DOCSLIB.ORG

Mazes for Superintelligent Ginzburg Projection

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mazes for Superintelligent Ginzburg Projection Izidor Hafner Mazes for Superintelligent Ginzburg projection Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 2 Introduction Let as take an example. We are given a uniform polyhedron. 10 4 small rhombidodecahedron 910, 4, ÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ = 9 3 In Mathematica the polyhedron is given by a list of faces and with a list of koordinates of vertices [Roman E. Maeder, The Mathematica Programmer II, Academic Press1996]. The list of faces consists of a list of lists, where a face is represented by a list of vertices, which is given by a matrix. Let us show the first five faces: 81, 2, 6, 15, 26, 36, 31, 19, 10, 3< 81, 3, 9, 4< 81, 4, 11, 22, 29, 32, 30, 19, 13, 5< 81, 5, 8, 2< 82, 8, 17, 26, 38, 39, 37, 28, 16, 7< The nest two figures represent faces and vertices. The polyhedron is projected onto supescribed sphere and the sphere is projected by a cartographic projection. Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 3 1 4 11 21 17 7 2 8 6 5 20 19 30 16 15 3 14 10 12 18 26 34 39 37 29 23 9 24 13 28 27 41 36 38 22 25 31 33 40 42 35 32 5 8 17 24 13 3 1 2 6 15 26 36 31 19 10 9 4 7 14 27 38 47 43 30 20 12 25 49 42 18 11 16 39 54 32 21 23 2837 50 57 53 44 29 22 3433 35 4048 56 60 59 52 41 45 51 58 55 46 The problem is to find the path from the black dot to gray dot, where thick lines represent walls of a maze. Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 4 The solution is given by a list of faces passed from the black to gray dot. 3 2 13 1 4 9 12 10 11 5 7 8 6 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 5 Problems Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 6 1. 5 5 5 small ditrigonal icosidodecahedron 9 ÅÅÅÅÅ , 3, ÅÅÅÅÅ , 3, ÅÅÅÅÅ , 3= 2 2 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 7 2. 5 small icosicosidodecahedron 96, ÅÅÅÅÅ , 6, 3= 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 8 3. 5 small snub icosicosidodecahedron 93, ÅÅÅÅÅ , 3, 3, 3, 3= 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 9 4. 3 small dodecicosidodecahedron 910, ÅÅÅÅÅ , 10, 5= 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 10 5. 5 5 5 5 5 small stellated dodecahedron 9 ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ = 2 2 2 2 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 11 6. 1 great dodecahedron ÅÅÅÅÅ 85, 5, 5, 5, 5< 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 12 7. 5 5 dodecadodecahedron 9 ÅÅÅÅÅ , 5, ÅÅÅÅÅ , 5= 2 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 13 8. 5 truncated great dodecahedron 910, 10, ÅÅÅÅÅ = 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 14 9. 5 rhombidodecadodecahedron 94, ÅÅÅÅÅ , 4, 5= 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 15 10. 10 4 small rhombidodecahedron 910, 4, ÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ = 9 3 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 16 11. 5 snub dodecadodecahedron 93, 3, ÅÅÅÅÅ , 3, 5= 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 17 12. 5 5 5 ditrigonal dodecadodecahedron 9 ÅÅÅÅÅ , 5, ÅÅÅÅÅ , 5, ÅÅÅÅÅ , 5= 3 3 3 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 18 13. 1 great icosahedron ÅÅÅÅÅ 83, 3, 3, 3, 3< 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 19 14. 5 5 great icosidodecahedron 9 ÅÅÅÅÅ , 3, ÅÅÅÅÅ , 3= 2 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 20 15. 5 great truncated icosahedron 96, 6, ÅÅÅÅÅ = 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 21 16. 6 4 rhombicosahedron 96, 4, ÅÅÅÅÅ , ÅÅÅÅÅ = 5 3 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 22 17. 