DOCSLIB.ORG

Naming Archimedean and Catalan Polyhedra and Tilings

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Naming Archimedean and Catalan Polyhedra and Tilings -21- Naming Archimedean and Catalan Polyhedra and Tilings The book in which Archimedes enumerated the polyhedra that have regular faces and equivalent vertices is unfortunately lost; however, its contents were reconstructed by Kepler, from whom the tradi- tional names descend. In this chapter we explain these “Keplerian” names for the Archimedean and Catalan solids and extend them to the analogous tessellations of two- and three-dimensional Euclidean space. We shall describe the polyhedra in dual pairs indicated by the ar- rows, and at the same time give our abbreviations for them, starting with the five Platonic (or regular) ones: TC↔ OD↔ I Tetrahedron Cube Octahedron Dodecahedron Icosahedron Etymologically, the Greek stem “hedr-” is cognate with the Latin “sede-” and the English “seat,” so that, for instance, “dodecahedron” really means “twelve-seater.” (opposite page) The Archimedean solids and their duals can be nicely arranged so that their edges are mutually tangent, at their intersections, to a common sphere. 283 © 2016 by Taylor & Francis Group, LLC 284 21. Naming Archimedean and Catalan Polyhedra and Tilings Truncation and “Kis”ing These are followed by their “truncated” and “kis-” versions. Here, truncation means cutting off the corners in such a way that each regular n-gonal face is replaced by a regular 2n-gonal one. The dual operation is to erect a pyramid on each face, thus replacing a regular m-gon by m isoceles triangles. These give five Archimedean and five Catalan solids: truncated truncated truncated truncated truncated Tetrahedron Cube Octahedron Dodecahedron Icosahedron tT tC tO tD tI kT kC kO kD kI kisTetrahedron kisCube kisOctahedron kisDodecahedron kisIcosahedron The names used by Kepler for the Catalan ones were rather longer, namely, triakis tetrakis triakis pentakis triakis Downloaded by [University of Bergen Library] at 04:55 26 October 2016 tetrahedron hexahedron octahedron dodecahedron icosahedron and were usually printed as single words. In these, “kis” meant “times,” so that Kepler’s “tetrakishexahedron” was literally a “(4 × 6)-seater.” We retranslate “kis” as “multiplied,” allowing us to ab- breviate this to “kiscube,” meaning “multiplied cube.”1 1Beware! This is not the same as the “mucube” of later chapters. © 2016 by Taylor & Francis Group, LLC Marriage and Children 285 Marriage and Children We marry a regular polyhedron P and its dual Q by placing them so that corresponding edges intersect at right angles. Then, their daughter polyhedron is the Archimedean polyhedron that is their intersection, whose dual, their son, is the Catalan one that is their Polyhedral Sex convex hull. This gives rise to only two new Archimedean and two new Catalan solids, which we can respectively truncate and “kis” to We call a polyhedron with get two more of each: V vertices and F faces male if V>F(its virility ex- cedes its feminity) and fe- male if F>V.Thosewith V = F are hermaphrodites. truncated truncated CubOctahedron IcosiDodecahedron CubOctahedron IcosiDodecahedron CO ID tCO tID R12 R30 kR12 kR30 Rhombic Rhombic kisRhombic kisRhombic dodecahedron triacontahedron dodecahedron triacontahedron Downloaded by [University of Bergen Library] at 04:55 26 October 2016 The daughter and son of two mutually dual tetrahedra are the reg- ular octahedron and cube, so that particular “marriage” leads to no new Archimedean and Catalan solids. However, we can incestuously marry the two Rhombic solids R12 and R30 to their respective duals CO and ID, producing two more Archimedean daughters and two more Catalan sons: © 2016 by Taylor & Francis Group, LLC 286 21. Naming Archimedean and Catalan Polyhedra and Tilings Trapezium and Trapezoid If you look in a large enough dictionary you will probably find the asser- tions that “trapezium” (Br.) = “trapezoid” (U.S.) while RhombiCubOctahedron RhombIcosiDodecahedron “trapezoid” (Br.) = “trapez- RCO RID ium” (U.S.)