DOCSLIB.ORG

Tangled Platonic Polyhedra

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Tangled Platonic Polyhedra Tangled Platonic Polyhedra Toen Castle, V. Robins and S. T. Hyde Last typeset on February 7, 2010 Abstract A method to find all embeddings of a given graph on the genus-one two-dimensional torus is used to construct a variety of entangled Platonic polyhedra graphs. These graph embeddings contain knotted and/or linked subgraphs of arbitrary complexity, the simplest of which include the ‘trefoil’ knot and the ‘Hopf’ and ‘Whitehead’ links. Analyses of knotting and linking, point group symmetries and an energy function are presented for embeddings of the topological tetrahedron, cube and octahedron. 1 Contents 1 Introduction 3 1.1 The Big Picture (surfaces in space) . 7 2 Method - generalising Platonic Polyhedra 10 2.1 Universal cover . 11 2.2 2-cell embeddings . 12 2.3 Tilings of the torus dissolving to polyhedron nets . 15 2.4 Tangling . 17 2.5 Knots and links . 18 2.6 Energy . 22 2.7 Symmetry . 22 3 Tetrahedron 25 3.1 2-cell embeddings of the tetrahedron . 26 3.2 The tetrahedron in the universal covers: <8, 4> and <9, 3> .... 30 3.3 Comparison of tetrahedral isotopes . 31 4 Cube 34 4.1 2-cell embeddings of the cube . 34 4.2 The cube in its universal cover . 35 4.3 Symmetries of tangled cubes . 39 4.4 High symmetry cube isotopes . 40 4.5 Honeycomb symmetries . 42 4.6 Cube isotope knots, links and energies . 43 5 Octahedron 49 5.1 2-cell embeddings of the octahedron . 49 5.2 The toroidal octahedron in its universal cover . 49 5.3 High symmetry toroidal octahedra isotopes . 52 5.4 Properties of toroidal octahedra . 52 6 Further constructions 60 6.1 Non- 2-cell torus graph embeddings . 60 6.2 Duals . 61 6.3 Chirality . 64 7 Conclusion 64 2 1 Introduction Polyhedra have long been of interest to mathematicians and philosophers, the fas- cination aided by the spatial, geometric, and combinatorial patterns commensurate with their simple structures. One notable early discussion of regular polyhedra was by Plato in his Timaeus [1]. He identified the ‘classical elements’ of fire, air, earth, water and quintessence or aether with the five regular polyhedra - tetrahedra, oc- trahedra, cubes, icosahedra and dodecahedra - as universal building blocks in his early ‘theory of everything’. Just as the understanding of the constitutive nature of elements has since grown, so too has the conception of polyhedra expanded. Polyhedra were initially envisaged as solid bodies with planar polygonal faces, such as used intrinsically by the ancient Egyptians in the design for their pyramids. The polyhedra now known as Platonic polyhedra due to the interest recorded in them by Plato, are defined by being regular, as each face is a congruent regular polygon, with the same number of faces meeting around each vertex. Later the concept had developed so that in the fifteenth century, Leonardo da Vinci famously illustrated Fra Luca Bartolomeo de Pacioli’s De divina proportione with an image of a rhombicuboctahedron that emphasised only the edges and vertices; omitting the faces and solid body (see Fig. 1). Keppler broadened the concept further by considering stellated polyhedra [2], before the idea was extended further still to infinite polyhedra [3] and myriad other still broader generalisations. Each gen- eralisation allows interesting new structures to be created as the principles of the Platonic polyhedra are relaxed or extended. Within the mathematical literature, the definitions of polyhedra have expanded to mirror this growing variety of structures. We follow the conventions of tiling theory, as described by Grunbaum¨ [5], in identifying the face of a polyhedron with the cyclically ordered set of vertices and connecting edges that surrounds it. Using this definition allows us to maintain the intuitive link with regular polyhedra while creating topologically interesting finite structures, embedded in oriented manifolds of genus greater than zero. This definition is suitably broad to allow a freedom of structure that is not constrained to convexity, planar faces, or topologically spheri- cal polyhedra. In turn, this freedom allows the presence of knots and links, as well as other entanglement modes [6] within the polyhedra. Interestingly, topologically toroidal polyhedra can be created within a definition more restrictive than ours, such as the dual pair of the Szilassi and Csasz´ ar´ polyhedra which have planar faces and straight edges [7, 8]. However, these requirements of straightness and linearity need to be dropped to allow a fuller variety of structures. Our method generates topologically distinct ways in which graphs (sets of vertices and edges with all edges joining pairs of vertices) can be embedded in space. Some of these ways are topologically trivial, whereas up to ambient isotopy 3 Figure 1: A ‘skeletal’ ployhedron is composed of just the edges and corners, and omits the faces and solid body. This illustration of a rhombicuboctahedron allows viewing of both the front and back elements[4]. 