Cubic Circular, Issue 7 & 8
Total Page：16
File Type：pdf, Size：1020Kb
Load more
Recommended publications
-
Final Poster
Associating Finite Groups with Dessins d’Enfants Luis Baeza, Edwin Baeza, Conner Lawrence, and Chenkai Wang Abstract Platonic Solids Rotation Group Dn: Regular Convex Polygon Approach Each finite, connected planar graph has an automorphism group G;such Following Magot and Zvonkin, reduce to easier cases using “hypermaps” permutations can be extended to automorphisms of the Riemann sphere φ : P1(C) P1(C), then composing β = φ f where S 2(R) P1(C). In 1984, Alexander Grothendieck, inspired by a result of f : 1( ) ! 1( )isaBely˘ımapasafunctionofeither◦ zn or ' P C P C Gennadi˘ıBely˘ıfrom 1979, constructed a finite, connected planar graph 4 zn/(zn +1)! 2 such that Aut(f ) Z or Aut(f ) D ,respectively. ' n ' n ∆β via certain rational functions β(z)=p(z)/q(z)bylookingatthe inverse image of the interval from 0 to 1. The automorphisms of such a Hypermaps: Rotation Group Zn graph can be identified with the Galois group Aut(β)oftheassociated 1 1 rational function β : P (C) P (C). In this project, we investigate how Rigid Rotations of the Platonic Solids I Wheel/Pyramids (J1, J2) ! w 3 (w +8) restrictive Grothendieck’s concept of a Dessin d’Enfant is in generating all n 2 I φ(w)= 1 1 z +1 64 (w 1) automorphisms of planar graphs. We discuss the rigid rotations of the We have an action : PSL2(C) P (C) P (C). β(z)= : v = n + n, e =2 n, f =2 − n ◦ ⇥ 2 !n 2 4 zn · Platonic solids (the tetrahedron, cube, octahedron, icosahedron, and I Zn = r r =1 and Dn = r, s s = r =(sr) =1 are the rigid I Cupola (J3, J4, J5) dodecahedron), the Archimedean solids, the Catalan solids, and the rotations of the regular convex polygons,with 4w 4(w 2 20w +105)3 I φ(w)= − ⌦ ↵ ⌦ 1 ↵ Rotation Group A4: Tetrahedron 3 2 Johnson solids via explicit Bely˘ımaps. -
7 LATTICE POINTS and LATTICE POLYTOPES Alexander Barvinok
7 LATTICE POINTS AND LATTICE POLYTOPES Alexander Barvinok INTRODUCTION Lattice polytopes arise naturally in algebraic geometry, analysis, combinatorics, computer science, number theory, optimization, probability and representation the- ory. They possess a rich structure arising from the interaction of algebraic, convex, analytic, and combinatorial properties. In this chapter, we concentrate on the the- ory of lattice polytopes and only sketch their numerous applications. We briefly discuss their role in optimization and polyhedral combinatorics (Section 7.1). In Section 7.2 we discuss the decision problem, the problem of finding whether a given polytope contains a lattice point. In Section 7.3 we address the counting problem, the problem of counting all lattice points in a given polytope. The asymptotic problem (Section 7.4) explores the behavior of the number of lattice points in a varying polytope (for example, if a dilation is applied to the polytope). Finally, in Section 7.5 we discuss problems with quantifiers. These problems are natural generalizations of the decision and counting problems. Whenever appropriate we address algorithmic issues. For general references in the area of computational complexity/algorithms see [AB09]. We summarize the computational complexity status of our problems in Table 7.0.1. TABLE 7.0.1 Computational complexity of basic problems. PROBLEM NAME BOUNDED DIMENSION UNBOUNDED DIMENSION Decision problem polynomial NP-hard Counting problem polynomial #P-hard Asymptotic problem polynomial #P-hard∗ Problems with quantifiers unknown; polynomial for ∀∃ ∗∗ NP-hard ∗ in bounded codimension, reduces polynomially to volume computation ∗∗ with no quantifier alternation, polynomial time 7.1 INTEGRAL POLYTOPES IN POLYHEDRAL COMBINATORICS We describe some combinatorial and computational properties of integral polytopes. -
Computational Design Framework 3D Graphic Statics
Computational Design Framework for 3D Graphic Statics 3D Graphic for Computational Design Framework Computational Design Framework for 3D Graphic Statics Juney Lee Juney Lee Juney ETH Zurich • PhD Dissertation No. 25526 Diss. ETH No. 25526 DOI: 10.3929/ethz-b-000331210 Computational Design Framework for 3D Graphic Statics A thesis submitted to attain the degree of Doctor of Sciences of ETH Zurich (Dr. sc. ETH Zurich) presented by Juney Lee 2015 ITA Architecture & Technology Fellow Supervisor Prof. Dr. Philippe Block Technical supervisor Dr. Tom Van Mele Co-advisors Hon. D.Sc. William F. Baker Prof. Allan McRobie PhD defended on October 10th, 2018 Degree confirmed at the Department Conference on December 5th, 2018 Printed in June, 2019 For my parents who made me, for Dahmi who raised me, and for Seung-Jin who completed me. Acknowledgements I am forever indebted to the Block Research Group, which is truly greater than the sum of its diverse and talented individuals. The camaraderie, respect and support that every member of the group has for one another were paramount to the completion of this dissertation. I sincerely thank the current and former members of the group who accompanied me through this journey from close and afar. I will cherish the friendships I have made within the group for the rest of my life. I am tremendously thankful to the two leaders of the Block Research Group, Prof. Dr. Philippe Block and Dr. Tom Van Mele. This dissertation would not have been possible without my advisor Prof. Block and his relentless enthusiasm, creative vision and inspiring mentorship. -
On a Remarkable Cube of Pyrite, Carrying Crys- Tallized Gold and Galena of Unusual Habit