! Although only the first of T24 T60 these is still true, it is inter- Tetragonal Tetragonal esting to see how this curi- icosikaitetrahedron hexacontahedron ous situation came about. Proclus, a commentator on Euclid, used “trapezion” for a quadrilateral with (just) two parallel sides and “trapezoid” for a general four sided polygon with (typically) no parallel Strictly speaking, the solids we get by truncating CO and ID sides. All the European languages except English are not Archimedean since they have unequal edges, but they can have maintained this usage. be reformed to become so by distorting them suitably, and Kepler’s However, in English the names refer to the reformed versions. Their duals, the two Catalan words “trapezium” and “trapezoid” were acci- solids we have described as kisrhombic have traditionally been called dentally interchanged in the hexakis octahedron and icosahedron. We prefer to use “hexakis” Hutton’s Mathematical only for the replacement of a hexagon (rather than a triangle) by six Dictionary of 1800 and this switched usage has triangles. persisted in the U.S.A. but Since the traditional names here have often been misunderstood, was corrected in England we shall explain them. The prefix “rhombi-” in rhombicuboctahe- between 1875 and 1900. The end result has been dron, for instance, does not refer to a supposed operation of “rhombi- that British “trapezium” and truncation” but is an abbreviation for “rhombic dodecahedron,” U.S. “trapezoid” have sur- vived as synonyms, while one of the two polyhedra R12 and CO of which RCO is the British “trapezoid” and U.S. daughter. “trapezium” have been ob- In the names of the two Catalan solids, “icosi-kai-tetra” and Downloaded by [University of Bergen Library] at 04:55 26 October 2016 solete for more than a cen- tury, the word “quadrilat- “hexaconta” mean “twenty-plus-four” and “sixty,” respectively, and eral” having been reintro- “tetragonal” refers to the four-cornered-ness of the faces. Tradition- duced to replace them. ally, the adjective has been “trapezoidal,” but this is based on an obsolete meaning of the word “trapezoid” (see sidebar). The two “rhombi” solids have some square faces that come from their father “rhombic” polyhedra, and we can divide each such square into two triangles in such a way as to increase the number of faces at each vertex from four to five and then reform the resulting two solids so that all their faces are regular. © 2016 by Taylor & Francis Group, LLC Coxeter’s Semi-Snub Operation 287 Kepler’s term for the first of the resulting two solids was “cubus simus” in which the second Latin word means “rounded” or “flat- tened.” The “simian” apes are those with flattened noses. The traditional English names are snub cube (sC) and snub dodecahe- dron (sD), although they could equally be described as the snub octahedron and icosahedron. Their duals have respectively 24 and 60 pentagonal faces: snub Cube snub Dodecahedron sC sD P24 P60 Pentagonal Pentagonal icosikaitetrahedron hexacontahedron Coxeter’s Semi-Snub Operation Coxeter pointed out that since the snub cube is equally the snub octahedron, it would be more sensible to regard it as being derived from the cuboctahedron, by a new type of snubbing operation. We call this semi-snubbing, abbreviated to “ssnub,” because it is only Downloaded by [University of Bergen Library] at 04:55 26 October 2016 half of the operation that leads from the cube to its ordinary snub: C → CO → ssCO = sC. Regarded as the semi-snub cuboctahedron, the snub cube is ob- tained by distorting its daughter rhombicuboctahedron by twisting © 2016 by Taylor & Francis Group, LLC 288 21. Naming Archimedean and Catalan Polyhedra and Tilings the regularly embedded (regular) squares in one direction, say, clock- wise, which automatically twists the triangles counterclockwise and turns the non-regularly embedded (half-regular) squares into skew quadrilaterals that can be filled with two triangles. However, this is only a topological description; the canonical snub cube is obtained by a deformation that makes all the faces regular. (If we continue this twisting, until the skew quadrilaterals have no width at all, we obtain the cuboctahedron.) The semi-snub can be defined topologically for an even-valence polyhedron in a similar way, but its faces can all be made regular only for some very special polyhedra. A notable example is the semi-snub mucube of Chapter 23. Euclidean Plane Tessellations Not many of the corresponding tilings of the Euclidean plane have previously received names. However, quadrille (Q), which we can interpret as “quadrangular grille,” has been used for the standard tiling by squares. Based on this we propose deltille (Δ) for that by equilateral triangles and