Geometries and Transformations
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Semi-Regular Tilings of the Hyperbolic Plane
SEMI-REGULAR TILINGS OF THE HYPERBOLIC PLANE BASUDEB DATTA AND SUBHOJOY GUPTA Abstract. A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, and uniqueness, of a semi-regular tiling with a given vertex-type, and pose some open questions. 1. Introduction A tiling of a surface is a partition into (topological) polygons (the tiles) which are non- overlapping (interiors are disjoint) and such that tiles which touch, do so either at exactly one vertex, or along exactly one common edge. The vertex-type of a vertex v is a cyclic tuple of integers [k1; k2;:::; kd] where d is the degree (or valence) of v, and each ki (for 1 i d) is the number of sides (the size) of the i-th polygon around v, in either clockwise or≤ counter-clockwise≤ order. A semi-regular tiling on a surface of constant curvature (eg., the round sphere, the Euclidean plane or the hyperbolic plane) is one in which each polygon is regular, each edge is a geodesic, and the vertex-type is identical for each vertex (see Figure 1). Two tilings are equivalent if they are combinatorially isomorphic, that is, there is a homeomorphism of the surface to itself that takes vertices, edges and tiles of one tiling to those of the other. In fact, two semi-regular tilings of the hyperbolic plane are equivalent if and only if there is an isometry of the hyperbolic plane that realizes the combinatorial isomorphism between them (see Lemma 2.5). -
Journal of Student Writing the Journal of Student Writing
2019 THe Journal of Student Writing THe Journal of Student Writing 2019 Supervising Editor Bradley Siebert Managing Editors Jennifer Pacioianu Ande Davis Consulting Editor Muffy Walter The Angle is produced with the support of the Washburn University English Department. All contributors must be students at Washburn University. Prizewinners in each category were awarded a monetary prize. Works published here remain the intellectual property of their creators. Further information and submission guidelines are available on our website at washburn.edu/angle. To contact the managing editors, email us at [email protected]. Table of Contents First Year Writing Tragedies Build Bridges Ethan Nelson . 3 Better Than I Deserve Gordon Smith . 7 The Power of Hope Whit Downing . 11 Arts and Humanities A Defense of the Unethical Status of Deceptive Placebo Use in Research Gabrielle Kentch . 15 Personality and Prejudice: Defining the MBTI types in Jane Austen’s novels Madysen Mooradian . 27 “Washburn—Thy Strength Revealed”: Washburn University and the 1966 Tornado Taylor Nickel . 37 Washburn Student Handbooks/Planners and How They’ve Changed Kayli Goodheart . 45 The Heart of a Musician Savannah Workman . 63 Formation of Identity in 1984 Molly Murphy . 67 Setting Roots in the Past Mikaela Cox. 77 Natural and Social Sciences The Effects of Conspiracy Exposure on Politically Cooperative Behavior Lydia Shontz, Katy Chase, and Tomohiro Ichikawa . 91 A New Major is Born: The Life of Computer Science at Washburn University Alex Montgomery . 111 Kinesiology Tape: Physiological or Psychological? Mikaela Cox . .121 Breaking w/the Fifth, an Introduction to Spherical Geometry Mary P. Greene . 129 Water Bears Sydney Watters . -
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. -
The Sphere by Itself: Can We Distribute Points on It Evenly?