5 great snub icosidodecahedron 93, 3, ÅÅÅÅÅ , 3, 3= 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 23 18. pentagonal prism 84, 4, 5< Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 24 19. pentagonal antiprism 83, 3, 3, 5< Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 25 20. 5 pentagrammic prism 94, 4, ÅÅÅÅÅ = 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 26 21. 5 pentagrammic antiprism 93, 3, 3, ÅÅÅÅÅ = 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 27 22. 5 pentagrammic crossed antiprism 93, 3, 3, ÅÅÅÅÅ = 3 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 28 Solutions Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 29 1. 5 5 5 small ditrigonal icosidodecahedron 9 ÅÅÅÅÅ , 3, ÅÅÅÅÅ , 3, ÅÅÅÅÅ , 3= 2 2 2 12 13 1 11 14 16 2 10 15 3 7 9 6 4 8 5 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 30 2. 5 small icosicosidodecahedron 96, ÅÅÅÅÅ , 6, 3= 2 21 16 20 221 19 17 15 18 14 2 12 4 11 13 3 9 7 5 10 8 6 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 31 3. 5 small snub icosicosidodecahedron 93, ÅÅÅÅÅ , 3, 3, 3, 3= 2 11 10 17 16 13 12 9 8 18 15 14 7 20 19 5 6 30 3 4 28 29 21 2 22 23 1 27 24 26 25 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 32 4. 3 small dodecicosidodecahedron 910, ÅÅÅÅÅ , 10, 5= 2 15 6 5 11 1013 14 7 4 3 129 18 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 33 5. 5 5 5 5 5 small stellated dodecahedron 9 ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ , ÅÅÅÅÅ = 2 2 2 2 2 4 3 1 5 6 12 2 8 7 11 10 9 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 34 6. 1 great dodecahedron ÅÅÅÅÅ 85, 5, 5, 5, 5< 2 9 1 6 4 7 11 10 212 3 8 5 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 35 7. 5 5 dodecadodecahedron 9 ÅÅÅÅÅ , 5, ÅÅÅÅÅ , 5= 2 2 125 1021 14 15 2 13 116 198 20 9 163 14 17 187 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 36 8. 5 truncated great dodecahedron 910, 10, ÅÅÅÅÅ = 2 10 8 12 3 6 4 111 2 7 13 149 5 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 37 9. 5 rhombidodecadodecahedron 94, ÅÅÅÅÅ , 4, 5= 2 5 202 18 16 148 126 1 3 19 17 15 13 7 4 10 21 9 11 22 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 38 10. 10 4 small rhombidodecahedron 910, 4, ÅÅÅÅÅÅÅÅÅ , ÅÅÅÅÅ = 9 3 12 9 13 11 8 10 7 4 6 2 5 1 3 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 39 11. 5 snub dodecadodecahedron 93, 3, ÅÅÅÅÅ , 3, 5= 2 24 15 23 2 12 14 2216 13 21 11 17 3 18 5 20 6 1 4 10 19 7 9 8 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 40 12. 5 5 5 ditrigonal dodecadodecahedron 9 ÅÅÅÅÅ , 5, ÅÅÅÅÅ , 5, ÅÅÅÅÅ , 5= 3 3 3 1217 14 23 2 1118 9 1316 145 203 1922 107 158 216 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 41 13. 1 great icosahedron ÅÅÅÅÅ 83, 3, 3, 3, 3< 2 14 10 18 8 12 6 16 3 5 20 19 1 7 11 2 17 4 13 15 9 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 42 14. 5 5 great icosidodecahedron 9 ÅÅÅÅÅ , 3, ÅÅÅÅÅ , 3= 2 2 4 9 1 13 7 18 8 14 5 3 15 17 10 2 12 6 11 16 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 43 15. 5 great truncated icosahedron 96, 6, ÅÅÅÅÅ = 2 5 8 3 13 11 14 15 7 94 6 10 1 12 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 44 16. 6 4 rhombicosahedron 96, 4, ÅÅÅÅÅ , ÅÅÅÅÅ = 5 3 9 18 17 4 10 5 11 6 8 12 3 1 16 2 15 13 7 14 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 45 17. 5 great snub icosidodecahedron 93, 3, ÅÅÅÅÅ , 3, 3= 2 14 13 21 1 26 27 4 5 15 9 8 22 20 12 24 16 25 3 2 11 28 6 7 23 10 19 29 17 18 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 46 18. pentagonal prism 84, 4, 5< 2 3 5 1 4 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 47 19. pentagonal antiprism 83, 3, 3, 5< 7 8 6 2 1 5 3 4 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 48 20. 5 pentagrammic prism 94, 4, ÅÅÅÅÅ = 2 5 1 3 7 4 6 2 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 49 21. 5 pentagrammic antiprism 93, 3, 3, ÅÅÅÅÅ = 2 3 2 4 8 1 5 7 6 Cell[ TextData[ {"I. Hafner, Mazes for Superintelligent}], "Header"] 50 22. 5 pentagrammic crossed antiprism 93, 3, 3, ÅÅÅÅÅ = 3 3 8 2 4 1 7 5 6.