hextille (H) for that by regular hexagons. In these, the termination “tille” may be regarded as an amalgam of “tile” and “grille.” The following table gives the resulting “Ke- plerian” names for the “Archimedean” and “Catalan” tilings of the plane. We have encountered their figures already, in Table 19.1 on pages 262–263. Symbol Archimedean Face Code Catalan Symbol Tiling Tiling Downloaded by [University of Bergen Library] at 04:55 26 October 2016 Q quadrille 4444 quadrille Q Δ deltille 333333 hextille H H hextille 666 deltille Δ tQ tr. quadrille 488 kisquadrille kQ tH tr. hextille 31212 kisdeltille kΔ HΔ hexadeltille 3636 rhombille R∞ tHΔ tr. hexadeltille 4612 kisrhombille kR∞ RHΔ rhombihexadeltille 3464 tetrille T∞ 4 sQ snub quadrille 43343 4-fold pentille P∞ 4 isQ isosnub quadrille 44333 iso(4-)pentille i P∞ 6 sH snub hextille 63333 6-fold pentille P∞ © 2016 by Taylor & Francis Group, LLC Additional Data 289 Additional Data We pause for additional data on the Archimedean polyhedra and planar tilings. Vertex Figures To any vertex V of a polyhedron, we associate the vertex figure, which is customarily obtained by slicing the polyhedron by a plane suitably near to V and perpendicular to the line joining V to the center of the polyhedron. Usually, each edge of the vertex figure is marked with the number of sides of the corresponding face of the polyhedron.
Load more

Recommended publications
  • On the Archimedean Or Semiregular Polyhedra
    ON THE ARCHIMEDEAN OR SEMIREGULAR POLYHEDRA Mark B. Villarino Depto. de Matem´atica, Universidad de Costa Rica, 2060 San Jos´e, Costa Rica May 11, 2005 Abstract We prove that there are thirteen Archimedean/semiregular polyhedra by using Euler’s polyhedral formula. Contents 1 Introduction 2 1.1 RegularPolyhedra .............................. 2 1.2 Archimedean/semiregular polyhedra . ..... 2 2 Proof techniques 3 2.1 Euclid’s proof for regular polyhedra . ..... 3 2.2 Euler’s polyhedral formula for regular polyhedra . ......... 4 2.3 ProofsofArchimedes’theorem. .. 4 3 Three lemmas 5 3.1 Lemma1.................................... 5 3.2 Lemma2.................................... 6 3.3 Lemma3.................................... 7 4 Topological Proof of Archimedes’ theorem 8 arXiv:math/0505488v1 [math.GT] 24 May 2005 4.1 Case1: fivefacesmeetatavertex: r=5. .. 8 4.1.1 At least one face is a triangle: p1 =3................ 8 4.1.2 All faces have at least four sides: p1 > 4 .............. 9 4.2 Case2: fourfacesmeetatavertex: r=4 . .. 10 4.2.1 At least one face is a triangle: p1 =3................ 10 4.2.2 All faces have at least four sides: p1 > 4 .............. 11 4.3 Case3: threefacesmeetatavertes: r=3 . ... 11 4.3.1 At least one face is a triangle: p1 =3................ 11 4.3.2 All faces have at least four sides and one exactly four sides: p1 =4 6 p2 6 p3. 12 4.3.3 All faces have at least five sides and one exactly five sides: p1 =5 6 p2 6 p3 13 1 5 Summary of our results 13 6 Final remarks 14 1 Introduction 1.1 Regular Polyhedra A polyhedron may be intuitively conceived as a “solid figure” bounded by plane faces and straight line edges so arranged that every edge joins exactly two (no more, no less) vertices and is a common side of two faces.
    [Show full text]
  • Archimedean Solids
    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2008 Archimedean Solids Anna Anderson University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Anderson, Anna, "Archimedean Solids" (2008). MAT Exam Expository Papers. 4. https://digitalcommons.unl.edu/mathmidexppap/4 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Archimedean Solids Anna Anderson In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor July 2008 2 Archimedean Solids A polygon is a simple, closed, planar figure with sides formed by joining line segments, where each line segment intersects exactly two others. If all of the sides have the same length and all of the angles are congruent, the polygon is called regular. The sum of the angles of a regular polygon with n sides, where n is 3 or more, is 180° x (n – 2) degrees. If a regular polygon were connected with other regular polygons in three dimensional space, a polyhedron could be created. In geometry, a polyhedron is a three- dimensional solid which consists of a collection of polygons joined at their edges. The word polyhedron is derived from the Greek word poly (many) and the Indo-European term hedron (seat).
