DOCSLIB.ORG

Star Polyhedra

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Star Polyhedra Regular polytopes with Hd symmetry Johannes Kepler’s mathematical stars Gabriel Pallier 1 / 34 Outline Background on convex and star polyhedra Star polytopes and platonic Riemann surfaces Schläfli-Hess polytopes 2 / 34 Denote by fpg the convex regular p-gon. The integer p is the number of 0-cells (vertices) and of 1-cells (edges). Convex regular polygons There exists infinitely many convex regular polygons: those are in one-to-one correspondence with elements of Z>3. 3 / 34 Denote by fpg the convex regular p-gon. The integer p is the number of 0-cells (vertices) and of 1-cells (edges). Convex regular polygons There exists infinitely many convex regular polygons: those are in one-to-one correspondence with elements of Z>3. Figure: The convex regular 3-gon. 3 / 34 Convex regular polygons There exists infinitely many convex regular polygons: those are in one-to-one correspondence with elements of Z>3. Figure: The convex regular 4-gon. Denote by fpg the convex regular p-gon. The integer p is the number of 0-cells (vertices) and of 1-cells (edges). 3 / 34 Convex regular polygons There exists infinitely many convex regular polygons: those are in one-to-one correspondence with elements of Z>3. Figure: The convex regular 5-gon. Denote by fpg the convex regular p-gon. The integer p is the number of 0-cells (vertices) and of 1-cells (edges). 3 / 34 Convex regular polygons There exists infinitely many convex regular polygons: those are in one-to-one correspondence with elements of Z>3. Figure: The convex regular 5-gon. Denote by fpg the convex regular p-gon. The integer p is the number of 0-cells (vertices) and of 1-cells (edges). 3 / 34 I There exists p 2 Z>3 such that all 2-cells (faces) are fpg’s, and I There exists q 2 Z>3 such that exactly q faces meet at each vertex. Such a polyhedron is denoted by fp; qg. Figure: The regular icosahedron f3; 5g and dodecahedron f5; 3g. There exists finitely many such polyhedra: the five Plato solids. Convex regular polyhedra Definition A real convex polyhedron is said to be regular if 4 / 34 I There exists q 2 Z>3 such that exactly q faces meet at each vertex. Such a polyhedron is denoted by fp; qg. Figure: The regular icosahedron f3; 5g and dodecahedron f5; 3g. There exists finitely many such polyhedra: the five Plato solids. Convex regular polyhedra Definition A real convex polyhedron is said to be regular if I There exists p 2 Z>3 such that all 2-cells (faces) are fpg’s, and 4 / 34 Figure: The regular icosahedron f3; 5g and dodecahedron f5; 3g. There exists finitely many such polyhedra: the five Plato solids. Convex regular polyhedra Definition A real convex polyhedron is said to be regular if I There exists p 2 Z>3 such that all 2-cells (faces) are fpg’s, and I There exists q 2 Z>3 such that exactly q faces meet at each vertex. Such a polyhedron is denoted by fp; qg. 4 / 34 There exists finitely many such polyhedra: the five Plato solids. Convex regular polyhedra Definition A real convex polyhedron is said to be regular if I There exists p 2 Z>3 such that all 2-cells (faces) are fpg’s, and I There exists q 2 Z>3 such that exactly q faces meet at each vertex. Such a polyhedron is denoted by fp; qg. Figure: The regular icosahedron f3; 5g and dodecahedron f5; 3g. 4 / 34 Convex regular polyhedra Definition A real convex polyhedron is said to be regular if I There exists p 2 Z>3 such that all 2-cells (faces) are fpg’s, and I There exists q 2 Z>3 such that exactly q faces meet at each vertex. Such a polyhedron is denoted by fp; qg. Figure: The regular icosahedron f3; 5g and dodecahedron f5; 3g. There exists finitely many such polyhedra: the five Plato solids. 4 / 34 2 The isometry group Wp;q of fp; qg acts on S with a fundamental spherical triangle 4 of angles π=p; π=q; π=2. Thanks to Gauss-Bonnet theorem (or even its 16th century precursor by Girard), π π π Area(4) = + + − π: 2 p q As Area(4) must be nonnegative, a necessary (and in fact sufficient) condition for fp; qg to define a convex polyhedron is 1 1 1 + > (spherical group eq.) p q 2 Why finitely many ? Equivalently, real convex polyhedra are regular tilings of the geometric sphere S2. 