Chiral Polyhedra Derived from Coxeter Diagrams and Quaternions
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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 -
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. -
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. -
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. -
A Note About the Metrics Induced by Truncated Dodecahedron and Truncated Icosahedron
INTERNATIONAL JOURNAL OF GEOMETRY Vol. 6 (2017), No. 2, 5 - 16 A Note About The Metrics Induced by Truncated Dodecahedron And Truncated Icosahedron Ozcan¨ Geli¸sgen,Temel Ermi¸sand Ibrahim G¨unaltılı Abstract. Polyhedrons have been studied by mathematicians and ge- ometers during many years, because of their symmetries. There are only five regular convex polyhedra known as the Platonic solids. Semi-regular convex polyhedron which are composed of two or more types of regular polygons meeting in identical vertices are called Archimedean solids. There are some relations between metrics and polyhedra. For example, it has been shown that cube, octahedron, deltoidal icositetrahedron are maximum, taxi- cab, Chinese Checker's unit sphere, respectively. In this study, we introduce two new metrics, and show that the spheres of the 3-dimensional analytical space furnished by these metrics are truncated dodecahedron and truncated icosahedron. Also we give some properties about these metrics. 1. Introduction What is a polyhedron? This question is interestingly hard to answer simply! A polyhedron is a three-dimensional figure made up of polygons. When discussing polyhedra one will use the terms faces, edges and vertices. Each polygonal part of the polyhedron is called a face. A line segment along which two faces come together is called an edge. A point where several edges and faces come together is called a vertex. Traditional polyhedra consist of flat faces, straight edges, and vertices. There are many thinkers that worked on polyhedra among the ancient Greeks. Early civilizations worked out mathematics as problems and their solutions. Polyhedrons have been studied by mathematicians, scientists during many years, because of their symmetries. -
Dron and Rhombic Triacontahedron
KoG•19–2015 Z. Can, O.¨ Gelis¸gen, R. Kaya: On the Metrics Induced by Icosidodecahedron .... Original scientific paper ZEYNEP CAN Accepted 11. 5. 2015. O¨ ZCAN GELIS¸GEN RUSTEM¨ KAYA On the Metrics Induced by Icosidodecahe- dron and Rhombic Triacontahedron On the Metrics Induced by Icosidodecahedron O metrici induciranoj ikosadodekaedrom i and Rhombic Triacontahedron trijakontaedrom ABSTRACT SAZETAKˇ The theory of convex sets is a vibrant and classical field Teorija konveksnih skupova je vitalno i klasiˇcnopodruˇcje of modern mathematics with rich applications. If every moderne matematike s bogatom primjenom. Ako se sve points of a line segment that connects any two points toˇcke duˇzine,koja spaja bilo koje dvije toˇcke skupa, nalaze of the set are in the set, then it is convex. The more u tom skupu, tada je taj skup konveksan. Sve se viˇse geometric aspects of convex sets are developed introduc- geometrijskih aspekata o konveksnim skupovima razvija ing some notions, but primarily polyhedra. A polyhedra, uvode´ci neke pojmove, ponajprije poliedre. Konveksni n when it is convex, is an extremely important special solid poliedar je iznimno vaˇznoposebno tijelo u R . Neki primje- n in R . Some examples of convex subsets of Euclidean ri konveksnih podskupova euklidskog trodimenzionalnog 3-dimensional space are Platonic Solids, Archimedean prostora su Platonova tijela, Arhimedova tijela, tijela du- Solids and Archimedean Duals or Catalan Solids. In this alna Arhimedovim tijelima i Catalanova tijela. U ovom study, we give two new metrics to be their spheres an ˇclankuprikazujemo dvije metrike koje su sfere Arhime- archimedean solid icosidodecahedron and its archimedean dovom tijelu ikosadodekaedru i njemu dualnom tijelu, tri- dual rhombic triacontahedron. -
From Klein's Platonic Solids to Kepler's
Dessin d'Enfants Examples due to Magot and Zvonkin Moduli Spaces From Klein's Platonic Solids to Kepler's Archimedean Solids: Elliptic Curves and Dessins d'Enfants Part II Edray Herber Goins Department of Mathematics Purdue University September 7, 2012 Number Theory Seminar From Klein's Platonic Solids to Kepler's Archimedean Solids Dessin d'Enfants Examples due to Magot and Zvonkin Moduli Spaces Abstract In 1884, Felix Klein wrote his influential book, \Lectures on the Icosahedron," where he explained how to express the roots of any quintic polynomial in terms of elliptic modular functions. His idea was to relate rotations of the icosahedron with the automorphism group of 5-torsion points on a suitable elliptic curve. In fact, he created a theory which related rotations of each of the five regular solids (the tetrahedron, cube, octahedron, icosahedron, and dodecahedron) with the automorphism groups of 3-, 4-, and 5-torsion points. Using modern language, the functions which relate the rotations with elliptic curves are Bely˘ımaps. In 1984, Alexander Grothendieck introduced the concept of a Dessin d'Enfant in order to understand Galois groups via such maps. We will complete a circle of ideas by reviewing Klein's theory with an emphasis on the octahedron; explaining how to realize the five regular solids (the Platonic solids) as well as the thirteen semi-regular solids (the Archimedean solids) as Dessins d'Enfant; and discussing how the corresponding Bely˘ımaps relate to moduli spaces of elliptic curves. Number Theory Seminar From Klein's -