    [Show full text]
  • Arxiv:Math/9906062V1 [Math.MG] 10 Jun 1999 Udo Udmna Eerh(Rn 96-01-00166)
    Embedding the graphs of regular tilings and star-honeycombs into the graphs of hypercubes and cubic lattices ∗ Michel DEZA CNRS and Ecole Normale Sup´erieure, Paris, France Mikhail SHTOGRIN Steklov Mathematical Institute, 117966 Moscow GSP-1, Russia Abstract We review the regular tilings of d-sphere, Euclidean d-space, hyperbolic d-space and Coxeter’s regular hyperbolic honeycombs (with infinite or star-shaped cells or vertex figures) with respect of possible embedding, isometric up to a scale, of their skeletons into a m-cube or m-dimensional cubic lattice. In section 2 the last remaining 2-dimensional case is decided: for any odd m ≥ 7, star-honeycombs m m {m, 2 } are embeddable while { 2 ,m} are not (unique case of non-embedding for dimension 2). As a spherical analogue of those honeycombs, we enumerate, in section 3, 36 Riemann surfaces representing all nine regular polyhedra on the sphere. In section 4, non-embeddability of all remaining star-honeycombs (on 3-sphere and hyperbolic 4-space) is proved. In the last section 5, all cases of embedding for dimension d> 2 are identified. Besides hyper-simplices and hyper-octahedra, they are exactly those with bipartite skeleton: hyper-cubes, cubic lattices and 8, 2, 1 tilings of hyperbolic 3-, 4-, 5-space (only two, {4, 3, 5} and {4, 3, 3, 5}, of those 11 have compact both, facets and vertex figures). 1 Introduction arXiv:math/9906062v1 [math.MG] 10 Jun 1999 We say that given tiling (or honeycomb) T has a l1-graph and embeds up to scale λ into m-cube Hm (or, if the graph is infinite, into cubic lattice Zm ), if there exists a mapping f of the vertex-set of the skeleton graph of T into the vertex-set of Hm (or Zm) such that λdT (vi, vj)= ||f(vi), f(vj)||l1 = X |fk(vi) − fk(vj)| for all vertices vi, vj, 1≤k≤m ∗This work was supported by the Volkswagen-Stiftung (RiP-program at Oberwolfach) and Russian fund of fundamental research (grant 96-01-00166).
    [Show full text]
  • A Tourist Guide to the RCSR
    A tourist guide to the RCSR Some of the sights, curiosities, and little-visited by-ways Michael O'Keeffe, Arizona State University RCSR is a Reticular Chemistry Structure Resource available at http://rcsr.net. It is open every day of the year, 24 hours a day, and admission is free. It consists of data for polyhedra and 2-periodic and 3-periodic structures (nets). Visitors unfamiliar with the resource are urged to read the "about" link first. This guide assumes you have. The guide is designed to draw attention to some of the attractions therein. If they sound particularly attractive please visit them. It can be a nice way to spend a rainy Sunday afternoon. OKH refers to M. O'Keeffe & B. G. Hyde. Crystal Structures I: Patterns and Symmetry. Mineral. Soc. Am. 1966. This is out of print but due as a Dover reprint 2019. POLYHEDRA Read the "about" for hints on how to use the polyhedron data to make accurate drawings of polyhedra using crystal drawing programs such as CrystalMaker (see "links" for that program). Note that they are Cartesian coordinates for (roughly) equal edge. To make the drawing with unit edge set the unit cell edges to all 10 and divide the coordinates given by 10. There seems to be no generally-agreed best embedding for complex polyhedra. It is generally not possible to have equal edge, vertices on a sphere and planar faces. Keywords used in the search include: Simple. Each vertex is trivalent (three edges meet at each vertex) Simplicial. Each face is a triangle.
