E8 Physics and Quasicrystals Icosidodecahedron and Rhombic Triacontahedron Vixra 1301.0150 Frank Dodd (Tony) Smith Jr
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Systematic Narcissism
Systematic Narcissism Wendy W. Fok University of Houston Systematic Narcissism is a hybridization through the notion of the plane- tary grid system and the symmetry of narcissism. The installation is based on the obsession of the physical reflection of the object/subject relation- ship, created by the idea of man-made versus planetary projected planes (grids) which humans have made onto the earth. ie: Pierre Charles L’Enfant plan 1791 of the golden section projected onto Washington DC. The fostered form is found based on a triacontahedron primitive and devel- oped according to the primitive angles. Every angle has an increment of 0.25m and is based on a system of derivatives. As indicated, the geometry of the world (or the globe) is based off the planetary grid system, which, according to Bethe Hagens and William S. Becker is a hexakis icosahedron grid system. The formal mathematical aspects of the installation are cre- ated by the offset of that geometric form. The base of the planetary grid and the narcissus reflection of the modular pieces intersect the geometry of the interfacing modular that structures the installation BACKGROUND: Poetics: Narcissism is a fixation with oneself. The Greek myth of Narcissus depicted a prince who sat by a reflective pond for days, self-absorbed by his own beauty. Mathematics: In 1978, Professors William Becker and Bethe Hagens extended the Russian model, inspired by Buckminster Fuller’s geodesic dome, into a grid based on the rhombic triacontahedron, the dual of the icosidodeca- heron Archimedean solid. The triacontahedron has 30 diamond-shaped faces, and possesses the combined vertices of the icosahedron and the dodecahedron. -
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). -
Rhombic Triacontahedron
Rhombic Triacontahedron Figure 1 Rhombic Triacontahedron. Vertex labels as used for the corresponding vertices of the 120 Polyhedron. COPYRIGHT 2007, Robert W. Gray Page 1 of 11 Encyclopedia Polyhedra: Last Revision: August 5, 2007 RHOMBIC TRIACONTAHEDRON Figure 2 Icosahedron (red) and Dodecahedron (yellow) define the rhombic Triacontahedron (green). Figure 3 “Long” (red) and “short” (yellow) face diagonals and vertices. COPYRIGHT 2007, Robert W. Gray Page 2 of 11 Encyclopedia Polyhedra: Last Revision: August 5, 2007 RHOMBIC TRIACONTAHEDRON Topology: Vertices = 32 Edges = 60 Faces = 30 diamonds Lengths: 15+ ϕ = 2 EL ≡ Edge length of rhombic Triacontahedron. ϕ + 2 EL = FDL ≅ 0.587 785 252 FDL 2ϕ 3 − ϕ = DVF ϕ 2 FDL ≡ Long face diagonal = DVF ≅ 1.236 067 977 DVF ϕ 2 = EL ≅ 1.701 301 617 EL 3 − ϕ 1 FDS ≡ Short face diagonal = FDL ≅ 0.618 033 989 FDL ϕ 2 = DVF ≅ 0.763 932 023 DVF ϕ 2 2 = EL ≅ 1.051 462 224 EL ϕ + 2 COPYRIGHT 2007, Robert W. Gray Page 3 of 11 Encyclopedia Polyhedra: Last Revision: August 5, 2007 RHOMBIC TRIACONTAHEDRON DFVL ≡ Center of face to vertex at the end of a long face diagonal 1 = FDL 2 1 = DVF ≅ 0.618 033 989 DVF ϕ 1 = EL ≅ 0.850 650 808 EL 3 − ϕ DFVS ≡ Center of face to vertex at the end of a short face diagonal 1 = FDL ≅ 0.309 016 994 FDL 2 ϕ 1 = DVF ≅ 0.381 966 011 DVF ϕ 2 1 = EL ≅ 0.525 731 112 EL ϕ + 2 ϕ + 2 DFE = FDL ≅ 0.293 892 626 FDL 4 ϕ ϕ + 2 = DVF ≅ 0.363 271 264 DVF 21()ϕ + 1 = EL 2 ϕ 3 − ϕ DVVL = FDL ≅ 0.951 056 516 FDL 2 = 3 − ϕ DVF ≅ 1.175 570 505 DVF = ϕ EL ≅ 1.618 033 988 EL COPYRIGHT 2007, Robert W. -
Construction of Great Icosidodecahedron This Paper
