Ivory Polyhedra Turned on a Lathe
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Final Poster
Associating Finite Groups with Dessins d’Enfants Luis Baeza, Edwin Baeza, Conner Lawrence, and Chenkai Wang Abstract Platonic Solids Rotation Group Dn: Regular Convex Polygon Approach Each finite, connected planar graph has an automorphism group G;such Following Magot and Zvonkin, reduce to easier cases using “hypermaps” permutations can be extended to automorphisms of the Riemann sphere φ : P1(C) P1(C), then composing β = φ f where S 2(R) P1(C). In 1984, Alexander Grothendieck, inspired by a result of f : 1( ) ! 1( )isaBely˘ımapasafunctionofeither◦ zn or ' P C P C Gennadi˘ıBely˘ıfrom 1979, constructed a finite, connected planar graph 4 zn/(zn +1)! 2 such that Aut(f ) Z or Aut(f ) D ,respectively. ' n ' n ∆β via certain rational functions β(z)=p(z)/q(z)bylookingatthe inverse image of the interval from 0 to 1. The automorphisms of such a Hypermaps: Rotation Group Zn graph can be identified with the Galois group Aut(β)oftheassociated 1 1 rational function β : P (C) P (C). In this project, we investigate how Rigid Rotations of the Platonic Solids I Wheel/Pyramids (J1, J2) ! w 3 (w +8) restrictive Grothendieck’s concept of a Dessin d’Enfant is in generating all n 2 I φ(w)= 1 1 z +1 64 (w 1) automorphisms of planar graphs. We discuss the rigid rotations of the We have an action : PSL2(C) P (C) P (C). β(z)= : v = n + n, e =2 n, f =2 − n ◦ ⇥ 2 !n 2 4 zn · Platonic solids (the tetrahedron, cube, octahedron, icosahedron, and I Zn = r r =1 and Dn = r, s s = r =(sr) =1 are the rigid I Cupola (J3, J4, J5) dodecahedron), the Archimedean solids, the Catalan solids, and the rotations of the regular convex polygons,with 4w 4(w 2 20w +105)3 I φ(w)= − ⌦ ↵ ⌦ 1 ↵ Rotation Group A4: Tetrahedron 3 2 Johnson solids via explicit Bely˘ımaps. -
Chiral Polyhedra Derived from Coxeter Diagrams and Quaternions
SQU Journal for Science, 16 (2011) 82-101 © 2011 Sultan Qaboos University Chiral Polyhedra Derived from Coxeter Diagrams and Quaternions Mehmet Koca, Nazife Ozdes Koca* and Muna Al-Shueili Department of Physics, College of Science, Sultan Qaboos University, P.O.Box 36, Al- Khoud, 123 Muscat, Sultanate of Oman, *Email: [email protected]. متعددات السطوح اليدوية المستمدة من أشكال كوكزتر والرباعيات مهمت كوجا، نزيفة كوجا و منى الشعيلي ملخص: هناك متعددات أسطح ٌدوٌان أرخمٌدٌان: المكعب المسطوح والثنعشري اﻷسطح المسطوح مع أشكالها الكتﻻنٌة المزدوجة: رباعٌات السطوح العشرونٌة الخماسٌة مع رباعٌات السطوح السداسٌة الخماسٌة. فً هذا البحث نقوم برسم متعددات السطوح الٌدوٌة مع مزدوجاتها بطرٌقة منظمة. نقوم باستخدام المجموعة الجزئٌة الدورانٌة الحقٌقٌة لمجموعات كوكستر ( WAAAWAWB(1 1 1 ), ( 3 ), ( 3 و WH()3 ﻻشتقاق المدارات الممثلة لﻷشكال. هذه الطرٌقة تقودنا إلى رسم المتعددات اﻷسطح التالٌة: رباعً الوجوه وعشرونً الوجوه و المكعب المسطوح و الثنعشري السطوح المسطوح على التوالً. لقد أثبتنا أنه بإمكان رباعً الوجوه وعشرونً الوجوه التحول إلى صورتها المرآتٌة بالمجموعة الثمانٌة الدورانٌة الحقٌقٌة WBC()/32. لذلك ﻻ ٌمكن تصنٌفها كمتعددات أسطح ٌدوٌة. من المﻻحظ أٌضا أن رؤوس المكعب المسطوح ورؤوس الثنعشري السطوح المسطوح ٌمكن اشتقاقها من المتجهات المجموعة خطٌا من الجذور البسٌطة وذلك باستخدام مجموعة الدوران الحقٌقٌة WBC()/32 و WHC()/32 على التوالً. من الممكن رسم مزدوجاتها بجمع ثﻻثة مدارات من المجموعة قٌد اﻻهتمام. أٌضا نقوم بإنشاء متعددات اﻷسطح الشبه منتظمة بشكل عام بربط متعددات اﻷسطح الٌدوٌة مع صورها المرآتٌة. كنتٌجة لذلك نحصل على مجموعة البٌروهٌدرال كمجموعة جزئٌة من مجموعة الكوكزتر WH()3 ونناقشها فً البحث. الطرٌقة التً نستخدمها تعتمد على الرباعٌات. ABSTRACT: There are two chiral Archimedean polyhedra, the snub cube and snub dodecahedron together with their dual Catalan solids, pentagonal icositetrahedron and pentagonal hexacontahedron. -
On Some Properties of Distance in TO-Space