    [Show full text]
  • Regular Polyhedra Through Time
    Fields Institute I. Hubard Polytopes, Maps and their Symmetries September 2011 Regular polyhedra through time The greeks were the first to study the symmetries of polyhedra. Euclid, in his Elements showed that there are only five regular solids (that can be seen in Figure 1). In this context, a polyhe- dron is regular if all its polygons are regular and equal, and you can find the same number of them at each vertex. Figure 1: Platonic Solids. It is until 1619 that Kepler finds other two regular polyhedra: the great dodecahedron and the great icosahedron (on Figure 2. To do so, he allows \false" vertices and intersection of the (convex) faces of the polyhedra at points that are not vertices of the polyhedron, just as the I. Hubard Polytopes, Maps and their Symmetries Page 1 Figure 2: Kepler polyhedra. 1619. pentagram allows intersection of edges at points that are not vertices of the polygon. In this way, the vertex-figure of these two polyhedra are pentagrams (see Figure 3). Figure 3: A regular convex pentagon and a pentagram, also regular! In 1809 Poinsot re-discover Kepler's polyhedra, and discovers its duals: the small stellated dodecahedron and the great stellated dodecahedron (that are shown in Figure 4). The faces of such duals are pentagrams, and are organized on a \convex" way around each vertex. Figure 4: The other two Kepler-Poinsot polyhedra. 1809. A couple of years later Cauchy showed that these are the only four regular \star" polyhedra. We note that the convex hull of the great dodecahedron, great icosahedron and small stellated dodecahedron is the icosahedron, while the convex hull of the great stellated dodecahedron is the dodecahedron.
    [Show full text]
  • Uniform Panoploid Tetracombs
    Uniform Panoploid Tetracombs George Olshevsky TETRACOMB is a four-dimensional tessellation. In any tessellation, the honeycells, which are the n-dimensional polytopes that tessellate the space, Amust by definition adjoin precisely along their facets, that is, their ( n!1)- dimensional elements, so that each facet belongs to exactly two honeycells. In the case of tetracombs, the honeycells are four-dimensional polytopes, or polychora, and their facets are polyhedra. For a tessellation to be uniform, the honeycells must all be uniform polytopes, and the vertices must be transitive on the symmetry group of the tessellation. Loosely speaking, therefore, the vertices must be “surrounded all alike” by the honeycells that meet there. If a tessellation is such that every point of its space not on a boundary between honeycells lies in the interior of exactly one honeycell, then it is panoploid. If one or more points of the space not on a boundary between honeycells lie inside more than one honeycell, the tessellation is polyploid. Tessellations may also be constructed that have “holes,” that is, regions that lie inside none of the honeycells; such tessellations are called holeycombs. It is possible for a polyploid tessellation to also be a holeycomb, but not for a panoploid tessellation, which must fill the entire space exactly once. Polyploid tessellations are also called starcombs or star-tessellations. Holeycombs usually arise when (n!1)-dimensional tessellations are themselves permitted to be honeycells; these take up the otherwise free facets that bound the “holes,” so that all the facets continue to belong to two honeycells. In this essay, as per its title, we are concerned with just the uniform panoploid tetracombs.
    [Show full text]
  • 15 BASIC PROPERTIES of CONVEX POLYTOPES Martin Henk, J¨Urgenrichter-Gebert, and G¨Unterm
    15 BASIC PROPERTIES OF CONVEX POLYTOPES Martin Henk, J¨urgenRichter-Gebert, and G¨unterM. Ziegler INTRODUCTION Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their im- portance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry to linear and combinatorial optimiza- tion. In this chapter we try to give a short introduction, provide a sketch of \what polytopes look like" and \how they behave," with many explicit examples, and briefly state some main results (where further details are given in subsequent chap- ters of this Handbook). We concentrate on two main topics: • Combinatorial properties: faces (vertices, edges, . , facets) of polytopes and their relations, with special treatments of the classes of low-dimensional poly- topes and of polytopes \with few vertices;" • Geometric properties: volume and surface area, mixed volumes, and quer- massintegrals, including explicit formulas for the cases of the regular simplices, cubes, and cross-polytopes. We refer to Gr¨unbaum [Gr¨u67]for a comprehensive view of polytope theory, and to Ziegler [Zie95] respectively to Gruber [Gru07] and Schneider [Sch14] for detailed treatments of the combinatorial and of the convex geometric aspects of polytope theory. 15.1 COMBINATORIAL STRUCTURE GLOSSARY d V-polytope: The convex hull of a finite set X = fx1; : : : ; xng of points in R , n n X i X P = conv(X) := λix λ1; : : : ; λn ≥ 0; λi = 1 : i=1 i=1 H-polytope: The solution set of a finite system of linear inequalities, d T P = P (A; b) := x 2 R j ai x ≤ bi for 1 ≤ i ≤ m ; with the extra condition that the set of solutions is bounded, that is, such that m×d there is a constant N such that jjxjj ≤ N holds for all x 2 P .