Load more

Recommended publications
  • Cons=Ucticn (Process)
    DOCUMENT RESUME ED 038 271 SE 007 847 AUTHOR Wenninger, Magnus J. TITLE Polyhedron Models for the Classroom. INSTITUTION National Council of Teachers of Mathematics, Inc., Washington, D.C. PUB DATE 68 NOTE 47p. AVAILABLE FROM National Council of Teachers of Mathematics,1201 16th St., N.V., Washington, D.C. 20036 ED RS PRICE EDRS Pr:ce NF -$0.25 HC Not Available from EDRS. DESCRIPTORS *Cons=ucticn (Process), *Geometric Concepts, *Geometry, *Instructional MateriAls,Mathematical Enrichment, Mathematical Models, Mathematics Materials IDENTIFIERS National Council of Teachers of Mathematics ABSTRACT This booklet explains the historical backgroundand construction techniques for various sets of uniformpolyhedra. The author indicates that the practical sianificanceof the constructions arises in illustrations for the ideas of symmetry,reflection, rotation, translation, group theory and topology.Details for constructing hollow paper models are provided for thefive Platonic solids, miscellaneous irregular polyhedra and somecompounds arising from the stellation process. (RS) PR WON WITHMICROFICHE AND PUBLISHER'SPRICES. MICROFICHEREPRODUCTION f ONLY. '..0.`rag let-7j... ow/A U.S. MOM Of NUM. INCIII01 a WWII WIC Of MAW us num us us ammo taco as mums NON at Ot Widel/A11011 01116111111 IT.P01115 OF VOW 01 OPENS SIAS SO 101 IIKISAMIT IMRE Offlaat WC Of MANN POMO OS POW. OD PROCESS WITH MICROFICHE AND PUBLISHER'S PRICES. reit WeROFICHE REPRODUCTION Pvim ONLY. (%1 00 O O POLYHEDRON MODELS for the Classroom MAGNUS J. WENNINGER St. Augustine's College Nassau, Bahama. rn ErNATIONAL COUNCIL. OF Ka TEACHERS OF MATHEMATICS 1201 Sixteenth Street, N.W., Washington, D. C. 20036 ivetmIssromrrIPRODUCE TmscortnIGMED Al"..Mt IAL BY MICROFICHE ONLY HAS IEEE rano By Mat __Comic _ TeachMar 10 ERIC MID ORGANIZATIONS OPERATING UNSER AGREEMENTS WHIM U.
    [Show full text]
  • Bridges Conference Proceedings Guidelines Word
    Bridges 2019 Conference Proceedings Helixation Rinus Roelofs Lansinkweg 28, Hengelo, the Netherlands; [email protected] Abstract In the list of names of the regular and semi-regular polyhedra we find many names that refer to a process. Examples are ‘truncated’ cube and ‘stellated’ dodecahedron. And Luca Pacioli named some of his polyhedral objects ‘elevations’. This kind of name-giving makes it easier to understand these polyhedra. We can imagine the model as a result of a transformation. And nowadays we are able to visualize this process by using the technique of animation. Here I will to introduce ‘helixation’ as a process to get a better understanding of the Poinsot polyhedra. In this paper I will limit myself to uniform polyhedra. Introduction Most of the Archimedean solids can be derived by cutting away parts of a Platonic solid. This operation , with which we can generate most of the semi-regular solids, is called truncation. Figure 1: Truncation of the cube. We can truncate the vertices of a polyhedron until the original faces of the polyhedron become regular again. In Figure 1 this process is shown starting with a cube. In the third object in the row, the vertices are truncated in such a way that the square faces of the cube are transformed to regular octagons. The resulting object is the Archimedean solid, the truncated cube. We can continue the process until we reach the fifth object in the row, which again is an Archimedean solid, the cuboctahedron. The whole process or can be represented as an animation. Also elevation and stellation can be described as processes.
    [Show full text]
  • [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.