Aksaray University Aksaray J. Sci. Eng. Journal of Science and Engineering Volume 4, Issue 2, pp. 113-126 e-ISSN: 2587-1277 doi: 10.29002/asujse.688279 http://dergipark.gov.tr/asujse http://asujse.aksaray.edu.tr Available online at Research Article On Some Properties of Distance in TO-Space Zeynep Can* Department of Mathematics, Faculty of Science and Letters, Aksaray University, Aksaray 68100, Turkey ▪Received Date: Feb 12, 2020 ▪Revised Date: Dec 21, 2020 ▪Accepted Date: Dec 24, 2020 ▪Published Online: Dec 25, 2020 Abstract The aim of this work is to investigate some properties of the truncated octahedron metric introduced in the space in further studies on metric geometry. With this metric, the 3- dimensional analytical space is a Minkowski geometry which is a non-Euclidean geometry in a finite number of dimensions. In a Minkowski geometry, the unit ball is a certain symmetric closed convex set instead of the usual sphere in Euclidean space. The unit ball of the truncated octahedron geometry is a truncated octahedron which is an Archimedean solid. In this study, 2 first, metric properties of truncated octahedron distance, 푑푇푂, in ℝ has been examined by 3 metric approach. Then, by using synthetic approach some distance formulae in ℝ푇푂, 3- dimensional analytical space furnished with the truncated octahedron metric has been found. Keywords Metric, Convex polyhedra, Truncated octahedron, Distance of a point to a line, Distance of a point to a plane, Distance between two lines *Corresponding Author: Zeynep Can, [email protected], 0000-0003-2656-5555 2017-2020©Published by Aksaray University 113 Zeynep Can (2020). -
[ENTRY POLYHEDRA] Authors: Oliver Knill: December 2000 Source: Translated Into This Format from Data Given In
ENTRY POLYHEDRA [ENTRY POLYHEDRA] Authors: Oliver Knill: December 2000 Source: Translated into this format from data given in http://netlib.bell-labs.com/netlib tetrahedron The [tetrahedron] is a polyhedron with 4 vertices and 4 faces. The dual polyhedron is called tetrahedron. cube The [cube] is a polyhedron with 8 vertices and 6 faces. The dual polyhedron is called octahedron. hexahedron The [hexahedron] is a polyhedron with 8 vertices and 6 faces. The dual polyhedron is called octahedron. octahedron The [octahedron] is a polyhedron with 6 vertices and 8 faces. The dual polyhedron is called cube. dodecahedron The [dodecahedron] is a polyhedron with 20 vertices and 12 faces. The dual polyhedron is called icosahedron. icosahedron The [icosahedron] is a polyhedron with 12 vertices and 20 faces. The dual polyhedron is called dodecahedron. small stellated dodecahedron The [small stellated dodecahedron] is a polyhedron with 12 vertices and 12 faces. The dual polyhedron is called great dodecahedron. great dodecahedron The [great dodecahedron] is a polyhedron with 12 vertices and 12 faces. The dual polyhedron is called small stellated dodecahedron. great stellated dodecahedron The [great stellated dodecahedron] is a polyhedron with 20 vertices and 12 faces. The dual polyhedron is called great icosahedron. great icosahedron The [great icosahedron] is a polyhedron with 12 vertices and 20 faces. The dual polyhedron is called great stellated dodecahedron. truncated tetrahedron The [truncated tetrahedron] is a polyhedron with 12 vertices and 8 faces. The dual polyhedron is called triakis tetrahedron. cuboctahedron The [cuboctahedron] is a polyhedron with 12 vertices and 14 faces. The dual polyhedron is called rhombic dodecahedron. -
Spectral Realizations of Graphs