Chapter III The sphere by itself: can we distribute points on it evenly? III.1. The metric of the sphere and spherical trigonometry As we shall see throughout this chapter, the geometry of the “ordinary sphere” S2 two dimensional in a space of three dimensions harbors many pitfalls. It’s much more subtle than we might think, given the nice roundness and all the sym- metries of the object. Its geometry is indeed not made easier at least for certain questions by its being round, compact, and bounded, in contrast to the Euclidean plane. Sect. III.3 will be the most representative in this regard; but, much simpler and more fundamental, we encounter the “impossible” problem of maps of Earth, which we will scarcely mention, except in Sect. III.3; see also 18.1 of [B]. One of the reasons for the difficulties the sphere poses is that its group of isometries is not at all commutative, whereas the Euclidean plane admits a commutative group of trans- lations. But this group of rotations about the origin of E3 the orthogonal group O.3/ is crucial for our lives. Here is the place to quote (Gromov, 1988b): “O.3/ pervades all the essential properties of the physical world. But we remain intellectually blind to this symmetry, even if we encounter it frequently and use it in everyday life, for instance when we experience or engender mechanical movements, such as walking. This is due in part to the non commutativity of O.3/,whichis difficult to grasp.” Since spherical geometry is not much treated in the various curricula, we permit ourselves at the outset some very elementary recollections, which are treated in detail in Chap. -
Detc2020-17855
Proceedings of the ASME 2020 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2020 August 16-19, 2020, St. Louis, USA DETC2020-17855 DRAFT: GENERATIVE INFILLS FOR ADDITIVE MANUFACTURING USING SPACE-FILLING POLYGONAL TILES Matthew Ebert∗ Sai Ganesh Subramanian† J. Mike Walker ’66 Department of J. Mike Walker ’66 Department of Mechanical Engineering Mechanical Engineering Texas A&M University Texas A&M University College Station, Texas 77843, USA College Station, Texas 77843, USA Ergun Akleman‡ Vinayak R. Krishnamurthy Department of Visualization J. Mike Walker ’66 Department of Texas A&M University Mechanical Engineering College Station, Texas 77843, USA Texas A&M University College Station, Texas 77843, USA ABSTRACT 1 Introduction We study a new class of infill patterns, that we call wallpaper- In this paper, we present a geometric modeling methodology infills for additive manufacturing based on space-filling shapes. for generating a new class of infills for additive manufacturing. To this end, we present a simple yet powerful geometric modeling Our methodology combines two fundamental ideas, namely, wall- framework that combines the idea of Voronoi decomposition space paper symmetries in the plane and Voronoi decomposition to with wallpaper symmetries defined in 2-space. We first provide allow for enumerating material patterns that can be used as infills. a geometric algorithm to generate wallpaper-infills and design four special cases based on selective spatial arrangement of seed points on the plane. Second, we provide a relationship between 1.1 Motivation & Objectives the infill percentage to the spatial resolution of the seed points for Generation of infills is an essential component in the additive our cases thus allowing for a systematic way to generate infills manufacturing pipeline [1, 2] and significantly affects the cost, at the desired volumetric infill percentages. -
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. -
259 ) on the Equations of Loci Traced Upon the Surface of The
259 ) On the Equations of Loci traced upon the Surface of the Sphere, as expressed by Spherical Co-ordinates. By THOMAS STE- PHENS DAVIES, Esq. F.R.S.ED. F.R.A.S. (Read 16th January 1832J 1 HE modern system of analytical geometry of three dimensions originated with CLAIRAULT, and received its final form from the hands of MONGE. DESCARTES, it is true, had remarked, that the orthogonal projections of a curve anyhow situated in space, upon two given rectangular planes, determined the magnitude, species, and position of that curve; but this is, in fact, only an appro- priation to scientific purposes of a principle which must have been employed from the earliest period of architectural delinea- tion—the orthography and ichnographyyor the ground-plan and section of the system of represented lines. Had DESCARTES, however, done more than make the suggestion—had he pointed out the particular aspect under which it could have been ren- dered available to geometrical research—had he furnished a suit- able notation and methods of investigation—and, finally, had he given a few examples, calculated to render his analytical pro- cesses intelligible to other mathematicians;—then, indeed, this branch of science would have owed him deeper obligations than it can now be said to do. Still I do not wish to be understood to undervalue the labours of that extraordinary man, indirectly at least, upon this subject; for it is certain, that, though he did succeed in placing the inquiry in its true position, yet it is ulti- mately to his method of treating plane loci, by means of indeter- minate algebraical equations, that we owe every thing we know in 260 Mr DAVIES on the Equations of Loci the Geometry of Three Dimensions, beyond its simplest elements, as well as the most interesting and important portions of the doc- trine of plane loci itself. -
