrahedron yields the rhombic icosahedron . This in turn has a zone of eight faces, which can be removed so as to yield the two halves of a rhombic dodecahedron (p = 4) -- not Keplers ... PENROSE'S KITES AND DARTS: 7 legal arrangements of tiles around a vertex Donald was kite dart unaware of Robert Ammann’s 3-D Penrose analogues.
    [Show full text]
  • Assembling 10 Rhombic Hexahedrons Into a Rhombic Icosahedron
    Assembling 10 Rhombic Hexahedrons into a Rhombic Icosahedron Jen-chung Chuan In this workshop with Cabri 3D we are to assemble 10 rhombic hexahedrons to form a rhombic icosahedron. A. Construction of the rhombic icosahedron by taking the convex hull of 10 rhombi: 1) Five rhombi on top: these are the same as five faces on top of the rhombic triacontahedron, obtained by taking the convex hull of a regular dodecahedron with the icosahedron “in dual position”: 2) Five rhombi at the bottom: these are the reflections of the rhombi in 1) w.r.t. the upward translation of the center of triacontahedron by half length of the vertical edge: 3) The required rhombic icosahedron is formed by take the convex hull of the 10 rhombi: B. Construction of two basic building blocks for the rhombic icosahedron: Flat block: the "flat" rhombic hexahedron Rounded block: the “rounded" rhombic hexahedron Each of the two blocks can be constructed as a prism with a face of rhombic tricontahedron as base. C. Two possible "growths” associated with blocks asssembly: Growth 1: Five rounded blocks followed by five flat blocks. 1) Starting with the rounded block, build four other congruent pieces by taking appropriate plane reflections. This completes the growth for the rounded blocks. 2) Build one flat block from the pocket formed by the rounded blocks. This "core" will serve as the only "hidden" pieces after the assembly. 3) Build four other flat blocks by taking appropriate plane reflections of the core. Growth 2: This procedure reminds us of the Greedy Algorithm. 1) Starting with the (flat) core.
    [Show full text]
  • St10-Ocr.Pdf
    revue interdisciplinaire de g&om&rieappliquke aux probl&mesde structure et morphologieen design, architecture et gCnie PARUTION NUMtiRO: ISSUE NUMBER: 10 interdisciplinary journal on geometryapplied to problemsof structure and morphologyin architecture, designand engineering La revue est publiee par le groupe de recherche ‘Topologie The Journal is published by the Structural Topology L’Cquipe de ce numkro / Staff for this issue structurale’, avec la collaboration de I’Association math& research group, with the collaboration of the Association matique du Quebec et de 1’UniversitC du Quebec a mathematique du Quebec and the Universite du Quebec a Montreal. Montreal. Redaction/Editors Henry Crapo Jacques Arc hambault Cornit de direction / Management committee Janos J. Baracs Michel Fleury Walter Whiteley, president et representant du groupe de recherche ‘Topologie structurale’/president and representative of Jean-Luc Raymond the Structural Topology research group, Champlain Regional College, St. Lambert, Quebec. Walter Whi teley Janos J. Baracs, representant de son Ccole/representative of his school, Ecole d’architecture, Universite de Montreal. Traduction/Translation Henry Crapo Henry Crapo, redacteur en chef et membre ex officio/editor in chief and ex officio member, departement de mathemati- Jean-Luc Raymond ques, Universite du Quebec a Montreal. Francine Lefebvre Michel Fleury, reprtsentant de son dCpartement/representative of his department, departement de design, Universite du Jacques Archambault Quebec a Montreal. Typographic/Typesetting Pierre Leroux, representant de son departement/representative of his department, departement de mathematiques, Carole Deslandes UniversitC du Quebec a Montreal. Modulo Editeur Outremon t, Quebec Richard Pallascio, representant de 1’Association mathematique du Quebec/representative of the Association mathemati- que du Quebec, College Edouard-Montpetit, Longueuil, Quebec.
    [Show full text]