4 (stretching and deformations without passing edges through each other or them- selves) others are inherently knotted or interlinked. While the method is applicable to any graph, in each case considered in this paper, the graph will be that of a tetra- hedron, cube, or octahedron; but the embeddings of the constituent vertices and edges will vary between examples. If these graphs are considered in a chemical context – as atoms and bonds, or metal complexes and ligands – the results shed light on various topologically distinct conformational isomers that may be attain- able by the compound. As these isomers differ by their ambient isotopy, we follow the nomenclature of [9] in calling them isotopes. In literature regarding structural DNA manipulation [10] they are denoted topoisomers. A C D D B E F A G H G H B F E C Figure 2: A topologically simple cube and a tangled version. Both are shown reticulating underlying surfaces. There are different ways to quantify the ‘tangledness’ of a graph embedded in space. These methods are mostly imported from or inspired by knot theory: cycles within the graph embedding form loops which may be knotted or linked, and so can be described by knot and link tables, minimal crossing numbers, or knot polynomials etc. [11]. Indeed there has been some recent interest in the knots and links present in embedded graphs [12], however this work focuses on the knots and links unavoidably created in the embedding of sufficiently complex graphs, rather than the means of introducing controlled complexity into the embeddings of otherwise simple structures. An alternative to analysing the knots and links contained within an isotope is a generalisation of the knot polynomials called the Yamada polynomial [13] which provides an invariant (up to factors) of the entire embedded graph. This method 5 has been used to analyse the entanglement of the ‘theta graph’ – the graph with two vertices connected by three edges[14], however its utility for providing useful information about embeddings of larger graphs is unclear. Instead we prefer to use the hierarchy of graph embedding complexity provided by the minimum genus of the oriented surface necessary for a graph embedding as introduced in [9]. As the genus increases, so too does the complexity of knots [15], links and other entanglement modes [6] which manifest in the graph embedding. This hierarchy fits naturally with our method for generating these structures, which uses the 2-cell embeddings (see Section 2.2) of graphs to reticulate an ori- ented surface with the vertices and edges of the graph. A ‘reticulation’ of a surface indicates that the vertices and edges of a graph form a network on the surface, with edges only meeting or crossing at designated vertices. The use of 2-cell embed- dings is a common technique within graph theory [16]. It is used in [17] and with a different emphasis in [18]. As we only consider small (comparatively combinator- ically simple) graphs, we can omit many of the group theoretic aspects discussed in these works. As we are considering isotopes, within the equivalence class of ambient iso- topy, it suffices to work with a representative in a canonical form. As this paper focuses on the genus-one surface, we consider only the ‘doughnut’ manner of torus embedding in space. This avoids issues arising from self-interesection of the sur- face, and facilitates the analysis using its universal cover: an abstraction which represents the surface, irrespective of its specific shape in space. The universal cover of the torus is the plane, with a two-periodic translational symmetry echoing the longitudinal and meridional lines of the surface structure of the torus. The particular geometric placement of the points within the universal cover is determined to be the average of the placements of the connected points (barycen- tric). This gives a unique and canonical representation for a given isotopy class of graph surface embeddings [19]. It is also well-formed, in terms of points and lines having distinct placements, for all graphs we consider here. In the representation in the universal cover, the graph becomes a translational unit cell of a periodic re- peating pattern. In an alternative formulation, the surface can be cut open to form a surface patch reticulated with one unit of the repeating pattern in the universal cover. The manner that the patches fit together depends upon the periodicity of the universal cover. To find the actual placement of vertices, edges, and implicit faces in three-dimensional space, the surface is embedded in space in a suitable way, and then dissolved, leaving behind the points and lines, which can form a topologically tangled structure. While the isotopes in this paper generalise Platonic polyhedra by virtue of shar- ing their graphs, some can share another feature of Platonic polyhedra: equally 6 sized faces.