5 / 34 Thanks to Gauss-Bonnet theorem (or even its 16th century precursor by Girard), π π π Area(4) = + + − π: 2 p q As Area(4) must be nonnegative, a necessary (and in fact sufficient) condition for fp; qg to define a convex polyhedron is 1 1 1 + > (spherical group eq.) p q 2 Why finitely many ? Equivalently, real convex polyhedra are regular tilings of the geometric 2 2 sphere S . The isometry group Wp;q of fp; qg acts on S with a fundamental spherical triangle 4 of angles π=p; π=q; π=2. 5 / 34 As Area(4) must be nonnegative, a necessary (and in fact sufficient) condition for fp; qg to define a convex polyhedron is 1 1 1 + > (spherical group eq.) p q 2 Why finitely many ? Equivalently, real convex polyhedra are regular tilings of the geometric 2 2 sphere S . The isometry group Wp;q of fp; qg acts on S with a fundamental spherical triangle 4 of angles π=p; π=q; π=2. Thanks to Gauss-Bonnet theorem (or even its 16th century precursor by Girard), π π π Area(4) = + + − π: 2 p q 5 / 34 Why finitely many ? Equivalently, real convex polyhedra are regular tilings of the geometric 2 2 sphere S . The isometry group Wp;q of fp; qg acts on S with a fundamental spherical triangle 4 of angles π=p; π=q; π=2. Thanks to Gauss-Bonnet theorem (or even its 16th century precursor by Girard), π π π Area(4) = + + − π: 2 p q As Area(4) must be nonnegative, a necessary (and in fact sufficient) condition for fp; qg to define a convex polyhedron is 1 1 1 + > (spherical group eq.) p q 2 5 / 34 where the last equality accounts for χ(S2) = 2, after possibly triangulating the 2-cells. Cell numbers Using the expression for Area(4) and Euler’s formula, one can recover the full combinatorial data of fp; qg: denoting by fp; qg(k) the set of k-cells 4π 4q fp; qg(2) = = ; (Conv 2) 2p · Area(4) 2p + 2q − pq 4p fp; qg(0) = fq; pg(2) = ; (Conv 0) 2p + 2q − pq 2p + 2q + pq fp; qg(1) = fp; qg(2) + fp; qg(0) − 2 = ; (Conv 1) 2p + 2q − pq 6 / 34 Cell numbers Using the expression for Area(4) and Euler’s formula, one can recover the full combinatorial data of fp; qg: denoting by fp; qg(k) the set of k-cells 4π 4q fp; qg(2) = = ; (Conv 2) 2p · Area(4) 2p + 2q − pq 4p fp; qg(0) = fq; pg(2) = ; (Conv 0) 2p + 2q − pq 2p + 2q + pq fp; qg(1) = fp; qg(2) + fp; qg(0) − 2 = ; (Conv 1) 2p + 2q − pq where the last equality accounts for χ(S2) = 2, after possibly triangulating the 2-cells. 6 / 34 This still defines an abstract group Wp;q for any p; q 2 Z>3 with a representation in O(V ) for some real quadratic space V of dim 3. However I V has signature (1; 2) if 1=p + 1=q > 1=2, and I degenerates if 1=p + 1=q = 1=2, 2 which makes Wp;q act on the real hyperbolic plane H or on the affine euclidean plane E2, respectively. The fundamental triangle 4 has angles π=2, π=p, π=q, and its images by Wp;q form the chambers of a Coxeter complex. Coxeter’s kaleidoscope A presentation for Wp;q is 2 2 2 p q Wp;q = r; s; t j r ; s ; t ; (rs) ; (st) : (Cox) 7 / 34 However I V has signature (1; 2) if 1=p + 1=q > 1=2, and I degenerates if 1=p + 1=q = 1=2, 2 which makes Wp;q act on the real hyperbolic plane H or on the affine euclidean plane E2, respectively. The fundamental triangle 4 has angles π=2, π=p, π=q, and its images by Wp;q form the chambers of a Coxeter complex. Coxeter’s kaleidoscope A presentation for Wp;q is 2 2 2 p q Wp;q = r; s; t j r ; s ; t ; (rs) ; (st) : (Cox) This still defines an abstract group Wp;q for any p; q 2 Z>3 with a representation in O(V ) for some real quadratic space V of dim 3. 7 / 34 The fundamental triangle 4 has angles π=2, π=p, π=q, and its images by Wp;q form the chambers of a Coxeter complex. Coxeter’s kaleidoscope A presentation for Wp;q is 2 2 2 p q Wp;q = r; s; t j r ; s ; t ; (rs) ; (st) : (Cox) This still defines an abstract group Wp;q for any p; q 2 Z>3 with a representation in O(V ) for some real quadratic space V of dim 3. However I V has signature (1; 2) if 1=p + 1=q > 1=2, and I degenerates if 1=p + 1=q = 1=2, 2 which makes Wp;q act on the real hyperbolic plane H or on the affine euclidean plane E2, respectively.
