Article of Alexandra Fritz and Herwig Hauser
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Are Your Polyhedra the Same As My Polyhedra?
Are Your Polyhedra the Same as My Polyhedra? Branko Gr¨unbaum 1 Introduction “Polyhedron” means different things to different people. There is very little in common between the meaning of the word in topology and in geometry. But even if we confine attention to geometry of the 3-dimensional Euclidean space – as we shall do from now on – “polyhedron” can mean either a solid (as in “Platonic solids”, convex polyhedron, and other contexts), or a surface (such as the polyhedral models constructed from cardboard using “nets”, which were introduced by Albrecht D¨urer [17] in 1525, or, in a more mod- ern version, by Aleksandrov [1]), or the 1-dimensional complex consisting of points (“vertices”) and line-segments (“edges”) organized in a suitable way into polygons (“faces”) subject to certain restrictions (“skeletal polyhedra”, diagrams of which have been presented first by Luca Pacioli [44] in 1498 and attributed to Leonardo da Vinci). The last alternative is the least usual one – but it is close to what seems to be the most useful approach to the theory of general polyhedra. Indeed, it does not restrict faces to be planar, and it makes possible to retrieve the other characterizations in circumstances in which they reasonably apply: If the faces of a “surface” polyhedron are sim- ple polygons, in most cases the polyhedron is unambiguously determined by the boundary circuits of the faces. And if the polyhedron itself is without selfintersections, then the “solid” can be found from the faces. These reasons, as well as some others, seem to warrant the choice of our approach. -
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. -
Representing the Sporadic Archimedean Polyhedra As Abstract Polytopes$
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Discrete Mathematics 310 (2010) 1835–1844 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Representing the sporadic Archimedean polyhedra as abstract polytopesI Michael I. Hartley a, Gordon I. Williams b,∗ a DownUnder GeoSolutions, 80 Churchill Ave, Subiaco, 6008, Western Australia, Australia b Department of Mathematics and Statistics, University of Alaska Fairbanks, PO Box 756660, Fairbanks, AK 99775-6660, United States article info a b s t r a c t Article history: We present the results of an investigation into the representations of Archimedean Received 29 August 2007 polyhedra (those polyhedra containing only one type of vertex figure) as quotients of Accepted 26 January 2010 regular abstract polytopes. Two methods of generating these presentations are discussed, Available online 13 February 2010 one of which may be applied in a general setting, and another which makes use of a regular polytope with the same automorphism group as the desired quotient. Representations Keywords: of the 14 sporadic Archimedean polyhedra (including the pseudorhombicuboctahedron) Abstract polytope as quotients of regular abstract polyhedra are obtained, and summarised in a table. The Archimedean polyhedron Uniform polyhedron information is used to characterize which of these polyhedra have acoptic Petrie schemes Quotient polytope (that is, have well-defined Petrie duals). Regular cover ' 2010 Elsevier B.V. All rights reserved. Flag action Exchange map 1. Introduction Much of the focus in the study of abstract polytopes has been on the regular abstract polytopes. A publication of the first author [6] introduced a method for representing any abstract polytope as a quotient of regular polytopes. -
Representing the Sporadic Archimedean Polyhedra As Abstract Polytopes
REPRESENTING THE SPORADIC ARCHIMEDEAN POLYHEDRA AS ABSTRACT POLYTOPES MICHAEL I. HARTLEY AND GORDON I. WILLIAMS Abstract. We present the results of an investigation into the representations of Archimedean polyhedra (those polyhedra containing only one type of vertex gure) as quotients of regu- lar abstract polytopes. Two methods of generating these presentations are discussed, one of which may be applied in a general setting, and another which makes use of a regular poly- tope with the same automorphism group as the desired quotient. Representations of the 14 sporadic Archimedean polyhedra (including the pseudorhombicuboctahedron) as quotients of regular abstract polyhedra are obtained, and summarised in a table. The information is used to characterise which of these polyhedra have acoptic Petrie schemes (that is, have well-dened Petrie duals). 