Dispersion Relations of Periodic Quantum Graphs Associated with Archimedean Tilings (I)
Dispersion relations of periodic quantum graphs associated with Archimedean tilings (I) Yu-Chen Luo1, Eduardo O. Jatulan1,2, and Chun-Kong Law1 1 Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 80424. Email: [email protected] 2 Institute of Mathematical Sciences and Physics, University of the Philippines Los Banos, Philippines 4031. Email: [email protected] January 15, 2019 Abstract There are totally 11 kinds of Archimedean tiling for the plane. Applying the Floquet-Bloch theory, we derive the dispersion relations of the periodic quantum graphs associated with a number of Archimedean tiling, namely the triangular tiling (36), the elongated triangular tiling (33; 42), the trihexagonal tiling (3; 6; 3; 6) and the truncated square tiling (4; 82). The derivation makes use of characteristic functions, with the help of the symbolic software Mathematica. The resulting dispersion relations are surpris- ingly simple and symmetric. They show that in each case the spectrum is composed arXiv:1809.09581v2 [math.SP] 12 Jan 2019 of point spectrum and an absolutely continuous spectrum. We further analyzed on the structure of the absolutely continuous spectra. Our work is motivated by the studies on the periodic quantum graphs associated with hexagonal tiling in [13] and [11]. Keywords: characteristic functions, Floquet-Bloch theory, quantum graphs, uniform tiling, dispersion relation. 1 1 Introduction Recently there have been a lot of studies on quantum graphs, which is essentially the spectral problem of a one-dimensional Schr¨odinger operator acting on the edge of a graph, while the functions have to satisfy some boundary conditions as well as vertex conditions which are usually the continuity and Kirchhoff conditions. -
Supporting Information for Harnessing Instabilities for Design of Soft Reconfigurable Auxetic/Chiral Materials
Electronic Supplementary Material (ESI) for Soft Matter This journal is © The Royal Society of Chemistry 2013 Supporting Information for Harnessing instabilities for design of soft reconfigurable auxetic/chiral materials Jongmin Shim,1, 2 Sicong Shan,3, 2 Andrej Koˇsmrlj,4 Sung H. Kang,3 Elizabeth R. Chen,3 James C. Weaver,5 and Katia Bertoldi3, 6 1Department of Civil, Structural and Environmental Engineering, University at Buffalo, NY, 14260, USA 2J.S. and S.S. contributed equally to this work 3School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA 4Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 5Wyss Institute for Biologically Inspired Engineering, Harvard University, Cambridge, Massachusetts 02138, USA 6Kavli Institute, Harvard University, Cambridge, Massachusetts 02138, USA ANALYSIS - DESIGN PRINCIPLES From tilings of the 2D Euclidean plane to porous structures To identify possible periodic monodisperse circular hole arrangements in elastic plates where buckling can be exploited as a mechanism to reversibly switch between undeformed/expanded and deformed/compact configur- ations, we investigate the hole arrangements by consider- ing geometric constraints on the tilings (i.e., tessellations) of the 2D Euclidean plane. In order for all the monodisperse circular holes to close through buckling of the ligaments, the plates should meet the following requirements: (a) the center-to-center dis- tances of adjacent holes are identical, so that all the lig- aments are characterized by the same minimum width and undergo the first buckling mode in an approxim- ately uniform manner; (b) there is an even number of ligaments around every hole, so that the deformation induced by buckling leads to their closure. -
New Family of Tilings of Three-Dimensional Euclidean Space by Tetrahedra and Octahedra