Load more

Recommended publications
  • The Small Stellated Dodecahedron Code and Friends
    Downloaded from http://rsta.royalsocietypublishing.org/ on September 5, 2018 The small stellated dodecahedron code and rsta.royalsocietypublishing.org friends J. Conrad1,C.Chamberland2,N.P.Breuckmann3 and Research B. M. Terhal4,5 Cite this article: Conrad J, Chamberland C, 1JARA Institute for Quantum Information, RWTH Aachen University, Breuckmann NP,Terhal BM. 2018 The small Aachen 52056, Germany stellated dodecahedron code and friends. Phil. 2Institute for Quantum Computing and Department of Physics and Trans. R. Soc. A 376: 20170323. Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L http://dx.doi.org/10.1098/rsta.2017.0323 3G1 3Department of Physics and Astronomy, University College London, Accepted: 16 March 2018 London WC1E 6BT, UK 4QuTech, Delft University of Technology, PO Box 5046, 2600 GA Delft, One contribution of 17 to a discussion meeting The Netherlands issue ‘Foundations of quantum mechanics and 5Institute for Theoretical Nanoelectronics, Forschungszentrum their impact on contemporary society’. Juelich, 52425 Juelich, Germany Subject Areas: CC, 0000-0003-3239-5783 quantum computing, quantum physics We explore a distance-3 homological CSS quantum Keywords: code, namely the small stellated dodecahedron code, quantum error correction, fault tolerance, for dense storage of quantum information and we compare its performance with the distance-3 surface homological quantum codes code. The data and ancilla qubits of the small stellated dodecahedron code can be located on Author for correspondence: the edges respectively vertices of a small stellated B. M. Terhal dodecahedron, making this code suitable for three- e-mail: [email protected] dimensional connectivity. This code encodes eight logical qubits into 30 physical qubits (plus 22 ancilla qubits for parity check measurements) in contrast with one logical qubit into nine physical qubits (plus eight ancilla qubits) for the surface code.
    [Show full text]
  • Star Polyhedra: from St Mark's Basilica in Venice To
    STAR POLYHEDRA: FROM ST MARK’S BASILICA IN VENICE TO HUNGARIAN PROTESTANT CHURCHES Tibor TARNAI János KRÄHLING Sándor KABAI Full Professor Associate Professor Manager BUTE BUTE UNICONSTANT Co. Budapest, HUNGARY Budapest, HUNGARY Budapest, HUNGARY Summary Star polyhedra appear on some baroque churches in Rome to represent papal heraldic symbols, and on the top of protestant churches in Hungary to represent the Star of Bethlehem. These star polyhedra occur in many different shapes. In this paper, we will provide a morphological overview of these star polyhedra, and we will reconstruct them by using the program package Mathematica 6. Keywords : Star polyhedra; structural morphology; stellation; elevation; renaissance; baroque architecture; protestant churches; spire stars; Mathematica 6.0. 1. Introduction Star is an ancient symbol that was important already for the Egyptians. Its geometrical representation has been a planar star polygon or a polygramma. In reliefs, they have got certain spatial character, but still they remained 2-dimensional objects. Real 3-dimensional representations of stars are star polyhedra. They are obtained by extending the faces of convex polyhedra so that their planes intersect again [1]. This technique is called stellation . In a broader sense, those polyhedra are also called star polyhedra that are obtained by elevating the face centres of the polyhedra, which are, in this way, augmented by pyramids (the base of a pyramid is identical to the face of the polyhedron on which it stands). To our knowledge, star polyhedra do not occur in European culture until the Renaissance. The first example of a star polyhedron is a planar projection of a small stellated dodecahedron in the pavement mosaic of St Mark’s Basilica in Venice ( Fig.