Load more

Recommended publications
  • Symmetry As an Aesthetic Factor
    Comp. & Maths. with Appls, Vol. 12B, Nos. I/2, pp. 77-82. 1986 0886-9561/86 $3,1)0+ .00 Printed in Great Britain. © 1986 Pergamon Press Ltd. SYMMETRY AS AN AESTHETIC FACTOR HAROLD OSBORNEt Kreutzstrasse 12, 8640 Rappersvill SG, Switzerland Abstract--In classical antiquity symmetry meant commensurability and was believed to constitute a canon of beauty in nature as in art. This intellectualist conception of beauty persisted through the Middle Ages with the addition doctrine that the phenomenal world manifests an imperfect replica of the ideal symmetry of divine Creation. The concept of the Golden Section came to the fore at the Renaissance and has continued as a minority interest both for organic nature and for fine art. The modern idea of symmetry is based more loosely upon the balance of shapes or magnitudes and corresponds to a change from an intellectual to a perceptual attitude towards aesthetic experience. None of these theories of symmetry has turned out to be a principle by following which aesthetically satisfying works of art can be mechanically constructed. In contemporary theory the vaguer notion of organic unity has usurped the prominence formerly enjoyed by that of balanced symmetry. From classical antiquity the idea of symmetry in close conjunction with that of proportion dominated the studio practice of artists and the thinking of theorists. Symmetry was asserted to be the key to perfection in nature as in art. But the traditional concept was radically different from what we understand by symmetry today--so different that "symmetry" can no longer be regarded as a correct translation of the Greek word symmetria from which it derives--and some acquaintance with the historical background of these ideas is essential in order to escape from the imbroglio of confusion which has resulted from the widespread conflation of the two.
    [Show full text]
  • Equivelar Octahedron of Genus 3 in 3-Space
    Equivelar octahedron of genus 3 in 3-space Ruslan Mizhaev ([email protected]) Apr. 2020 ABSTRACT . Building up a toroidal polyhedron of genus 3, consisting of 8 nine-sided faces, is given. From the point of view of topology, a polyhedron can be considered as an embedding of a cubic graph with 24 vertices and 36 edges in a surface of genus 3. This polyhedron is a contender for the maximal genus among octahedrons in 3-space. 1. Introduction This solution can be attributed to the problem of determining the maximal genus of polyhedra with the number of faces - . As is known, at least 7 faces are required for a polyhedron of genus = 1 . For cases ≥ 8 , there are currently few examples. If all faces of the toroidal polyhedron are – gons and all vertices are q-valence ( ≥ 3), such polyhedral are called either locally regular or simply equivelar [1]. The characteristics of polyhedra are abbreviated as , ; [1]. 2. Polyhedron {9, 3; 3} V1. The paper considers building up a polyhedron 9, 3; 3 in 3-space. The faces of a polyhedron are non- convex flat 9-gons without self-intersections. The polyhedron is symmetric when rotated through 1804 around the axis (Fig. 3). One of the features of this polyhedron is that any face has two pairs with which it borders two edges. The polyhedron also has a ratio of angles and faces - = − 1. To describe polyhedra with similar characteristics ( = − 1) we use the Euler formula − + = = 2 − 2, where is the Euler characteristic. Since = 3, the equality 3 = 2 holds true.
