Polyhedral Forms Obtained by Combinig Lateral Sheet of CP II-10 and Truncated Dodecahedron
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Archimedean Solids
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2008 Archimedean Solids Anna Anderson University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Anderson, Anna, "Archimedean Solids" (2008). MAT Exam Expository Papers. 4. https://digitalcommons.unl.edu/mathmidexppap/4 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Archimedean Solids Anna Anderson In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor July 2008 2 Archimedean Solids A polygon is a simple, closed, planar figure with sides formed by joining line segments, where each line segment intersects exactly two others. If all of the sides have the same length and all of the angles are congruent, the polygon is called regular. The sum of the angles of a regular polygon with n sides, where n is 3 or more, is 180° x (n – 2) degrees. If a regular polygon were connected with other regular polygons in three dimensional space, a polyhedron could be created. In geometry, a polyhedron is a three- dimensional solid which consists of a collection of polygons joined at their edges. The word polyhedron is derived from the Greek word poly (many) and the Indo-European term hedron (seat). -
New Perspectives on Polyhedral Molecules and Their Crystal Structures Santiago Alvarez, Jorge Echeverria
New Perspectives on Polyhedral Molecules and their Crystal Structures Santiago Alvarez, Jorge Echeverria To cite this version: Santiago Alvarez, Jorge Echeverria. New Perspectives on Polyhedral Molecules and their Crystal Structures. Journal of Physical Organic Chemistry, Wiley, 2010, 23 (11), pp.1080. 10.1002/poc.1735. hal-00589441 HAL Id: hal-00589441 https://hal.archives-ouvertes.fr/hal-00589441 Submitted on 29 Apr 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Journal of Physical Organic Chemistry New Perspectives on Polyhedral Molecules and their Crystal Structures For Peer Review Journal: Journal of Physical Organic Chemistry Manuscript ID: POC-09-0305.R1 Wiley - Manuscript type: Research Article Date Submitted by the 06-Apr-2010 Author: Complete List of Authors: Alvarez, Santiago; Universitat de Barcelona, Departament de Quimica Inorganica Echeverria, Jorge; Universitat de Barcelona, Departament de Quimica Inorganica continuous shape measures, stereochemistry, shape maps, Keywords: polyhedranes http://mc.manuscriptcentral.com/poc Page 1 of 20 Journal of Physical Organic Chemistry 1 2 3 4 5 6 7 8 9 10 New Perspectives on Polyhedral Molecules and their Crystal Structures 11 12 Santiago Alvarez, Jorge Echeverría 13 14 15 Departament de Química Inorgànica and Institut de Química Teòrica i Computacional, 16 Universitat de Barcelona, Martí i Franquès 1-11, 08028 Barcelona (Spain). -
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 -
Wythoffian Skeletal Polyhedra
Wythoffian Skeletal Polyhedra by Abigail Williams B.S. in Mathematics, Bates College M.S. in Mathematics, Northeastern University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Dedication I would like to dedicate this dissertation to my Meme. She has always been my loudest cheerleader and has supported me in all that I have done. Thank you, Meme. ii Abstract of Dissertation Wythoff's construction can be used to generate new polyhedra from the symmetry groups of the regular polyhedra. In this dissertation we examine all polyhedra that can be generated through this construction from the 48 regular polyhedra. We also examine when the construction produces uniform polyhedra and then discuss other methods for finding uniform polyhedra. iii Acknowledgements I would like to start by thanking Professor Schulte for all of the guidance he has provided me over the last few years. He has given me interesting articles to read, provided invaluable commentary on this thesis, had many helpful and insightful discussions with me about my work, and invited me to wonderful conferences. I truly cannot thank him enough for all of his help. I am also very thankful to my committee members for their time and attention. Additionally, I want to thank my family and friends who, for years, have supported me and pretended to care everytime I start talking about math. Finally, I want to thank my husband, Keith. -
Operation-Based Notation for Archimedean Graph