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • 1 Mar 2014 Polyhedra, Complexes, Nets and Symmetry
    Polyhedra, Complexes, Nets and Symmetry Egon Schulte∗ Northeastern University Department of Mathematics Boston, MA 02115, USA March 4, 2014 Abstract Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are finite or infinite 3-periodic structures with interesting geometric, combinatorial, and algebraic properties. They can be viewed as finite or infinite 3-periodic graphs (nets) equipped with additional structure imposed by the faces, allowed to be skew, zig- zag, or helical. A polyhedron or complex is regular if its geometric symmetry group is transitive on the flags (incident vertex-edge-face triples). There are 48 regular polyhedra (18 finite polyhedra and 30 infinite apeirohedra), as well as 25 regular polygonal complexes, all infinite, which are not polyhedra. Their edge graphs are nets well-known to crystallographers, and we identify them explicitly. There also are 6 infinite families of chiral apeirohedra, which have two orbits on the flags such that adjacent flags lie in different orbits. 1 Introduction arXiv:1403.0045v1 [math.MG] 1 Mar 2014 Polyhedra and polyhedra-like structures in ordinary euclidean 3-space E3 have been stud- ied since the early days of geometry (Coxeter, 1973). However, with the passage of time, the underlying mathematical concepts and treatments have undergone fundamental changes. Over the past 100 years we can observe a shift from the classical approach viewing a polyhedron as a solid in space, to topological approaches focussing on the underlying maps on surfaces (Coxeter & Moser, 1980), to combinatorial approaches highlighting the basic incidence structure but deliberately suppressing the membranes that customarily span the faces to give a surface.
    [Show full text]
  • 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.
    [Show full text]
  • Registered by Australia Post Publication No. NBG6524
    Registered by Australia Post Publication No. NBG6524 The Australian UNIX* systems User Group Newsletter Volume 6 Number 5 April 1986 CONTENTS Editorial 2 Assessing Interactive Programs via Batch Processing 3 UNIX Fighting Fires 7 A Controlled, Productive Applications Environment at Esso Australia’s Production Department 19 CNFX - File Transfer Between UNIX systems and CSIRONET 30 Folding Regular Polyhedra 40 Initial Experiences with a Gould UTX/32 System in a University Environment46 An Introduction to the UNIFY Data Base Program 51 fsh - A Functional UNIX Command Interpreter 57 The Perth AUUG Meeting: A Visual Perspective 70 AUUG Meeting in Canberra 79 AUUG Membership and Subscription Forms 81 Copyright © 1985. AUUGN is the journal of the Australian UNIX systems User Group. Copying without fee is permitted provided that copies are not made or distributed for commercial advantage and credit to the source is given. Abstracting with credit is permitted. No~ other reproduction is permitted without the prior permission of the Australian UNIX systems User Group. * UNIX is a trademark of AT&T Bell Laboratories. Vol 6 No 5 AUUGN Editorial This issue is devoted entirely to the proceedings of the Perth conference. It contains eight of the sixteen papers presented; the others I have so far been unable to obtain. The group is also looking for a new newsletter editor as I find that I am unable to devote enough time to the exercise. If you are interested, contact me as soon as possible at the address and phone number appearing on the last page of this issue. Memberships and Subscriptions Membership and Subscription forms may be found at the end of this issue and all correspondence should be addressed to Greg Rose Honorary Secretary, AUUG PO Box 366 Kensington NSW 2033 Australia Next AUUG Meeting The next meeting will be in Canberra at the Australian National University.
    [Show full text]
  • Paper Models of Polyhedra Collection 4: Kepler-Poinsot Polyhedra
    Paper Models of Polyhedra Collection 4: Kepler-Poinsot Polyhedra - Great Stellated Dodecahedron - Small Stellated Dodecahedron - Great Icosahedron - Great Dodecahedron www.polyhedra.net Great Stellated Dodecahedron made out of one piece of paper. Cut the lines between the long and the short sides of the triangles. Fold the long lines backwards and fold the short lines forwards. Cut these lines www.polyhedra.net Great Stellated Dodecahedron made out of two pieces of paper. 2 Cut the lines between the long and the short sides of the triangles. Fold the long lines backwards and fold the short lines forwards. This is piece one. On the next page is piece two. 1 2 www.polyhedra.net 1 2 1 www.polyhedra.net Great Stellated Dodecahedron made out of five pieces of paper. Cut the lines between the long and the short sides of the triangles. Fold the long lines backwards and fold the short lines forwards. Glue part two to part one at the tap with 1 the number two. 2 6 www.polyhedra.net 2 3 1 www.polyhedra.net 3 4 2 www.polyhedra.net 4 5 3 www.polyhedra.net 5 1 4 www.polyhedra.net Small Stellated Dodecahedron (small version) Fold the long lines backwards Fold the short lines forwards www.polyhedra.net Small Stellated Dodecahedron (large version) Fold the long lines backwards 2 Fold the short lines forwards 1 3 www.polyhedra.net 2 1 www.polyhedra.net 3 1 4 www.polyhedra.net 3 4 www.polyhedra.net Great Icosahedron In 10 colors Small version Fold the dotted lines forwards www.polyhedra.net Fold the other lines backwards D Great Icosahedron In 10 colors A Large version Made out of 6 pieces C B E F Fold the dotted lines forwards www.polyhedra.net Fold the other lines backwards A B www.polyhedra.net C www.polyhedra.net A D www.polyhedra.net A A E www.polyhedra.net A F www.polyhedra.net Great Dodecahedron Small version www.polyhedra.net Fold the dotted lines forwards Fold the other lines backwards Great Dodecahedron Large version Made out of 6 pieces A B F C E D www.polyhedra.net B A C A www.polyhedra.net D A E A www.polyhedra.net F A www.polyhedra.net.