    [Show full text]
  • Crystallographic Topology and Its Applications * Carroll K
    4 Crystallographic Topology and Its Applications * Carroll K. Johnson and Michael N. Burnett Chemical and Analytical Sciences Division, Oak Ridge National Laboratory Oak Ridge, TN 37831-6197, USA [email protected] http://www. ornl.gov/ortep/topology. html William D. Dunbar Division of Natural Sciences & Mathematics, Simon’s Rock College Great Barrington, MA 01230, USA wdunbar @plato. simons- rock.edu http://www. simons- rock.edd- wdunbar Abstract two disciplines. Since both topology and crystallography have many subdisciplines, there are a number of quite Geometric topology and structural crystallography different intersection regions that can be called crystall o- concepts are combined to define a new area we call graphic topology; but we will confine this discussion to Structural Crystallographic Topology, which may be of one well delineated subarea. interest to both crystallographers and mathematicians. The structural crystallography of interest involves the In this paper, we represent crystallographic symmetry group theory required to describe symmetric arrangements groups by orbifolds and crystal structures by Morse func- of atoms in crystals and a classification of the simplest arrangements as lattice complexes. The geometric topol- tions. The Morse @fiction uses mildly overlapping Gau s- sian thermal-motion probability density functions cen- ogy of interest is the topological properties of crystallo- tered on atomic sites to form a critical net with peak, pass, graphic groups, represented as orbifolds, and the Morse pale, and pit critical points joined into a graph by density theory global analysis of critical points in symmetric gradient-flow separatrices. Critical net crystal structure functions. Here we are taking the liberty of calling global drawings can be made with the ORTEP-111 graphics pro- analysis part of topology.
    [Show full text]
  • De Divino Errore ‘De Divina Proportione’ Was Written by Luca Pacioli and Illustrated by Leonardo Da Vinci
    De Divino Errore ‘De Divina Proportione’ was written by Luca Pacioli and illustrated by Leonardo da Vinci. It was one of the most widely read mathematical books. Unfortunately, a strongly emphasized statement in the book claims six summits of pyramids of the stellated icosidodecahedron lay in one plane. This is not so, and yet even extensively annotated editions of this book never noticed this error. Dutchmen Jos Janssens and Rinus Roelofs did so, 500 years later. Fig. 1: About this illustration of Leonardo da Vinci for the Milanese version of the ‘De Divina Proportione’, Pacioli erroneously wrote that the red and green dots lay in a plane. The book ‘De Divina Proportione’, or ‘On the Divine Ratio’, was written by the Franciscan Fra Luca Bartolomeo de Pacioli (1445-1517). His name is sometimes written Paciolo or Paccioli because Italian was not a uniform language in his days, when, moreover, Italy was not a country yet. Labeling Pacioli as a Tuscan, because of his birthplace of Borgo San Sepolcro, may be more correct, but he also studied in Venice and Rome, and spent much of his life in Perugia and Milan. In service of Duke and patron Ludovico Sforza, he would write his masterpiece, in 1497 (although it is more correct to say the work was written between 1496 and 1498, because it contains several parts). It was not his first opus, because in 1494 his ‘Summa de arithmetic, geometrica, proportioni et proportionalita’ had appeared; the ‘Summa’ and ‘Divina’ were not his only books, but surely the most famous ones. For hundreds of years the books were among the most widely read mathematical bestsellers, their fame being only surpassed by the ‘Elements’ of Euclid.