1. Introduction Much of the focus in the study of abstract polytopes has been on the study of the regular abstract polytopes. A publication of the rst author [Har99a] introduced a method for rep- resenting any abstract polytope as a quotient of regular polytopes. In the current work we present the application of this technique to the familiar, but still interesting, Archimedean polyhedra and discuss implications for the general theory of such representations that arose in trying to systematically develop these representations. We discuss the theory and presen- tations of the thirteen classical (uniform) Archimedean polyhedra as well as the pseudorhom- bicuboctahedron, which we will refer to as the fourteen sporadic Archimedean polyhedra. In a separate study, we will present and discuss the presentations for the two innite families of uniform convex polyhedra, the prisms and antiprisms. -
Max Brücknerʼs Wunderkammer of Paper Polyhedra
Bridges 2019 Conference Proceedings Max Brücknerʼs Wunderkammer of Paper Polyhedra George W. Hart Wiarton, ON, Canada, [email protected] Abstract In 1900 the German mathematician Max Brückner published a book with photographs of 146 amazing paper polyhedron models. While containing little that was cutting-edge mathematically and not produced as fine art, the photographs have had an enormous influence on mathematical art ever since. The artist M.C. Escher was particularly influential in spreading Brückner's ideas. I argue that the import of the book can best be understood by seeing it as a Wunderkammer—a cabinet of curiosities—that excited wonder in the reader. This paper explores Brückner's work and its legacy from that perspective. Introduction and History Max Brückner (1860–1934) was a German mathematician who received his Ph.D. at Leipzig University in 1886 under the supervision of the renowned Felix Klein. Not a university professor, he taught first at a grammar school and then a gymnasium (an academic high school). Brückner is best known for his 1900 book Vielecke und Vielflache: Theorie und Geschichte (Polygons and Polyhedra: Theory and History), which summarized much of what was then known about polyhedra. It is illustrated with hundreds of engraved images and ten full sheets of photographic plates illustrating 146 spectacular paper models neatly arrayed on shelves. Figures 1–3 show Brückner's Tables IX, VIII, and X respectively [24, 3]. Figure 1: Table IX from Brückner (two full page leafs in the original). n. 23 is a compound of three cubes; n. 6 is five octahedra; n. -
Systematic Symmetries: an Inquiry Into the Infinite Via the Works of Mc Escher
SYSTEMATIC SYMMETRIES: AN INQUIRY INTO THE INFINITE VIA THE WORKS OF M.C. ESCHER A thesis submitted to the Kent State University Honors College in partial fulfillment of the requirements for University Honors by Anna Levina May, 2011 Thesis written by Anna Levina Approved by , Advisor , Chair, Department of Mathematical Sciences , Dean, Honors College ii TABLE OF CONTENTS LIST OF FIGURES . iv LIST OF TABLES . vi ACKNOWLEDGEMENTS . vii 1 Introduction . .1 2 M.C. Escher . .4 3 Escher's Tessellations . 10 4 The Quadrilateral Systems . 13 5 Triangle Systems . 21 6 Wallpaper Groups . 25 7 Hyperbolic Geometry: Coxeter, Circle Limits, and Constructions . 36 8 Conclusion . 57 9 Bibliography . 60 10 Appendix I . 64 11 Appendix II . 67 iii LIST OF FIGURES 1 Castrovalva . .5 2 A few of Escher's works. .8 3 Reptiles . 11 4 A Translation of a Llama . 14 5 A Reflected Gnome . 14 6 A Glide Reflection with a Message . 15 7 Polygons being transformed in a transversal direction . 15 8 Rhombi transformed in diagonal directions. 16 9 System Transition . 18 10 Escher utilized a 2{icosohedral tiling in the construction of Sky and Water I. 19 11 An example of 3{fold symmetry. 21 12 An example of 6{fold symmetry. 21 13 Triangle System Transitions . 23 14 A translation. 28 15 A rotation. 29 16 A reflection. 30 17 A glide reflection . 31 18 Orthogonal Circles . 38 19 Poincar´elines of the first and second kind. 39 20 The points A and B, their corresponding ideal points P and Q, and the line segments necessary for the cross ratio which determines distance . -
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. -