SPECTRAL REALIZATIONS OF GRAPHS B. D. S. \DON" MCCONNELL 1. Introduction 2 The boundary of the regular hexagon in R and the vertex-and-edge skeleton of the regular tetrahe- 3 dron in R , as geometric realizations of the combinatorial 6-cycle and complete graph on 4 vertices, exhibit a significant property: Each automorphism of the graph induces a \rigid" isometry of the figure. We call such a figure harmonious.1 Figure 1. A pair of harmonious graph realizations (assuming the latter in 3D). Harmonious realizations can have considerable value as aids to the intuitive understanding of the graph's structure, but such realizations are generally elusive. This note explains and explores a proposition that provides a straightforward way to generate an entire family of harmonious realizations of any graph: A matrix whose rows form an orthogonal basis of an eigenspace of a graph's adjacency matrix has columns that serve as coordinate vectors of the vertices of an harmonious realization of the graph. This is a (projection of a) spectral realization. The hundreds of diagrams in Section 42 illustrate that spectral realizations of graphs with a high degree of symmetry can have great visual appeal. Or, not: they may exist in arbitrarily-high- dimensional spaces, or they may appear as an uninspiring jumble of points in one-dimensional space. In most cases, they collapse vertices and even edges into single points, and are therefore only very rarely faithful. Nevertheless, spectral realizations can often provide useful starting points for visualization efforts. (A basic Mathematica recipe for computing (projected) spectral realizations appears at the end of Section 3.) Not every harmonious realization of a graph is spectral. -
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 -
Paper Models of Polyhedra
Paper Models of Polyhedra Gijs Korthals Altes Polyhedra are beautiful 3-D geometrical figures that have fascinated philosophers, mathematicians and artists for millennia Copyrights © 1998-2001 Gijs.Korthals Altes All rights reserved . It's permitted to make copies for non-commercial purposes only email: [email protected] Paper Models of Polyhedra Platonic Solids Dodecahedron Cube and Tetrahedron Octahedron Icosahedron Archimedean Solids Cuboctahedron Icosidodecahedron Truncated Tetrahedron Truncated Octahedron Truncated Cube Truncated Icosahedron (soccer ball) Truncated dodecahedron Rhombicuboctahedron Truncated Cuboctahedron Rhombicosidodecahedron Truncated Icosidodecahedron Snub Cube Snub Dodecahedron Kepler-Poinsot Polyhedra Great Stellated Dodecahedron Small Stellated Dodecahedron Great Icosahedron Great Dodecahedron Other Uniform Polyhedra Tetrahemihexahedron Octahemioctahedron Cubohemioctahedron Small Rhombihexahedron Small Rhombidodecahedron S mall Dodecahemiododecahedron Small Ditrigonal Icosidodecahedron Great Dodecahedron Compounds Stella Octangula Compound of Cube and Octahedron Compound of Dodecahedron and Icosahedron Compound of Two Cubes Compound of Three Cubes Compound of Five Cubes Compound of Five Octahedra Compound of Five Tetrahedra Compound of Truncated Icosahedron and Pentakisdodecahedron Other Polyhedra Pentagonal Hexecontahedron Pentagonalconsitetrahedron Pyramid Pentagonal Pyramid Decahedron Rhombic Dodecahedron Great Rhombihexacron Pentagonal Dipyramid Pentakisdodecahedron Small Triakisoctahedron Small Triambic -
Photovoltaic Cells and Systems
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Crossref SQU Journal for Science, 16 (2011) 82-101 © 2011 Sultan Qaboos University Chiral Polyhedra Derived from Coxeter Diagrams and Quaternions Mehmet Koca, Nazife Ozdes Koca* and Muna Al-Shueili Department of Physics, College of Science, Sultan Qaboos University, P.O.Box 36, Al- Khoud, 123 Muscat, Sultanate of Oman, *Email: [email protected]. متعددات السطوح اليدوية المستمدة من أشكال كوكزتر والرباعيات مهمت كوجا، نزيفة كوجا و منى الشعيلي ملخص: هناك متعددات أسطح ٌدوٌان أرخمٌدٌان: المكعب المسطوح والثنعشري اﻷسطح المسطوح مع أشكالها الكتﻻنٌة المزدوجة: رباعٌات السطوح العشرونٌة الخماسٌة مع رباعٌات السطوح السداسٌة الخماسٌة. فً هذا البحث نقوم برسم متعددات السطوح الٌدوٌة مع مزدوجاتها بطرٌقة منظمة. نقوم باستخدام