    [Show full text]
  • Fractal Wallpaper Patterns Douglas Dunham John Shier
    Fractal Wallpaper Patterns Douglas Dunham John Shier Department of Computer Science 6935 133rd Court University of Minnesota, Duluth Apple Valley, MN 55124 USA Duluth, MN 55812-3036, USA [email protected] [email protected] http://john-art.com/ http://www.d.umn.edu/˜ddunham/ Abstract In the past we presented an algorithm that can fill a region with an infinite sequence of randomly placed and progressively smaller shapes, producing a fractal pattern. In this paper we extend that algorithm to fill rectangles and triangles that tile the plane, which yields wallpaper patterns that are locally fractal in nature. This produces artistic patterns which have a pleasing combination of global symmetry and local randomness. We show several sample patterns. Introduction We have described an algorithm [2, 4, 6] that can fill a planar region with a series of progressively smaller randomly-placed motifs and produce pleasing patterns. Here we extend that algorithm to fill rectangles and triangles that can fill the plane, creating “wallpaper” patterns. Figure 1 shows a randomly created circle pattern with symmetry group p6mm. In order to create our wallpaper patterns, we fill a fundamental region Figure 1: A locally random circle fractal with global p6mm symmetry. for one of the 17 2-dimensional crystallographic groups with randomly placed, progressively smaller copies of a motif, such as a circle in Figure 1. The randomness generates a fractal pattern. Then copies of the filled fundamental region can be used to tile the plane, yielding a locally fractal, but globally symmetric pattern. In the next section we recall how the basic algorithm works and describe the modifications needed to create wallpaper patterns.
    [Show full text]
  • Representing Graphs by Polygons with Edge Contacts in 3D
    Representing Graphs by Polygons with Side Contacts in 3D∗ Elena Arseneva1, Linda Kleist2, Boris Klemz3, Maarten Löffler4, André Schulz5, Birgit Vogtenhuber6, and Alexander Wolff7 1 Saint Petersburg State University, Russia [email protected] 2 Technische Universität Braunschweig, Germany [email protected] 3 Freie Universität Berlin, Germany [email protected] 4 Utrecht University, the Netherlands [email protected] 5 FernUniversität in Hagen, Germany [email protected] 6 Technische Universität Graz, Austria [email protected] 7 Universität Würzburg, Germany orcid.org/0000-0001-5872-718X Abstract A graph has a side-contact representation with polygons if its vertices can be represented by interior-disjoint polygons such that two polygons share a common side if and only if the cor- responding vertices are adjacent. In this work we study representations in 3D. We show that every graph has a side-contact representation with polygons in 3D, while this is not the case if we additionally require that the polygons are convex: we show that every supergraph of K5 and every nonplanar 3-tree does not admit a representation with convex polygons. On the other hand, K4,4 admits such a representation, and so does every graph obtained from a complete graph by subdividing each edge once. Finally, we construct an unbounded family of graphs with average vertex degree 12 − o(1) that admit side-contact representations with convex polygons in 3D. Hence, such graphs can be considerably denser than planar graphs. 1 Introduction A graph has a contact representation if its vertices can be represented by interior-disjoint geometric objects1 such that two objects touch exactly if the corresponding vertices are adjacent.
    [Show full text]
  • Congruence Links for Discriminant D = −3
    Principal Congruence Links for Discriminant D = −3 Matthias Goerner UC Berkeley April 20th, 2011 Matthias Goerner (UC Berkeley) Principal Congruence Links April 20th, 2011 1 / 44 Overview Thurston congruence link, geometric description Bianchi orbifolds, congruence and principal congruence manifolds Results implying there are finitely many principal congruence links Overview for the case of discriminant D = −3 Preliminaries for the construction Construction of two more examples Open questions Matthias Goerner (UC Berkeley) Principal Congruence Links April 20th, 2011 2 / 44 Thurston congruence link −3 M2+ζ Complement is non-compact finite-volume hyperbolic 3-manifold. Tesselated by 28 regular ideal hyperbolic tetrahedra. Tesselation is \regular", i.e., symmetry group takes every tetrahedron to every other tetrahedron in all possible 12 orientations. Matthias Goerner (UC Berkeley) Principal Congruence Links April 20th, 2011 3 / 44 Cusped hyperbolic 3-manifolds Ideal hyperbolic tetrahedron does not include the vertices. Remove a small hororball. Ideal tetrahedron is topologically a truncated tetrahedron. Cut is a triangle with a Euclidean structure from hororsphere. Matthias Goerner (UC Berkeley) Principal Congruence Links April 20th, 2011 4 / 44 Cusped hyperbolic 3-manifolds have toroidal ends Truncated tetrahedra form interior of a 3-manifold M¯ with boundary. @M¯ triangulated by the Euclidean triangles. @M¯ is a torus. Ends (cusps) of hyperbolic manifold modeled on torus × interval. Matthias Goerner (UC Berkeley) Principal Congruence Links April 20th, 2011 5 / 44 Knot complements can be cusped hyperbolic 3-manifolds Cusp homeomorphic to a tubular neighborhood of a knot/link component. Figure-8 knot complement tesselated by two regular ideal tetrahedra. Hyperbolic metric near knot so dense that light never reaches knot.