Islamic Tilings and Polyhedra Teachers Booklet
���������������������� �������������������������������������� ������������������������������� ��������������� ���������������������������������������������������������� ���������������������������������������������� ���������������������������� Form, Shape and Space: An Exhibition of Tilings and Polyhedra Contents 1. Introduction to the Exhibition 2 2. Islamic ArtArt 8 2.1 Geometry in Islamic art 10 2.2 Symbolism in Islamic art 11 3. The Alhambra Palace 12 4. The Platonic Solids 13 5. Activities 18 5.1 Making shapes: Equilateral triangles, hexagons and six-pointed stars 18 using a compass and ruler 5.2 Making shapes: Squares, octagons, crosses and eight-pointed stars using 19 a compass and ruler 5.3 Making shapes: Paper-folding polygons 20 5.4 Making patterns: Islamic tiling patterns using folded paper shapes 22 5.5 Making polyhedra: Patterned Platonic polyhedra using pull-up nets 31 5.6 Making polyhedra: Make your own Patterned Pull-up nets 42 1 1. Introduction to the Exhibition Symmetry is a simple concept that has tremendous meaning. It is possibly the most signifi cant and elegant connection that transcends the boundaries between art, science and mathematics. Symmetry surrounds us, both in the natural world and in the world conceived by humans. Patterns and proportional relationships create a visual language expressing order and generating appealing, fascinating compositions. In the natural world symmetrical patterns can be found on every conceivable scale. Microscopic organisms, viruses and crystal structures, all exhibit the mathematical regularities of symmetry. One of the most practical applications of symmetry, drawing on concepts originating in the study of molecular crystal structures, is in the analysis and construction of regularly repeating patterns and tilings. Following certain geometrical rules, a wealth of patterns can be created, using the economical power of symmetry. -
Exploring Polyhedra and Discovering Euler's
Exploring Polyhedra and Discovering Euler’s Formula1 Leah Wrenn Berman and Gordon Williams Ursinus College Summary The activities and exercises collected here provide an introduction to Euler’s formula, an introduction to interesting related topics, and sources for further exploration. While Euler’s formula applies to any planar graph, a natural and accessible context for the study of Euler’s formula is the study of polyhedra. The article includes an introductionto Euler’s formula, four student activities, and two appendices containing useful information for the instructor, such as an inductive proof of Euler’s theorem and several other interesting results that may be proved using Euler’s theorem. Each activity includes a discussion of connections to discrete mathematics and notes to the instructor. Three of the activities have worksheets at the end of this article, followed by solutions and a template for a toroidal polyhedron. A brief introduction to Euler’s formula Theorem (Euler’s formula, polyhedral version). Let P be any polyhedron topologicallyequivalent to a sphere. Let V be the number of vertices, E the number of edges and F the number of faces of P . Then V E F 2. C D Agraph issaid tobe a simple graph if it is an undirected graph containingneither loops nor multiple edges. A graph is a plane graph if it is embedded in the plane withoutcrossing edges. A graph is said to be planar if it admits such an embedding. Theorem (Euler’s formula, graph version). Let G be any simple plane graph. Let V be the number of vertices, E the number of edges and F the number of faces (alternately, the number of connected sets in the complement) of G. -
Woven Polyhedra (Imagiro37).Nb 1
Woven Polyhedra (Imagiro37).nb 1 Origami Burrs and Woven Polyhedra by Robert J. Lang This article and its successors originally appeared in Imagiro, an amateur press alliance, in 2000. In 1999 I attended the 2nd Scandinavian Origami Convention in Stockholm, where one of the attendees was Herman van Goubergen. Herman had some new models to display. This was an occurence of some note, because Herman only invents about one new model per year, and every time he does, it's utterly unlike anything you've ever seen before in the origami world. This year was a particularly auspicious occasion: he had invented several models. Herman's models were modulars. But, in