Operation-Based Notation for Archimedean Graph Hidetoshi Nonaka Research Group of Mathematical Information Science Division of Computer Science, Hokkaido University N14W9, Sapporo, 060 0814, Japan ABSTRACT Table 1. The list of Archimedean solids, where p, q, r are the number of vertices, edges, and faces, respectively. We introduce three graph operations corresponding to Symbol Name of polyhedron p q r polyhedral operations. By applying these operations, thirteen Archimedean graphs can be generated from Platonic graphs that A(3⋅4)2 Cuboctahedron 12 24 14 are used as seed graphs. A4610⋅⋅ Great Rhombicosidodecahedron 120 180 62 A468⋅⋅ Great Rhombicuboctahedron 48 72 26 Keyword: Archimedean graph, Polyhedral graph, Polyhedron A 2 Icosidodecahedron 30 60 32 notation, Graph operation. (3⋅ 5) A3454⋅⋅⋅ Small Rhombicosidodecahedron 60 120 62 1. INTRODUCTION A3⋅43 Small Rhombicuboctahedron 24 48 26 A34 ⋅4 Snub Cube 24 60 38 Archimedean graph is a simple planar graph isomorphic to the A354 ⋅ Snub Dodecahedron 60 150 92 skeleton or wire-frame of the Archimedean solid. There are A38⋅ 2 Truncated Cube 24 36 14 thirteen Archimedean solids, which are semi-regular polyhedra A3⋅102 Truncated Dodecahedron 60 90 32 and the subset of uniform polyhedra. Seven of them can be A56⋅ 2 Truncated Icosahedron 60 90 32 formed by truncation of Platonic solids, and all of them can be A 2 Truncated Octahedron 24 36 14 formed by polyhedral operations defined as Conway polyhedron 4⋅6 A 2 Truncated Tetrahedron 12 18 8 notation [1-2]. 36⋅ The author has recently developed an interactive modeling system of uniform polyhedra including Platonic solids, Archimedean solids and Kepler-Poinsot solids, based on graph drawing and simulated elasticity, mainly for educational purpose [3-5]. -
Polyhedra from Symmetric Placement of Regular Polygons
Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons Craig S. Kaplan George W. Hart University of Washington http://www.georgehart.com Box 352350, Seattle, WA 98195 USA [email protected] [email protected] Abstract In the quest for new visually interesting polyhedra with regular faces, we define and present an infinite class of solids, constructed by placing regular polygons at the rotational axes of a polyhedral symmetry group. This new technique can be used to generate many existing polyhedra, including most of the Archimedean solids. It also yields novel families of attractive symmetric polyhedra. 1 Introduction Most interesting polyhedra arise via a process that attempts to generate or preserve some measure of order. We might ask that the polyhedron be as symmetric as possible, that the symmetries act transitively on its vertices, edges or faces, or that all its faces be regular polygons. Indeed, these questions have all been asked and the resulting families of polyhedra enumerated to satisfaction. The last case, that of convex polyhedra all of whose faces are regular polygons, are the well-known Johnson solids [3]. Eager to discover new polyhedra with regular faces, but aware that the list of Johnson solids has been proven complete, we are faced with the prospect of letting some constraints slip a bit in order to innovate. We achieve this innovation by dropping the constraint that all faces be regular, asking only that many faces be regular, and that the remaining faces occur in a small number of different shapes. In this paper we define an infinite set of convex polyhedra that we call symmetrohedra.Theyare parameterized by several discrete and continuous values. -
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). -
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. -
Truncated Truncated Dodecahedron and Truncated Truncated Icosahedron Spaces
Cumhuriyet Science Journal CSJ e-ISSN: 2587-246X Cumhuriyet Sci. J., Vol.40-2 (2019) 457-470 ISSN: 2587-2680 Truncated Truncated Dodecahedron and Truncated Truncated Icosahedron Spaces Ümit Ziya SAVCI1 1Kütahya Dumlupınar University, Educational Faculty, Department of Mathematics Education, Kütahya, Turkey Received: 01.03.2019; Accepted: 22.05.2019 http://dx.doi.org/10.17776/csj.534616 Abstract. The theory of convex sets is a vibrant and classical field of modern mathematics with rich applications. The more geometric aspects of convex sets are developed introducing some notions, but primarily polyhedra. A polyhedra, when it is convex, is an extremely important special solid in ℝ. Some examples of convex subsets of Euclidean 3-dimensional space are Platonic Solids, Archimedean Solids and Archimedean Duals or Catalan Solids. There are some relations between metrics and polyhedra. For example, it has been shown