    [Show full text]
  • Syddansk Universitet Review of "Henning, Herbert: La Divina
    Syddansk Universitet Review of "Henning, Herbert: La divina proportione und die Faszination des Schönen oder das Schöne in der Mathematik" Robering, Klaus Published in: Mathematical Reviews Publication date: 2014 Document version Accepted author manuscript Citation for pulished version (APA): Robering, K. (2014). Review of "Henning, Herbert: La divina proportione und die Faszination des Schönen oder das Schöne in der Mathematik". Mathematical Reviews. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 14. Feb. 2017 MR3059375 Henning, Herbert La divina proportione und die Faszination des Sch¨onenoder das Sch¨onein der Mathematik. (German) [The divine proportion and the fascination of beauty or beauty in mathematics] Mitt. Math. Ges. Hamburg 32 (2012), 49{62 00A66 (11B39 51M04) The author points out that there is a special relationship between mathematics and the arts in that both aim at the discovery and presentation of truth and beauty.
    [Show full text]
  • Leonardo Universal
    Leonardo Universal DE DIVINA PROPORTIONE Pacioli, legendary mathematician, introduced the linear perspective and the mixture of colors, representing the human body and its proportions and extrapolating this knowledge to architecture. Luca Pacioli demonstrating one of Euclid’s theorems (Jacobo de’Barbari, 1495) D e Divina Proportione is a holy expression commonly outstanding work and icon of the Italian Renaissance. used in the past to refer to what we nowadays call Leonardo, who was deeply interested in nature and art the golden section, which is the mathematic module mathematics, worked with Pacioli, the author of the through which any amount can be divided in two text, and was a determined spreader of perspectives uneven parts, so that the ratio between the smallest and proportions, including Phi in many of his works, part and the largest one is the same as that between such as The Last Supper, created at the same time as the largest and the full amount. It is divine for its the illustrations of the present manuscript, the Mona being unique, and triune, as it links three elements. Lisa, whose face hides a perfect golden rectangle and The fusion of art and science, and the completion of the Uomo Vitruviano, a deep study on the human 60 full-page illustrations by the preeminent genius figure where da Vinci proves that all the main body of the time, Leonardo da Vinci, make it the most parts were related to the golden ratio. Luca Pacioli credits that Leonardo da Vinci made the illustrations of the geometric bodies with quill, ink and watercolor.