    [Show full text]
  • 7 Dee's Decad of Shapes and Plato's Number.Pdf
    Dee’s Decad of Shapes and Plato’s Number i © 2010 by Jim Egan. All Rights reserved. ISBN_10: ISBN-13: LCCN: Published by Cosmopolite Press 153 Mill Street Newport, Rhode Island 02840 Visit johndeetower.com for more information. Printed in the United States of America ii Dee’s Decad of Shapes and Plato’s Number by Jim Egan Cosmopolite Press Newport, Rhode Island C S O S S E M R O P POLITE “Citizen of the World” (Cosmopolite, is a word coined by John Dee, from the Greek words cosmos meaning “world” and politês meaning ”citizen”) iii Dedication To Plato for his pursuit of “Truth, Goodness, and Beauty” and for writing a mathematical riddle for Dee and me to figure out. iv Table of Contents page 1 “Intertransformability” of the 5 Platonic Solids 15 The hidden geometric solids on the Title page of the Monas Hieroglyphica 65 Renewed enthusiasm for the Platonic and Archimedean solids in the Renaissance 87 Brief Biography of Plato 91 Plato’s Number(s) in Republic 8:546 101 An even closer look at Plato’s Number(s) in Republic 8:546 129 Plato shows his love of 360, 2520, and 12-ness in the Ideal City of “The Laws” 153 Dee plants more clues about Plato’s Number v vi “Intertransformability” of the 5 Platonic Solids Of all the polyhedra, only 5 have the stuff required to be considered “regular polyhedra” or Platonic solids: Rule 1. The faces must be all the same shape and be “regular” polygons (all the polygon’s angles must be identical).
    [Show full text]
  • Experimentation and Proof in a Solid Geometry Teaching Situation
    FLM 30(3) - November 2010_FLM 05/09/10 8:50 PM Page 36 EXPERIMENTATION AND PROOF IN A SOLID GEOMETRY TEACHING SITUATION DENIS TANGUAY, DENISE GRENIER [1] La déduction est à l’aveugle ce que l’intuition est au they think it is about measuring, estimating, induction paralytique : l’une avance et ne voit pas, l’autre voit et from individual cases, rather than rational scientific n’avance pas. – René Thom [2] process. … Given that there are so few definitions in the [recent UK] curriculum, it would hardly be surpris- Celia Hoyles (1997) has urged the community of mathe- ing if students are unable to distinguish premises and to matics educators to better consider curriculum organisation reason from these to any conclusion. (op. cit., p. 10) and sequencing when grappling with the crucial issue of teaching and learning proof. She characterized the evolution This article is developed from our contribution to ICMI-19 of the curricular approach to proof in the UK as a pendu- (Tanguay & Grenier, 2009), and partly supports this warning. lum’s swing [3], the extremes of which could be caricatured Through the experiment presented here, we show how experi- as follows: mentation and construction, in conjunction with some usual 1) At one extreme, its teaching being confined to classroom contracts about proof, may sometimes contribute to Euclidean geometry, proof is considered as the math- move students away from the relevant reasoning and proving. ematically ultimate way of validating a result, to be produced according to a specific two-column format A didactical assumption in isolation from exploring and constructing activi- For the design of the activity, our starting assumption was ties.