Construction of Great IcosiDodecahedron This paper describes how to create the Great Icosidodecahedron analytically as a 3D object. This uniform polyhedron is particularly beautiful and can be constructed with 12 pentagrams and 20 equilateral triangles. You start with a pentagon and create the net for a Icosidodecahedron without the triangles (fig 1). Extend the sides of the pentagons to create pentagrams and fold the sides into the same planes as the icosidodecahedron and you get the first half of the polyhedron (fig 2). Now for the equilateral triangles. Pick every combination of two pentagrams where a single vertex shares a common point (fig 3). Create two equilateral triangles connecting the common vertex and each of the legs of the two pentagrams. Do this to for all pentagrams and you will come up with 20 distinct intersecting triangles. The final result is in figure 4. We can create the model using a second method. Based on the geometry of the existing model, create a single cup (fig 5). Add the adjoining pentagram star (fig 6). Now take this shape and rotate- duplicate for each plane of the pentagon icosidodecahedron and you get the model of figure 4. 1 Construction of Great IcosiDodecahedron fig1 The net for the Icosidodecahedron without the triangle faces. I start with a pentagon, triangulate it with an extra point in the center. 2 Construction of Great IcosiDodecahedron fig 2 Extend the sides of the pentagons to make pentagrams then fold the net into the same planes as the Icosidodecahedron. This makes 12 intersecting stars (pentagrams). fig 3 Now add equilateral triangles to each pair of pentagrams, which touch in only one vertex. -
Enhancing Strategic Discourse Systematically Using Climate Metaphors Widespread Comprehension of System Dynamics in Weather Patterns As a Resource -- /
Alternative view of segmented documents via Kairos 24 August 2015 | Draft Enhancing Strategic Discourse Systematically using Climate Metaphors Widespread comprehension of system dynamics in weather patterns as a resource -- / -- Introduction Systematic global insight of memorable quality? Towards memorable framing of global climate of governance processes? Visual representations of globality of requisite variety for global governance Four-dimensional requisite for a time-bound global civilization? Comprehending the shapes of time through four-dimensional uniform polychora Five-fold ordering of strategic engagement with time Interplay of cognitive patterns in discourse on systemic change Five-fold cognitive dynamics of relevance to governance? Decision-making capacity versus Distinction-making capacity: embodying whether as weather From star-dom to whizdom to isdom? From space-ship design to time-ship embodiment as a requisite metaphor of governance References Prepared in anticipation of the United Nations Climate Change Conference (Paris, 2015) Introduction There is considerable familiarity with the dynamics of climate and weather through the seasons and in different locations. These provide a rich source of metaphor, widely shared, and frequently exploited as a means of communicating subtle insights into social phenomena, strategic options and any resistance to them. It is however now difficult to claim any coherence to the strategic discourse on which humanity and global governance is held to be dependent. This is evident in the deprecation of one political faction by another, the exacerbation of conflictual perceptions by the media, the daily emergence of intractable crises, and the contradictory assertions of those claiming expertise in one arena or another. The dynamics have been caricatured as blame-gaming, as separately discussed (Blame game? It's them not us ! 2015). -
A Tourist Guide to the RCSR
A tourist guide to the RCSR Some of the sights, curiosities, and little-visited by-ways Michael O'Keeffe, Arizona State University RCSR is a Reticular Chemistry Structure Resource available at http://rcsr.net. It is open every day of the year, 24 hours a day, and admission is free. It consists of data for polyhedra and 2-periodic and 3-periodic structures (nets). Visitors unfamiliar with the resource are urged to read the "about" link first. This guide assumes you have. The guide is designed to draw attention to some of the attractions therein. If they sound particularly attractive please visit them. It can be a nice way to spend a rainy Sunday afternoon. OKH refers to M. O'Keeffe & B. G. Hyde. Crystal Structures I: Patterns and Symmetry. Mineral. Soc. Am. 1966. This is out of print but due as a Dover reprint 2019. POLYHEDRA Read the "about" for hints