keeping with his modus operandi, they were very different from any other modulars I'd ever seen, making use of curves and novel locking mechanisms. Herman explained his motivation: "I was trying to come up with modulars that were not just another way of making a triangle by sticking flaps into pockets." And that got me to thinking about modulars, too. I periodically play around with modulars, attracted by their geometrical properties and three-dimensionality. The way it works is, I get intrigued by their structure and design one or two; but then I have to actually fold it, and after I've folded what seems like several thousand units I ask myself why I'm singing my wings against this particular flame and go back to folding something restful and relaxing, like color-changed mating mosquitos. What Herman got me to thinking was, "what other ways are there to build modulars?" This thought bumped into another question that's been rattling around my brain lately, which is what to do for the Gathering For Gardner 4 meeting next spring. -
Geometric Symmetry Slides
GEOMETRIC SYMMETRY SLIDES ORDER FOR FRIEZE GROUPS t L tg LΓ tm V mt C t2 N t2mg VΛ t2mm H ORDER FOR WALLPAPER GROUPS p1 p2 pm pg cm pmm pmg pgg cmm p4 p4m p4g p3 p3m1 p31m p6 p6m 1 GEOMETRIC SYMMETRY—SLIDES 010 (Cypriot Vase with the Name of Thales). 020 SONNET ABOUT THE RAIN - Dobrivoje Jevti , 1978. B. Pavlovi , “On symmetry and asymmetry in literature”, Comp. and Math. with Applics. 12B (1/2) (1986), 197–227, at p.208 = Istvan Hargittai, Symmetry Unifying Human Understanding, Pergamon 1986, 197–227, at p.208. 030 Hedgehog Edward Rolland McLachlan, The Cartoons of McLachlan, Private Eye 1973. 040 Palindrome. THE END - Frederic Brown. Martin Gardner, The Ambidextrous Universe, Penguin 1970, end of Chapter 5 (pp.50–51). 050 Palindrome BLACK AND WHITE by J.A. Lindon. Lived as a dog - o no! God, as a devil Doom lives ever, it’s astir, Eve’s evil mood, Live, O devil, revel ever, live, do evil! Do, O God, no evil deed, live on, do good! Howard W. Bergerson, Palindromes and Anagrams, Dover 1973, p.117. 060 m The human anatomy according to ideal principles. Venice, Academy. [N.B. Leonardo’s notebooks were written backwards, from right to left]. Bruno Santi, Leonardo, Constable 1978, p.41. 070 m Mixtec culture: a sacrificial obsidian knife with mosaic handle. A two-handled mosaic serpent (below). These objects are believed to have been among the treasure presented to Cort´es by Moctezuma. Hammond Innes, The Conquistadors, Collins 1969, repr. by Fontana 1972, p.110. -
“Platonic Stars” Construction of Algebraic Curves and Surfaces with Prescribed Symmetries and Singularities
\Platonic stars" Construction of algebraic curves and surfaces with prescribed symmetries and singularities. diploma thesis in mathematics by Alexandra Fritz submitted to the Faculty of Mathematics, Computer Science and Physics of the University of Innsbruck in partial fulfillment of the requirements for the degree of Diplom-Ingenieurin supervisor: V.-Prof. Dr. Herwig Hauser, Faculty of Mathematics, University Vienna Innsbruck, December 11, 2009 ii Contents Preface v 1 Preparation 1 1.1 Basic notations . 1 1.2 Some basics from group theory . 4 1.3 Polytopes . 6 1.4 Normal forms of simple singularities . 10 1.5 Some invariant theory . 13 2 Construction 21 2.1 Recipe . 22 2.2 Explanatory example: Octahedral and hexahedral stars. 24 2.3 Three generalizations . 26 3 Examples 29 3.1 Plane dihedral stars . 29 3.2 Platonic and Archimedean stars . 34 3.3 Catalan stars and \relatives" . 40 3.4 Dihedral stars . 44 3.5 Further examples . 48 A Technical details 51 A.1 The Archimedean and Catalan solids . 51 A.2 Generators of some symmetry groups. 52 A.3 Primary and secondary invariants . 53 A.4 Coordinate change . 54 A.5 Factorization of the primary invariants of Ih ...................... 56 A.6 SINGULAR input and output . 56 Bibliography 58 iii iv CONTENTS Preface And so I raise my glass to symmetry To the second hand and its accuracy To the actual size of everything The desert is the sand You can't hold it in your hand It won't bow to your demands There's no difference you can make (Bright Eyes \I Believe in Symmetry"1) Actually the title of the thesis already says it all: the aim of our work was to construct algebraic curves and surfaces with previously fixed properties, namely symmetries and singularities.