    [Show full text]
  • Hereditary Polyhedra with Planar Regular Faces∗
    ISSN 2590-9770 The Art of Discrete and Applied Mathematics 3 (2020) #P2.07 https://doi.org/10.26493/2590-9770.1281.0ad (Also available at http://adam-journal.eu) Hereditary polyhedra with planar regular faces∗ Egon Schultey Department of Mathematics, Northeastern University, Boston, MA 02115, USA Asia Ivic´ Weissz Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada In memory of Norman Johnson, our friend and colleague. Received 18 December 2018, accepted 6 July 2019, published online 10 August 2020 Abstract A skeletal polyhedron in Euclidean 3-space is called hereditary if the symmetries of each face extend to symmetries of the entire polyhedron. In this paper we describe the finite hereditary skeletal polyhedra which have regular convex polygons or regular star- polygons as faces. Keywords: Symmetries of polyhedra, geometric polyhedral, uniform polyhedra. Math. Subj. Class. (2020): 51M20, 52B05, 52B22 1 Introduction In the design of polyhedral structures with high symmetry it is quite natural to proceed from a highly symmetric structure of lower rank (or dimension) and ask for the symmetries of the lower rank structure to be preserved for the entire structure. The entire structure then inherits the symmetries of the lower rank structure. For example, the Platonic solids and ∗Special thanks go to Peter McMullen for pointing out the omission of a known uniform polyhedron from the list in an earlier version of the manuscript. We did know about the polyhedron but a bit of absentmindedness had caused us to forget to include it. His forthcoming paper [13] will describe an alternative approach to the enumeration presented here, and will also deal with the case of skew faces.
    [Show full text]
  • Geometric Realizations of Abstract Regular Polyhedra with Automorphism Group H3
    Ateneo de Manila University Archīum Ateneo Mathematics Faculty Publications Mathematics Department 2-5-2020 Geometric realizations of abstract regular polyhedra with automorphism group H3 Mark L. Loyola Jonn Angel L. Aranas Follow this and additional works at: https://archium.ateneo.edu/mathematics-faculty-pubs Part of the Geometry and Topology Commons electronic reprint ISSN: 2053-2733 journals.iucr.org/a Geometric realizations of abstract regular polyhedra with automorphism group H 3 Jonn Angel L. Aranas and Mark L. Loyola Acta Cryst. (2020). A76, 358–368 IUCr Journals CRYSTALLOGRAPHY JOURNALS ONLINE Copyright c International Union of Crystallography Author(s) of this article may load this reprint on their own web site or institutional repository provided that this cover page is retained. Republication of this article or its storage in electronic databases other than as specified above is not permitted without prior permission in writing from the IUCr. For further information see https://journals.iucr.org/services/authorrights.html Acta Cryst. (2020). A76, 358–368 Aranas and Loyola · Geometric realizations of abstract regular polyhedra research papers Geometric realizations of abstract regular polyhedra with automorphism group H3 ISSN 2053-2733 Jonn Angel L. Aranas and Mark L. Loyola* Department of Mathematics, Ateneo de Manila University, Katipunan Avenue, Loyola Heights, Quezon City 1108, Philippines. *Correspondence e-mail: [email protected] Received 18 October 2019 Accepted 4 February 2020 A geometric realization of an abstract polyhedron P is a mapping that sends an i-face to an open set of dimension i. This work adapts a method based on Wythoff construction to generate a full rank realization of an abstract regular Edited by J.-G.
    [Show full text]
  • The Stars Above Us: Regular and Uniform Polytopes up to Four Dimensions Allen Liu Contents
    The Stars Above Us: Regular and Uniform Polytopes up to Four Dimensions Allen Liu Contents 0. Introduction: Plato’s Friends and Relations a) Definitions: What are regular and uniform polytopes? 1. 2D and 3D Regular Shapes a) Why There are Five Platonic Solids 2. Uniform Polyhedra a) Solid 14 and Vertex Transitivity b)Polyhedron Transformations 3. Dice Duals 4. Filthy Degenerates: Beach Balls, Sandwiches, and Lines 5. Regular Stars a) Why There are Four Kepler-Poinsot Polyhedra b)Mirror-regular compounds c) Stellation and Greatening 6. 57 Varieties: The Uniform Star Polyhedra 7. A. Square to A. Cube to A. Tesseract 8. Hyper-Plato: The Six Regular Convex Polychora 9. Hyper-Archimedes: The Convex Uniform Polychora 10. Schläfli and Hess: The Ten Regular Star Polychora 11. 1849 and counting: The Uniform Star Polychora 12. Next Steps Introduction: Plato’s Friends and Relations Tetrahedron Cube (hexahedron) Octahedron Dodecahedron Icosahedron It is a remarkable fact, known since the time of ancient Athens, that there are five Platonic solids. That is, there are precisely five polyhedra with identical edges, vertices, and faces, and no self- intersections. While will see a formal proof of this fact in part 1a, it seems strange a priori that the club should be so exclusive. In this paper, we will look at extensions of this family made by relaxing some conditions, as well as the equivalent families in numbers of dimensions other than three. For instance, suppose that we allow the sides of a shape to pass through one another. Then the following figures join the ranks: i Great Dodecahedron Small Stellated Dodecahedron Great Icosahedron Great Stellated Dodecahedron Geometer Louis Poinsot in 1809 found these four figures, two of which had been previously described by Johannes Kepler in 1619.