that cube, octahedron, deltoidal icositetrahedron are maximum, taxicab, Chinese Checker’s unit sphere, respectively. In this study, we give two new metrics to be their spheres truncated truncated dodecahedron and truncated truncated icosahedron. Keywords: Polyhedron, Metric, Truncated Truncated Dodecahedron, Truncated Truncated Icosahedron. Truncated Truncated Dodecahedron ve Truncated Truncated Icosahedron Uzayları Özet. Konveks kümeler teorisi, zengin uygulamalara sahip modern matematiğin canlı ve klasik bir alanıdır. Konveks kümelerin geometrik yönleri, bazı kavramlar, fakat öncelikle çokyüzlülerin tanıtılmasıyla geliştirilmiştir. Konveks olduğunda bir çokyüzlü, ℝ de çok önemli bir özel cisimdir. Öklid 3 boyutlu uzayın konveks alt kümelerinin bazı örnekleri Platonik cisimler, Arşimet cisimleri ve Arşimet dualleri veya Katalan cisimleridir. Metriklerle çokyüzlüler arasında bazı ilişkiler vardır. Örneğin, küp, sekizyüzlü, deltoidal icositetrahedron'un sırasıyla, maksimum, Taksi, Çin dama uzaylarının birim küresi olduğu görülmektedir. -
The Buckyball Symmetries | Neverendingbooks Page 1 of 11
the buckyball symmetries | neverendingbooks Page 1 of 11 about me RSS : Posts Comments Email neverendingbooks lieven le bruyn's blog Type and hit enter to Search Written on June 27, 2008 at 9:04 am by lievenlb the buckyball symmetries Filed under level1 {6 comments } Trinities 1. Galois ’ last letter 2. Arnold ’s trinities 3. Arnold ’s trinities version 2.0 4. the buckyball symmetries 5. Klein ’s dessins d ’enfant and the buckyball 6. the buckyball curve The buckyball is without doubt the hottest mahematical object at the moment (at least in Europe). Recall that the buckyball (middle) is a mixed form of two Platonic solids the Icosahedron on the left and the Dodecahedron on the right. For those of you who don’t know anything about football, it is that other ball-game, best described via a quote from the English player Gary Lineker http://www.neverendingbooks.org/index.php/the-buckyball-symmetries.html 5/23/2011 the buckyball symmetries | neverendingbooks Page 2 of 11 “Football is a game for 22 people that run around, play the ball, and one referee who makes a slew of mistakes, and in the end Germany always wins.” We still have a few days left hoping for a better ending… Let’s do some bucky-maths : what is the rotation symmetry group of the buckyball? For starters, dodeca- and icosahedron are dual solids , meaning that if you take the center of every face of a dodecahedron and connect these points by edges when the corresponding faces share an edge, you’ll end up with the icosahedron (and conversely). -
Mathematics Master Class Name
Platonic Solids Mathematics Master Class Name Date Teacher Student Class Platonic and Archimedean Solids There are only five regular convex polyhedra that can exist in our three dimensional Euclidean space. These polyhedra were known since the ancient times. Plato (427-348 BC) wrote about them in his work Timaeus (written in about 340 BC) to identify five principles upon which everything is modelled: he identified these principles as fire, earth, air, water, and cosmos (or divine force). He identified each with the elementary principle Tetrahedron – fire Cube – earth Octahedron – air Icosahedron – water Dodecahedron – cosmos or divine force Euclid’s Elements (c. 300 BC) list all these solids in mathematical way, giving their properties in the last book, book XIII. Around the same time Archimedes was working on Platonic solids and discovered new polyhedra which are now called Archimedean polyhedra. The difference between platonic and Archimedean polyhedra is that, while the former are regular polyhedra containing only regular and same faces, the latter contain two or more different types of faces (which are also regular polygons). Have a look below and at the next page where all the Platonic and Archimedean polyhedra are listed. Platonic Solids cube dodecahedron icosahedron octahedron tetrahedron 2 Archimedean Polyhedra Their names are snub cube cuboctahedron snub dodecahedron great rhombicosidodecahedron truncated cube great rhombicuboctahedron truncated dodecahedron icosadodecahedron truncated icosahedron small rhombicosidodecahedron truncated octahedron small rhombicuboctahedron truncated tetrahedron For the following exercise you need to cut out the nets and make Platonic solids out of them. 3 4 5 6 7 8 “Polyhedra” is a Greek word meaning _________ ___________. -
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.