Regular and Irregular Chiral Polyhedra from Coxeter Diagrams Via Quaternions
Article Regular and Irregular Chiral Polyhedra from Coxeter Diagrams via Quaternions Nazife Ozdes Koca Department of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khoud 123, Muscat, Oman; [email protected] Received: 6 July 2017; Accepted: 2 August 2017; Published: 7 August 2017 Abstract: Vertices and symmetries of regular and irregular chiral polyhedra are represented by quaternions with the use of Coxeter graphs. A new technique is introduced to construct the chiral Archimedean solids, the snub cube and snub dodecahedron together with their dual Catalan solids, pentagonal icositetrahedron and pentagonal hexecontahedron. Starting with the proper subgroups of the Coxeter groups ( ⊕ ⊕) , () , () and () , we derive the orbits representing the respective solids, the regular and irregular forms of a tetrahedron, icosahedron, snub cube, and snub dodecahedron. Since the families of tetrahedra, icosahedra and their dual solids can be transformed to their mirror images by the proper rotational octahedral group, they are not considered as chiral solids. Regular structures are obtained from irregular solids depending on the choice of two parameters. We point out that the regular and irregular solids whose vertices are at the edge mid-points of the irregular icosahedron, irregular snub cube and irregular snub dodecahedron can be constructed. Keywords: Coxeter diagrams; irregular chiral polyhedra; quaternions; snub cube; snub dodecahedron 1. Introduction In fundamental physics, chirality plays a very important role. A Weyl spinor describing a massless Dirac particle is either in a left-handed state or in a right-handed state. Such states cannot be transformed to each other by the proper Lorentz transformations. Chirality is a well-defined quantum number for massless particles. -
Ivory Polyhedra Turned on a Lathe
Bridges 2010: Mathematics, Music, Art, Architecture, Culture Ivory Polyhedra Turned on a Lathe Tibor Tarnai Department of Structural Mechanics Budapest University of Technology and Economics Müegyetem rkp. 3. Budapest, H-1521 Hungary E-mail: [email protected] Abstract A morphological survey of nested ivory spheres and polyhedra made by turning in Europe between the 16 th and 18 th centuries is presented, different polyhedra are identified, and examples of related objects of art of known ivory collections are shown. These examples convincingly demonstrate the high level of geometrical knowledge of the old master turners who produced these objects. Ivory Polyhedra as Works of Art Nowadays souvenir shops in China sell “devil’s work balls” or “puzzle balls”, that is, multiple spherical nested shells made of jade, sandalwood or artificial ivory. During the Qing Dynasty, such multi-layer balls turned on a lathe were made of ivory with piercing and rich figurative carving. The manufacturing technology required radially drilled conical holes whose number is 12 or 14; 12 holes at the face centres of a regular dodecahedron, while 14 holes at the face centres of a cuboctahedron. For these Chinese balls, mainly the aesthetic appearance was important, and geometry was just something necessary to it. It is little known how their making started. The only thing that we know is that Chinese balls already existed in the 14 th century. The book [1] made mention of a three-layer ball. Similar ivory art objects appeared in Europe in the second half of the 16 th century [2]. Their manufacturing method [3] was developed in Germany. -
Measuring Sphericity a Measure of the Sphericity of a Convex Solid Was Introduced by G