    [Show full text]
  • Complementary Relationship Between the Golden Ratio and the Silver
    Complementary Relationship between the Golden Ratio and the Silver Ratio in the Three-Dimensional Space & “Golden Transformation” and “Silver Transformation” applied to Duals of Semi-regular Convex Polyhedra September 1, 2010 Hiroaki Kimpara 1. Preface The Golden Ratio and the Silver Ratio 1 : serve as the fundamental ratio in the shape of objects and they have so far been considered independent of each other. However, the author has found out, through application of the “Golden Transformation” of rhombic dodecahedron and the “Silver Transformation” of rhombic triacontahedron, that these two ratios are closely related and complementary to each other in the 3-demensional space. Duals of semi-regular polyhedra are classified into 5 patterns. There exist 2 types of polyhedra in each of 5 patterns. In case of the rhombic polyhedra, they are the dodecahedron and the triacontahedron. In case of the trapezoidal polyhedral, they are the icosidodecahedron and the hexecontahedron. In case of the pentagonal polyhedra, they are icositetrahedron and the hexecontahedron. Of these 2 types, the one with the lower number of faces is based on the Silver Ratio and the one with the higher number of faces is based on the Golden Ratio. The “Golden Transformation” practically means to replace each face of: (1) the rhombic dodecahedron with the face of the rhombic triacontahedron, (2) trapezoidal icosidodecahedron with the face of the trapezoidal hexecontahedron, and (3) the pentagonal icositetrahedron with the face of the pentagonal hexecontahedron. These replacing faces are all partitioned into triangles. Similarly, the “Silver Transformation” means to replace each face of: (1) the rhombic triacontahedron with the face of the rhombic dodecahedron, (2) trapezoidal hexecontahedron with the face of the trapezoidal icosidodecahedron, and (3) the pentagonal hexecontahedron with the face of the pentagonal icositetrahedron.
    [Show full text]
  • Exhausting All Exact Solutions of BPS Domain Wall Networks in Arbitrary Dimensions
    PHYSICAL REVIEW D 101, 105020 (2020) Exhausting all exact solutions of BPS domain wall networks in arbitrary dimensions † ‡ Minoru Eto ,1,2,* Masaki Kawaguchi ,1, Muneto Nitta,2,3, and Ryotaro Sasaki1 1Department of Physics, Yamagata University, Kojirakawa-machi 1-4-12, Yamagata, Yamagata 990-8560, Japan 2Research and Education Center for Natural Sciences, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan 3Department of Physics, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan (Received 15 April 2020; accepted 15 May 2020; published 28 May 2020) We obtain full moduli parameters for generic nonplanar Bogomol’nyi-Prasad-Sommerfield networks of domain walls in an extended Abelian-Higgs model with N complex scalar fields and exhaust all exact solutions in the corresponding CPN−1 model. We develop a convenient description by grid diagrams which are polytopes determined by mass parameters of the model. To illustrate the validity of our method, we work out nonplanar domain wall networks for lower N in 3 þ 1 dimensions. In general, the networks can have compact vacuum bubbles, which are finite vacuum regions surrounded by domain walls, when the polytopes of the grid diagrams have inner vertices, and the size of bubbles can be controlled by moduli parameters. We also construct domain wall networks with bubbles in the shapes of the Platonic, Archimedean, Catalan, and Kepler-Poinsot solids. DOI: 10.1103/PhysRevD.101.105020 I. INTRODUCTION equations called BPS equations. In such cases, one can often embed the theories to supersymmetric (SUSY) It sometimes happens that systems have multiple discrete vacua or ground states, which is inevitable when a discrete theories by appropriately adding fermion superpartners, symmetry is spontaneously broken.
    [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]