on how to use the polyhedron data to make accurate drawings of polyhedra using crystal drawing programs such as CrystalMaker (see "links" for that program). Note that they are Cartesian coordinates for (roughly) equal edge. To make the drawing with unit edge set the unit cell edges to all 10 and divide the coordinates given by 10. There seems to be no generally-agreed best embedding for complex polyhedra. It is generally not possible to have equal edge, vertices on a sphere and planar faces. Keywords used in the search include: Simple. Each vertex is trivalent (three edges meet at each vertex) Simplicial. Each face is a triangle. -
Icosahedral Polyhedra from 6 Lattice and Danzer's ABCK Tiling
Icosahedral Polyhedra from 퐷6 lattice and Danzer’s ABCK tiling Abeer Al-Siyabi, a Nazife Ozdes Koca, a* and Mehmet Kocab aDepartment of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khoud, 123 Muscat, Sultanate of Oman, bDepartment of Physics, Cukurova University, Adana, Turkey, retired, *Correspondence e-mail: [email protected] ABSTRACT It is well known that the point group of the root lattice 퐷6 admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group 퐻3 , its roots and weights are determined in terms of those of 퐷6 . Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of 퐷6 determined by a pair of integers (푚1, 푚2 ) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of 퐻3 and it is shown that the inflation of the tiles can be obtained as projections of the lattice vectors characterized by the pair of integers which are linear combinations of the integers (푚1, 푚2 ) with coefficients from Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the < 퐴퐵퐶퐾 > octahedral tilings in 3D space with icosahedral symmetry 퐻3 and those related transformations in 6D space with 퐷6 symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in 퐷6 .The tetrahedron K constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of “K-polyhedron”, “B-polyhedron” and “C-polyhedron” generated by the icosahedral group have been discussed. -
Section 100.00 Synergy
Table of Contents ● 100.00 SYNERGY ● 100.01 Introduction: Scenario of the Child ❍ 100.010 Awareness of the Child ❍ 100.020 Human Sense Awareness ❍ 100.030 Resolvability Limits ❍ 100.10 Subdivision of Tetrahedral Unity ■ 100.101 Synergetic Unity ■ 100.120 Icosa and Tetra ❍ 100.20 Scenario of the Child ❍ 100.30 Omnirational Subdividing ■ 100.3011 Necklace ■ 100.304 Cheese Tetrahedron ■ 100.310 Two Tetra into Cube ❍ 100.320 Modular Subdivision of the Cosmic Hierarchy ❍ 100.330 "Me" Ball ❍ 100.40 Finite Event Scenario ■ 100.41 Foldability of Triangles into Tetrahedra ■ 100.415 Unfoldable Limit ❍ 100.50 Constant Triangular Symmetry ❍ 100.60 Finite Episoding ● 101.00 Definition: Synergy ● 120.00 Mass Interattraction ● 130.00 Precession and Entropy ● 140.00 Corollary of Synergy: Principle of the Whole System ● 150.00 Synergy-of-Synergies ● 160.00 Generalized Design Science Exploration ● 180.00 Design Science and Human-Tolerance Limits Next Page Copyright © 1997 Estate of Buckminster Fuller 100.00 SYNERGY 100.01 Introduction: Scenario of the Child [100.01-100.63 Child as Explorer Scenario] 100.010 Awareness of the Child: The simplest descriptions are those expressed by only one word. The one word alone that describes the experience "life" is "awareness." Awareness requires an otherness of which the observer can be aware. The communication of awareness is both subjective and objective, from passive to active, from otherness to self, from self to otherness. Awareness = self + otherness Awareness = observer + observed 100.011 Awareness is the otherness saying to the observer, "See Me." Awareness is the observer saying to self, "I see the otherness." Otherness induces awareness of self. -
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). -
Convex Polytopes and Tilings with Few Flag Orbits
Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami 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 Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals. -
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