Integre Technical Publishing Co., Inc. College Mathematics Journal 42:2 January 6, 2011 11:00 a.m. aravind.tex page 98 How Spherical Are the Archimedean Solids and Their Duals? P. K. Aravind P. K. Aravind is a Professor of Physics at Worcester Polytechnic Institute. He received his B.Sc. and M.Sc. degrees in Physics from Delhi University in Delhi, India, and his Ph.D. degree in Physics from Northwestern University. His research in recent years has centered around Bell’s theorem and quantum information theory, and particularly the role that polytopes, such as the 600-cell, play in it. For the last fifteen years he has taught in a summer program called Frontiers, conducted by WPI for bright high school students who are interested in pursuing mathematics, science, or engineering in college. A molecule of C-60, or a “Buckyball”, consists of 60 carbon atoms arranged at the vertices of a truncated icosahedron (see Figure 1), with the atoms being held in place by covalent bonds. The Buckyball has a very spherical appearance and its sphericity is the basis of several of its applications, such as a nanoscale ball bearing or a molecular cage to trap other atoms [10]. The buckyball pattern also occurs on the surface of many soccer balls. The huge interest in the Buckyball leads one to ask whether the truncated icosahedron is the most spherical of the elementary geometrical forms or whether there are others that are superior to it. That is the question I address in this paper. The reader who is familiar with the Archimedean solids and their duals is invited to try and guess the most spherical among them before reading on. -
Geometric Analysis of the Some Rectified Archimedean Solids
MATHEMATICAL SCIENCES AND APPLICATIONS E-NOTES 8 (2)96-109 (2020) ©MSAEN HTTPS://DOI.ORG/10.36753/MATHENOT.643969 Geometric Analysis of the Some Rectified Archimedean Solids Spaces via Their Isometry Groups Temel Ermi¸s* Abstract There are two main motivations in this article. First, we give the new metrics and the metric spaces whose unit spheres are Rectified Archimedean Solids. Then, using the general technique which is quite simple, we show that isometry group of R3 endowed with these new metrics are the semi direct product of the translation group T (3) of R3 with the Euclidean symmetry groups of Rectified Archimedean Solids. Keywords: Polyhedrons; Metric geometry; Isometry group; Octahedral group; Icosahedral group; Rectified Archimedean solids. AMS Subject Classification (2020): Primary: 51B20; 51N25; Secondary: 51F99; 51K05; 51K99. *Corresponding author 1. Introduction The main idea of the geometric group theory is to treat finitely-generated groups as geometric objects. It is understandable efforts to bring together geometry and algebra. Thus, one can easily analyze the mathematical system thanks to transformations on this system. The transformation preserve designated features of the geometric structure. The set of transformations compose the groups consisting of the symmetries of geometric objects. The symmetry group for a physical object (or polyhedron) is the set of ways that object can be repositioned so that it maintains its symmetry, or looks the same. Any object of any dimension has such a group of symmetries. The set of three-dimensional objects called polyhedrons possess many beautiful symmetries and accordingly form interesting symmetry groups under their rotations. So, the excellent symmetry of the polyhedrons (or solids) have made them perfect models for the studying on symmetries. -
Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation
S S symmetry Article Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation Lorenz Demey 1,* and Hans Smessaert 2 1 Center for Logic and Analytic Philosophy, KU Leuven, 3000 Leuven, Belgium 2 Research Group on Formal and Computational Linguistics, KU Leuven, 3000 Leuven, Belgium; [email protected] * Correspondence: [email protected]; Tel.: +32-16-328855 Academic Editor: Neil Y. Yen Received: 28 August 2017; Accepted: 27 September 2017; Published: 29 September 2017 Abstract: Aristotelian diagrams visualize the logical relations among a finite set of objects. These diagrams originated in philosophy, but recently, they have also been used extensively in artificial intelligence, in order to study (connections between) various knowledge representation formalisms. In this paper, we develop the idea that Aristotelian diagrams can be fruitfully studied as geometrical entities. In particular, we focus on four polyhedral Aristotelian diagrams for the Boolean algebra B4, viz. the rhombic dodecahedron, the tetrakis hexahedron, the tetraicosahedron and the nested tetrahedron. After an in-depth investigation of the geometrical properties and interrelationships of these polyhedral diagrams, we analyze the correlation (or lack thereof) between logical (Hamming) and geometrical (Euclidean) distance in each of these diagrams. The outcome of this analysis is that the Aristotelian rhombic dodecahedron and tetrakis hexahedron exhibit the strongest degree of correlation between logical and geometrical distance; the tetraicosahedron performs worse; and the nested tetrahedron has the lowest degree of correlation. Finally, these results are used to shed new light on the relative strengths and weaknesses of these polyhedral Aristotelian diagrams, by appealing to the congruence principle from cognitive research on diagram design.