    [Show full text]
  • Ad Quadratum Construction and Study of the Regular Polyhedra
    Ad Quadratum Construction and Study of the Regular Polyhedra by Jean Le Mée The Louis and Jeanette Brooks Engineering Design Center Department of Mechanical Engineering Studies in the Design Process Albert Nerken School of Engineering and Design Education The Cooper Union for the Advancement of Science & Art Series A, Number 8 New York, NY 10003 November 2001 Ad Quadratum Construction and Study of the Regular Polyhedra by Jean Le Mée The Louis and Jeanette Brooks Engineering Design Center Department of Mechanical Engineering Studies in the Design Process Albert Nerken School of Engineering and Design Education The Cooper Union for the Advancement of Science & Art Series A, Number 8 New York, NY 10003 November 2001 Copyright ¤ 2001 by Jean Le Mée iii Table of Contents Ad Quadratum Construction and Study of the Regular Polyhedra Acknowledgements ..................................................................................................vi Preface.......................................................................................................................vii Introduction..............................................................................................................1 Generation of the Platonic Forms ..........................................................................3 1. A view from the Center .............................................................................3 2. Complementary views ...............................................................................5 Platonic Forms and Circumsphere.........................................................................5
    [Show full text]
  • The Small Stellated Dodecahedron Code and Friends
    The Small Stellated Dodecahedron Code and rspa.royalsocietypublishing.org Friends Research J. Conrad1, C. Chamberland2, N. P. Breuckmann3, B. M. Terhal4;5 Article submitted to journal 1 JARA Institute for Quantum Information, RWTH Aachen University, Aachen 52056, Germany Subject Areas: 2 Institute for Quantum Computing and Department of quantum information Physics and Astronomy, University of Waterloo, Keywords: Waterloo, Ontario, N2L 3G1, Canada 3 quantum error correction, Department of Physics and Astronomy, University fault-tolerance, homological quantum College London, London WC1E 6BT, UK 4 codes QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands 5 Author for correspondence: Institute for Theoretical Nanoelectronics, Barbara M. Terhal Forschungszentrum Juelich, D-52425 Juelich, e-mail: [email protected] Germany We explore a distance-3 homological CSS quantum code, namely the small stellated dodecahedron code, for dense storage of quantum information and we compare its performance with the distance-3 surface code. The data and ancilla qubits of the small stellated dodecahedron code can be located on the edges resp. vertices of a small stellated dodecahedron, making this code suitable for 3D connectivity. This code encodes 8 logical qubits into 30 physical qubits (plus 22 ancilla qubits for parity check measurements) as compared to 1 logical qubit into 9 physical qubits (plus 8 ancilla qubits) for the surface code. We develop fault- tolerant parity check circuits and a decoder for this code, allowing us to numerically assess the circuit- based pseudo-threshold. arXiv:1712.07666v2 [quant-ph] 29 Mar 2018 c The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/ by/4.0/, which permits unrestricted use, provided the original author and source are credited.