    [Show full text]
  • Five-Vertex Archimedean Surface Tessellation by Lanthanide-Directed Molecular Self-Assembly
    Five-vertex Archimedean surface tessellation by lanthanide-directed molecular self-assembly David Écijaa,1, José I. Urgela, Anthoula C. Papageorgioua, Sushobhan Joshia, Willi Auwärtera, Ari P. Seitsonenb, Svetlana Klyatskayac, Mario Rubenc,d, Sybille Fischera, Saranyan Vijayaraghavana, Joachim Reicherta, and Johannes V. Bartha,1 aPhysik Department E20, Technische Universität München, D-85478 Garching, Germany; bPhysikalisch-Chemisches Institut, Universität Zürich, CH-8057 Zürich, Switzerland; cInstitute of Nanotechnology, Karlsruhe Institute of Technology, D-76344 Eggenstein-Leopoldshafen, Germany; and dInstitut de Physique et Chimie des Matériaux de Strasbourg (IPCMS), CNRS-Université de Strasbourg, F-67034 Strasbourg, France Edited by Kenneth N. Raymond, University of California, Berkeley, CA, and approved March 8, 2013 (received for review December 28, 2012) The tessellation of the Euclidean plane by regular polygons has by five interfering laser beams, conceived to specifically address been contemplated since ancient times and presents intriguing the surface tiling problem, yielded a distorted, 2D Archimedean- aspects embracing mathematics, art, and crystallography. Sig- like architecture (24). nificant efforts were devoted to engineer specific 2D interfacial In the last decade, the tools of supramolecular chemistry on tessellations at the molecular level, but periodic patterns with surfaces have provided new ways to engineer a diversity of sur- distinct five-vertex motifs remained elusive. Here, we report a face-confined molecular architectures, mainly exploiting molecular direct scanning tunneling microscopy investigation on the cerium- recognition of functional organic species or the metal-directed directed assembly of linear polyphenyl molecular linkers with assembly of molecular linkers (5). Self-assembly protocols have terminal carbonitrile groups on a smooth Ag(111) noble-metal sur- been developed to achieve regular surface tessellations, includ- face.
    [Show full text]
  • Snubs, Alternated Facetings, & Stott-Coxeter-Dynkin Diagrams
    Symmetry: Culture and Science Vol. 21, No.4, 329-344, 2010 SNUBS, ALTERNATED FACETINGS, & STOTT-COXETER-DYNKIN DIAGRAMS Dr. Richard Klitzing Non-professional mathematician, (b. Laupheim, BW., Germany, 1966). Address: Kantstrasse 23, 89522 Heidenheim, Germany. E-mail: [email protected] . Fields of interest: Polytopes, geometry, mathematical quasi-crystallography. Publications: (2002) Axial Symmetrical Edge-Facetings of Uniform Polyhedra. In: Symmetry: Cult. & Sci., vol. 13, nos. 3- 4, pp. 241-258. (2001) Convex Segmentochora. In: Symmetry: Cult. & Sci., vol. 11, nos. 1-4, pp. 139-181. (1996) Reskalierungssymmetrien quasiperiodischer Strukturen. [Symmetries of Rescaling for Quasiperiodic Structures, Ph.D. Dissertation, in German], Eberhard-Karls-Universität Tübingen, or Hamburg, Verlag Dr. Kovač, ISBN 978-3-86064-428- 7. – (Former ones: mentioned therein.) Abstract: The snub cube and the snub dodecahedron are well-known polyhedra since the days of Kepler. Since then, the term “snub” was applied to further cases, both in 3D and beyond, yielding an exceptional species of polytopes: those do not bow to Wythoff’s kaleidoscopical construction like most other Archimedean polytopes, some appear in enantiomorphic pairs with no own mirror symmetry, etc. However, they remained stepchildren, since they permit no real one-step access. Actually, snub polytopes are meant to be derived secondarily from Wythoffians, either from omnitruncated or from truncated polytopes. – This secondary process herein is analysed carefully and extended widely: nothing bars a progress from vertex alternation to an alternation of an arbitrary class of sub-dimensional elements, for instance edges, faces, etc. of any type. Furthermore, this extension can be coded in Stott-Coxeter-Dynkin diagrams as well.