المجموعة الجزئٌة الدورانٌة الحقٌقٌة لمجموعات كوكستر ( WAAAWAWB(1 1 1 ), ( 3 ), ( 3 و WH()3 ﻻشتقاق المدارات الممثلة لﻷشكال. هذه الطرٌقة تقودنا إلى رسم المتعددات اﻷسطح التالٌة: رباعً الوجوه وعشرونً الوجوه و المكعب المسطوح و الثنعشري السطوح المسطوح على التوالً. لقد أثبتنا أنه بإمكان رباعً الوجوه وعشرونً الوجوه التحول إلى صورتها المرآتٌة بالمجموعة الثمانٌة الدورانٌة الحقٌقٌة WBC()/32. لذلك ﻻ ٌمكن تصنٌفها كمتعددات أسطح ٌدوٌة. من المﻻحظ أٌضا أن رؤوس المكعب المسطوح ورؤوس الثنعشري السطوح المسطوح ٌمكن اشتقاقها من المتجهات المجموعة خطٌا من الجذور البسٌطة وذلك باستخدام مجموعة الدوران الحقٌقٌة WBC()/32 و WHC()/32 على التوالً. من الممكن رسم مزدوجاتها بجمع ثﻻثة مدارات من المجموعة قٌد اﻻهتمام. أٌضا نقوم بإنشاء متعددات اﻷسطح الشبه منتظمة بشكل عام بربط متعددات اﻷسطح الٌدوٌة مع صورها المرآتٌة. كنتٌجة لذلك نحصل على مجموعة البٌروهٌدرال كمجموعة جزئٌة من مجموعة الكوكزتر WH()3 ونناقشها فً البحث. -
One Shape to Make Them All
Bridges Finland Conference Proceedings Dual Models: One Shape to Make Them All Mircea Draghicescu ITSPHUN LLC [email protected] Abstract We show how a potentially infinite number of 3D decorative objects can be built by connecting copies of a single geometric element made of a flexible material. Each object is an approximate model of a compound of some polyhe- dron and its dual. We present the underlying geometry and show a few such constructs. The geometric element used in the constructions can be made from a variety of materials and can have many shapes that give different artistic effects. In the workshop, we will build a few models that the participants can take home. Introduction Wireframe models of polyhedra can be built by connecting linear elements that represent polyhedral edges. We describe here a different modeling method where each atomic construction element represents not an edge, but a pair of edges, one belonging to some polyhedron P and the other one to P 0, the dual of P ; the resulting object is a model of the compound of P and P 0. Using the same number of construction elements as the number of edges of P and P 0, the model of the compound exhibits a richer structure and, due to the additional connections, is stronger than the models of P and P 0 alone. The model has at least the symmetry of P and P 0, and can have a higher symmetry if P is self-dual. The construction process challenges the maker to think simultaneously of both P and P 0 and can be used as a teaching aid in the study of polyhedra and duality. -
Construction of Fractals Based on Catalan Solids Andrzej Katunina*
I.J. Mathematical Sciences and Computing, 2017, 4, 1-7 Published Online November 2017 in MECS (http://www.mecs-press.net) DOI: 10.5815/ijmsc.2017.04.01 Available online at http://www.mecs-press.net/ijmsc Construction of Fractals based on Catalan Solids Andrzej Katunina* aInstitute of Fundamentals of Machinery Design, Silesian University of Technology, 18A Konarskiego Street, 44-100 Gliwice, Poland Received: 17 June 2017; Accepted: 18 September 2017; Published: 08 November 2017 Abstract The deterministic fractals play an important role in computer graphics and mathematical sciences. The understanding of construction of such fractals, especially an ability of fractals construction from various types of polytopes is of crucial importance in several problems related both to the pure mathematical issues as well as some issues of theoretical physics. In the present paper the possibility of construction of fractals based on the Catalan solids is presented and discussed. The method and algorithm of construction of polyhedral strictly deterministic fractals is presented. It is shown that the fractals can be constructed only from a limited number of the Catalan solids due to the specific geometric properties of these solids. The contraction ratios and fractal dimensions are presented for existing fractals with adjacent contractions constructed based on the Catalan solids. Index Terms: Deterministic fractals, iterated function system, Catalan solids. © 2017 Published by MECS Publisher. Selection and/or peer review under responsibility of the Research Association of Modern Education and Computer Science 1. Introduction The fractals, due to their self-similar nature, become playing an important role in many scientific and technical applications. -