    [Show full text]
  • Polyhedron Modelling and Symmetry Groups
    Polyhedron modelling and symmetry groups I. Prok Department of Geometry, TU Budapest Abstract The theory of the regular and semi-regular polyhedra is a classical topic of geometry. However, in many cases (e.g. the regular and semi-regular star polyhedra), it seams to be a difficult problem to generate their numerical model by a CAD system. In this paper we shall present a method for constructing the data structure of these polyhedra on the basis of their symmetry group. Also a collection of regular and semi-regular solids will be visualized by computer. 1. Introduction denote the regular star pentagon the so-called pentagram. A solid is said to be regular if its faces are congruent regular The theory of regular and semi-regular polyhedra is a classi- polygons surrounding the vertices alike. Then the vertex fig- cal topic of geometry 1; 2; 3. The five Platonic and the thirteen ures are also regular polygons. A regular solid is denoted Archimedian solids are the most famous representatives of by a pair of numbers in braces: fp;qg means that the faces this type of polyhedra, but the four regular star solids and are regular p-gons and the vertex figures are regular q-gons. numerous other convex and non-convex semi-regular solids Transposing the role of the faces and vertices we can get also belong to this family. More generally, in this paper we the dual pair fq; pg of fp;qg. There are five convex regu- shall consider those polyhedra whose symmetries form the lar solids (the well-known Platonic solids): the tetrahedron symmetry group of a Platonic solid (or its orientation pre- f3;3g, the octahedron f3;4g, the cube (hexahedron) f4;3g, serving subgroup).
    [Show full text]
  • Historical Notes on Star Geometry in Mathematics, Art and Nature
    Historical notes on star geometry in mathematics, art and nature Aldo Brigaglia, Nicla Palladino, Maria Alessandra Vaccaro - Gamma: “I can. Look at this Counterexample 3: a star-polyhedron -I shall call it urchin. This consists of 12 star-pentagons. It has 12 vertices, 30 edges, and 12 pentagonal faces-you may check it if you like by counting. Thus the Descartes- Euler thesis is not true at all, since for this polyhedron V - E + F = - 6”. - Delta: “Why do you think that your 'urchin' is a polyhedron?” - Gamma: “Do you not see? This is a polyhedron, whose faces are the twelve star- pentagons”. - Delta: “But then you do not even know what a polygon is! A star-pentagon is certainly not a polygon!” In the above dialogue from [25], Imre Lakatos used the example of star polyhedra to describe the complex path definition – proof – refutation – new definition that mathematical thought threaded before reaching a consensus on fundamental points of the main mathematical topics. In this paper we will try to examine how in the history of polyhedra (and in particular star polyhedra), a long period of «discov- ery» of individual types due to the observation of natural objects or due to artistic imagination preceded (and was connected with) the mathematical solution fixing the “right” definitions. Such long period of discovery—we will argue—influenced further investigations on nature and art. The paper will start from the thirteen cen- tury and will end with the publications of Pappo’s work [30] and Kepler’s “Har- monices Mundi” which provided solid mathematical foundations to the subject.
    [Show full text]
  • Researches on Polyhedra, Part I A.-L
    Researches on polyhedra, Part I A.-L. Cauchy Translated into English by Guy Inchbald, © 2006 from the original: A.-L. Cauchy, “Recherches sur les polyèdres. Première partie,” Journal de l’École Polytechnique, Cahier 16, t. 9 (1813), pp. 68-74. Translator’s note Augustin-Louis Cauchy was the first to prove that the four regular star polyhedra, described by Poinsot, are the full set. He derived them by stellating the Platonic (regular convex) solids. His proof draws on the ability to rotate a copy of a regular solid and then superimpose it exactly on the original, setting the foundations for what we would nowadays call symmetry orbits. The Journal de l’École Polytechnique was organised in a confusing way. One or more consecutive Cahiers (Books) were gathered in a single Tome (sometimes translated as “Volume”), while existing bound Volumes (e.g. in the ENSAM archive at Aix-en-Provence) apparently may also gather one or more consecutive Cahiers though not necessarily corresponding to the division into Tomes. Cauchy’s writings in Cahier 16 form two consecutive but independent Memoirs, each divided into a few words of introduction followed by two main Parts. The present paper was reprinted in Série 2, Tome 1 of his Oeuvres Complètes, published in 1905 and re-published electronically by the Académie des sciences in 1995. This translation is made from the electronic (1995) version. Only the brief introduction and Part I of the first Memoir are translated here. I have tried to translate the original fairly literally, while sometimes taking liberties in the interest of readability in the English idiom.
    [Show full text]