    [Show full text]
  • Convex Polytopes and Tilings with Few Flag Orbits
    Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals.
    [Show full text]
  • Coloring Uniform Honeycombs
    Bridges 2009: Mathematics, Music, Art, Architecture, Culture Coloring Uniform Honeycombs Glenn R. Laigo, [email protected] Ma. Louise Antonette N. De las Peñas, [email protected] Mathematics Department, Ateneo de Manila University Loyola Heights, Quezon City, Philippines René P. Felix, [email protected] Institute of Mathematics, University of the Philippines Diliman, Quezon City, Philippines Abstract In this paper, we discuss a method of arriving at colored three-dimensional uniform honeycombs. In particular, we present the construction of perfect and semi-perfect colorings of the truncated and bitruncated cubic honeycombs. If G is the symmetry group of an uncolored honeycomb, a coloring of the honeycomb is perfect if the group H consisting of elements that permute the colors of the given coloring is G. If H is such that [ G : H] = 2, we say that the coloring of the honeycomb is semi-perfect . Background In [7, 9, 12], a general framework has been presented for coloring planar patterns. Focus was given to the construction of perfect colorings of semi-regular tilings on the hyperbolic plane. In this work, we will extend the method of coloring two dimensional patterns to obtain colorings of three dimensional uniform honeycombs. There is limited literature on colorings of three-dimensional honeycombs. We see studies on colorings of polyhedra; for instance, in [17], a method of coloring shown is by cutting the polyhedra and laying it flat to produce a pattern on a two-dimensional plane. In this case, only the faces of the polyhedra are colored. In [6], enumeration problems on colored patterns on polyhedra are discussed and solutions are obtained by applying Burnside's counting theorem.
    [Show full text]
  • Local Symmetry Preserving Operations on Polyhedra
    Local Symmetry Preserving Operations on Polyhedra Pieter Goetschalckx Submitted to the Faculty of Sciences of Ghent University in fulfilment of the requirements for the degree of Doctor of Science: Mathematics. Supervisors prof. dr. dr. Kris Coolsaet dr. Nico Van Cleemput Chair prof. dr. Marnix Van Daele Examination Board prof. dr. Tomaž Pisanski prof. dr. Jan De Beule prof. dr. Tom De Medts dr. Carol T. Zamfirescu dr. Jan Goedgebeur © 2020 Pieter Goetschalckx Department of Applied Mathematics, Computer Science and Statistics Faculty of Sciences, Ghent University This work is licensed under a “CC BY 4.0” licence. https://creativecommons.org/licenses/by/4.0/deed.en In memory of John Horton Conway (1937–2020) Contents Acknowledgements 9 Dutch summary 13 Summary 17 List of publications 21 1 A brief history of operations on polyhedra 23 1 Platonic, Archimedean and Catalan solids . 23 2 Conway polyhedron notation . 31 3 The Goldberg-Coxeter construction . 32 3.1 Goldberg ....................... 32 3.2 Buckminster Fuller . 37 3.3 Caspar and Klug ................... 40 3.4 Coxeter ........................ 44 4 Other approaches ....................... 45 References ............................... 46 2 Embedded graphs, tilings and polyhedra 49 1 Combinatorial graphs .................... 49 2 Embedded graphs ....................... 51 3 Symmetry and isomorphisms . 55 4 Tilings .............................. 57 5 Polyhedra ............................ 59 6 Chamber systems ....................... 60 7 Connectivity .......................... 62 References
    [Show full text]