New family of tilings of three-dimensional Euclidean space by tetrahedra and octahedra John H. Conwaya, Yang Jiaob, and Salvatore Torquatob,c,1 aDepartment of Mathematics, Princeton University, Princeton, NJ 08544; bPrinceton Institute for the Science and Technology of Materials, Princeton University, Princeton, NJ 08544; and cDepartment of Chemistry, Department of Physics, Princeton Center for Theoretical Science, Program in Computational and Applied Mathematics, Princeton University, Princeton, NJ 08544 Edited* by Ronald L. Graham, University of California, La Jolla, CA, and approved May 17, 2011 (received for review April 7, 2011) It is well known that two regular tetrahedra can be combined with a single regular octahedron to tile (complete fill) three-dimensional Euclidean space R3. This structure was called the “octet truss” by Buckminster Fuller. It was believed that such a tiling, which is the Delaunay tessellation of the face-centered cubic (fcc) lattice, and its closely related stacking variants, are the only tessellations of R3 that involve two different regular polyhedra. Here we identify and analyze a unique family comprised of a noncountably infinite number of periodic tilings of R3 whose smallest repeat tiling unit Fig. 1. The three regular tilings of the plane: (A) A portion of the tiling by triangles with a fundamental cell containing two triangles with two different consists of one regular octahedron and six smaller regular tetrahe- orientations (shown by different shadings). (B) A portion of the tiling by dra. We first derive an extreme member of this unique tiling family squares. (C) A portion of the tiling by hexagons. by showing that the “holes” in the optimal lattice packing of octa- hedra, obtained by Minkowski over a century ago, are congruent regular polygons and the remaining two of which are made of tetrahedra. -
Collection Volume I
Collection volume I PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Thu, 29 Jul 2010 21:47:23 UTC Contents Articles Abstraction 1 Analogy 6 Bricolage 15 Categorization 19 Computational creativity 21 Data mining 30 Deskilling 41 Digital morphogenesis 42 Heuristic 44 Hidden curriculum 49 Information continuum 53 Knowhow 53 Knowledge representation and reasoning 55 Lateral thinking 60 Linnaean taxonomy 62 List of uniform tilings 67 Machine learning 71 Mathematical morphology 76 Mental model 83 Montessori sensorial materials 88 Packing problem 93 Prior knowledge for pattern recognition 100 Quasi-empirical method 102 Semantic similarity 103 Serendipity 104 Similarity (geometry) 113 Simulacrum 117 Squaring the square 120 Structural information theory 123 Task analysis 126 Techne 128 Tessellation 129 Totem 137 Trial and error 140 Unknown unknown 143 References Article Sources and Contributors 146 Image Sources, Licenses and Contributors 149 Article Licenses License 151 Abstraction 1 Abstraction Abstraction is a conceptual process by which higher, more abstract concepts are derived from the usage and classification of literal, "real," or "concrete" concepts. An "abstraction" (noun) is a concept that acts as super-categorical noun for all subordinate concepts, and connects any related concepts as a group, field, or category. Abstractions may be formed by reducing the information content of a concept or an observable phenomenon, typically to retain only information which is relevant for a particular purpose. For example, abstracting a leather soccer ball to the more general idea of a ball retains only the information on general ball attributes and behavior, eliminating the characteristics of that particular ball. -
Analytic Methods for Simulated Light Transport
ANALYTIC METHODS FOR SIMULATED LIGHT TRANSPORT James Richard Arvo Yale University This thesis presents new mathematical and computational to ols for the simula tion of light transp ort in realistic image synthesis New algorithms are presented for exact computation of direct il lumination eects related to light emission shadow ing and rstorder scattering from surfaces New theoretical results are presented for the analysis of global il lumination algorithms which account for all interreec tions of light among surfaces of an environment First a closedform expression is derived for the irradiance Jacobian which is the derivative of a vector eld representing radiant energy ux The expression holds for diuse p olygonal scenes and correctly accounts for shadowing or partial o cclusion Three applications of the irradiance Jacobian are demonstrated lo cat ing lo cal irradiance extrema direct computation of isolux contours and surface mesh generation Next the concept of irradiance is generalized to tensors of arbitrary order A recurrence relation for irradiance tensors is derived that extends a widely used formula published by Lamb ert in Several formulas with applications in com puter graphics are derived from this recurrence relation and are indep endently veried using a new Monte Carlo metho d for sampling spherical triangles The formulas extend the range of nondiuse eects that can be computed in closed form to include illumination from directional area light sources and reections from and transmissions through glossy surfaces Finally new analysis