    [Show full text]
  • Single Digits
    ...................................single digits ...................................single digits In Praise of Small Numbers MARC CHAMBERLAND Princeton University Press Princeton & Oxford Copyright c 2015 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved The second epigraph by Paul McCartney on page 111 is taken from The Beatles and is reproduced with permission of Curtis Brown Group Ltd., London on behalf of The Beneficiaries of the Estate of Hunter Davies. Copyright c Hunter Davies 2009. The epigraph on page 170 is taken from Harry Potter and the Half Blood Prince:Copyrightc J.K. Rowling 2005 The epigraphs on page 205 are reprinted wiht the permission of the Free Press, a Division of Simon & Schuster, Inc., from Born on a Blue Day: Inside the Extraordinary Mind of an Austistic Savant by Daniel Tammet. Copyright c 2006 by Daniel Tammet. Originally published in Great Britain in 2006 by Hodder & Stoughton. All rights reserved. Library of Congress Cataloging-in-Publication Data Chamberland, Marc, 1964– Single digits : in praise of small numbers / Marc Chamberland. pages cm Includes bibliographical references and index. ISBN 978-0-691-16114-3 (hardcover : alk. paper) 1. Mathematical analysis. 2. Sequences (Mathematics) 3. Combinatorial analysis. 4. Mathematics–Miscellanea. I. Title. QA300.C4412 2015 510—dc23 2014047680 British Library
    [Show full text]
  • On Three-Dimensional Space Groups
    On Three-Dimensional Space Groups John H. Conway Department of Mathematics Princeton University, Princeton NJ 08544-1000 USA e-mail: [email protected] Olaf Delgado Friedrichs Department of Mathematics Bielefeld University, D-33501 Bielefeld e-mail: [email protected] Daniel H. Huson ∗ Applied and Computational Mathematics Princeton University, Princeton NJ 08544-1000 USA e-mail: [email protected] William P. Thurston Department of Mathematics University of California at Davis e-mail: [email protected] February 1, 2008 Abstract A entirely new and independent enumeration of the crystallographic space groups is given, based on obtaining the groups as fibrations over the plane crystallographic groups, when this is possible. For the 35 “irreducible” groups for which it is not, an independent method is used that has the advantage of elucidating their subgroup relationships. Each space group is given a short “fibrifold name” which, much like the orbifold names for two-dimensional groups, while being only specified up to isotopy, contains enough information to allow the construction of the group from the name. 1 Introduction arXiv:math/9911185v1 [math.MG] 23 Nov 1999 There are 219 three-dimensional crystallographic space groups (or 230 if we distinguish between mirror images). They were independently enumerated in the 1890’s by W. Barlow in England, E.S. Federov in Russia and A. Sch¨onfliess in Germany. The groups are comprehensively described in the International Tables for Crystallography [Hah83]. For a brief definition, see Appendix I. Traditionally the enumeration depends on classifying lattices into 14 Bravais types, distinguished by the symmetries that can be added, and then adjoining such symmetries in all possible ways.
    [Show full text]
  • Non-Fundamental Trunc-Simplex Tilings and Their Optimal Hyperball Packings and Coverings in Hyperbolic Space I
    Non-fundamental trunc-simplex tilings and their optimal hyperball packings and coverings in hyperbolic space I. For families F1-F4 Emil Moln´ar1, Milica Stojanovi´c2, Jen}oSzirmai1 1Budapest University of Tehnology and Economics Institute of Mathematics, Department of Geometry, H-1521 Budapest, Hungary [email protected], [email protected] 2Faculty of Organizational Sciences, University of Belgrade 11040 Belgrade, Serbia [email protected] Abstract Supergroups of some hyperbolic space groups are classified as a contin- uation of our former works. Fundamental domains will be integer parts of truncated tetrahedra belonging to families F1 - F4, for a while, by the no- tation of E. Moln´aret al. in 2006. As an application, optimal congruent hyperball packings and coverings to the truncation base planes with their very good densities are computed. This covering density is better than the conjecture of L. Fejes T´othfor balls and horoballs in 1964. 1 Introduction arXiv:2003.13631v1 [math.MG] 30 Mar 2020 1.1 Short history Hyperbolic space groups are isometry groups, acting discontinuously on hyperbolic 3-space H3 with compact fundamental domains. It will be investigated some series of such groups by looking for their fundamental domains. Face pairing identifications 2000 Mathematics Subject Classifications. 51M20, 52C22, 20H15, 20F55. Key words and Phrases. hyperbolic space group, fundamental domain, isometries, truncated simplex, Poincar`ealgorithm. 1 2 Moln´ar,Stojanovi´c,Szirmai on a given polyhedron give us generators and relations for a space group by the algorithmically generalized Poincar´eTheorem [4], [6], [11] (as in subsection 1.2). The simplest fundamental domains are 3-simplices (tetrahedra) and their integer parts by inner symmetries.
    [Show full text]