A Partial Glossary of Spanish Geological Terms Exclusive of Most Cognates
U.S. DEPARTMENT OF THE INTERIOR U.S. GEOLOGICAL SURVEY A Partial Glossary of Spanish Geological Terms Exclusive of Most Cognates by Keith R. Long Open-File Report 91-0579 This report is preliminary and has not been reviewed for conformity with U.S. Geological Survey editorial standards or with the North American Stratigraphic Code. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government. 1991 Preface In recent years, almost all countries in Latin America have adopted democratic political systems and liberal economic policies. The resulting favorable investment climate has spurred a new wave of North American investment in Latin American mineral resources and has improved cooperation between geoscience organizations on both continents. The U.S. Geological Survey (USGS) has responded to the new situation through cooperative mineral resource investigations with a number of countries in Latin America. These activities are now being coordinated by the USGS's Center for Inter-American Mineral Resource Investigations (CIMRI), recently established in Tucson, Arizona. In the course of CIMRI's work, we have found a need for a compilation of Spanish geological and mining terminology that goes beyond the few Spanish-English geological dictionaries available. Even geologists who are fluent in Spanish often encounter local terminology oijerga that is unfamiliar. These terms, which have grown out of five centuries of mining tradition in Latin America, and frequently draw on native languages, usually cannot be found in standard dictionaries. There are, of course, many geological terms which can be recognized even by geologists who speak little or no Spanish. -
Truncated Cube from Spheres
CPS Geometry Part 4 – Archimedean Solids 18. The Cuboctahedron in CPS 19. Truncated Tetrahedron in CPS 20. Truncated Octahedron in CPS 21. Truncated Cube in CPS 22. Truncated Icosahedron in CPS Nick Trif Ottawa, Ontario, Canada – 2018 www.platonicstructures.com CPS Geometry Part 4 – Archimedean Solids – 21: Truncated Cube from Spheres YouTube: https://youtu.be/1kJ6yEC-QcQ In the classical geometry, a truncated cube is generated from a cube using the same approach used for generating the truncated tetrahedron or the truncated octahedron. In the classical geometry, a truncated cube is generated from a cube using the same approach used for generating the truncated tetrahedron or the truncated octahedron. 1. Divide each edge of the cube in three parts; In the classical geometry, a truncated cube is generated from a cube using the same approach used for generating the truncated tetrahedron or the truncated octahedron. 1. Divide each edge of the cube in three parts; 2. Cut the vertexes of the cube with planes determined by the divisions done in the first step. In the case of the cube, the division will not have to generate three equal parts. Applying simple geometry, one can determine the relationship between the size of the initial cube AD and the size of the truncated cube BC. The presence of square 2 shows that these two dimensions are incommensurable with each other. For this reason, we will use another approach to identify the Truncated Cube pattern in CPS. We will do that by looking first to the Catalan Octahedron, the dual of the Truncated Cube.