ON A REMARKABLE CUBE OF PYRITE, CARRYING CRYS- TALLIZED GOLD AND GALENA OF UNUSUAL HABIT By JOSEPH E. POGUE Assistant Curator, Division of Mineralogy, U. S. National Museum With One Plate The intergrowth or interpenetration of two or more minerals, especially if these be well crystallized, often shows a certain mutual crystallographic control in the arrangement of the individuals, sug- gestive of interacting molecular forces. Occasionally a crystal upon nearly completing its growth exerts what may be termed "surface affinit}'," in that it seems to attract molecules of composition differ- ent from its own and causes these to crystallize in positions bearing definite crystallographic relations to the host crystal, as evidenced, for example, by the regular arrangement of marcasite on calcite, chalcopyrite on galena, quartz on fluorite, and so on. Of special interest, not only because exhibiting the features mentioned above, but also on account of the unusual development of the individuals and the great beauty of the specimen, is a large cube of pyrite, studded with crystals of native gold and partly covered by plates of galena, acquired some years ago by the U. S. National Museum. This cube measures about 2 inches (51 mm.) along its edge, and is prominently striated, as is often the case with pyrite. It contains something more than 130 crystals of gold attached to its surface, has about one-fourth of its area covered with galena, and upon one face shows an imperfect crystal of chalcopyrite. The specimen came into the possession of the National Museum in 1906 and was ob- tained from the Snettisham District, near Juneau, Southeast Alaska. -
A Parallelepiped Based Approach
3D Modelling Using Geometric Constraints: A Parallelepiped Based Approach Marta Wilczkowiak, Edmond Boyer, and Peter Sturm MOVI–GRAVIR–INRIA Rhˆone-Alpes, 38330 Montbonnot, France, [email protected], http://www.inrialpes.fr/movi/people/Surname Abstract. In this paper, efficient and generic tools for calibration and 3D reconstruction are presented. These tools exploit geometric con- straints frequently present in man-made environments and allow cam- era calibration as well as scene structure to be estimated with a small amount of user interactions and little a priori knowledge. The proposed approach is based on primitives that naturally characterize rigidity con- straints: parallelepipeds. It has been shown previously that the intrinsic metric characteristics of a parallelepiped are dual to the intrinsic charac- teristics of a perspective camera. Here, we generalize this idea by taking into account additional redundancies between multiple images of multiple parallelepipeds. We propose a method for the estimation of camera and scene parameters that bears strongsimilarities with some self-calibration approaches. Takinginto account prior knowledgeon scene primitives or cameras, leads to simpler equations than for standard self-calibration, and is expected to improve results, as well as to allow structure and mo- tion recovery in situations that are otherwise under-constrained. These principles are illustrated by experimental calibration results and several reconstructions from uncalibrated images. 1 Introduction This paper is about using partial information on camera parameters and scene structure, to simplify and enhance structure from motion and (self-) calibration. We are especially interested in reconstructing man-made environments for which constraints on the scene structure are usually easy to provide. -
Area, Volume and Surface Area
The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 11 AREA, VOLUME AND SURFACE AREA A guide for teachers - Years 8–10 June 2011 YEARS 810 Area, Volume and Surface Area (Measurement and Geometry: Module 11) For teachers of Primary and Secondary Mathematics 510 Cover design, Layout design and Typesetting by Claire Ho The Improving Mathematics Education in Schools (TIMES) Project 2009‑2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. © The University of Melbourne on behalf of the international Centre of Excellence for Education in Mathematics (ICE‑EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution‑NonCommercial‑NoDerivs 3.0 Unported License. http://creativecommons.org/licenses/by‑nc‑nd/3.0/ The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 11 AREA, VOLUME AND SURFACE AREA A guide for teachers - Years 8–10 June 2011 Peter Brown Michael Evans David Hunt Janine McIntosh Bill Pender Jacqui Ramagge YEARS 810 {4} A guide for teachers AREA, VOLUME AND SURFACE AREA ASSUMED KNOWLEDGE • Knowledge of the areas of rectangles, triangles, circles and composite figures. • The definitions of a parallelogram and a rhombus. • Familiarity with the basic properties of parallel lines. • Familiarity with the volume of a rectangular prism. • Basic knowledge of congruence and similarity. • Since some formulas will be involved, the students will need some experience with substitution and also with the distributive law. -
Evaporationin Icparticles
The JapaneseAssociationJapanese Association for Crystal Growth (JACG){JACG) Small Metallic ParticlesProduced by Evaporation in lnert Gas at Low Pressure Size distributions, crystal morphologyand crystal structures KazuoKimoto physiosLaboratory, Department of General Educatien, Nago)ra University Various experimental results ef the studies on fine 1. Introduction particles produced by evaperation and subsequent conden$ation in inert gas at low pressure are reviewed. Small particles of metals and semi-metals can A brief historical survey is given and experimental be produced by evaporation and subsequent arrangements for the production of the particles are condensation in the free space of an inert gas at described. The structure of the stnoke, the qualitative low pressure, a very simple technique, recently particle size clistributions, small particle statistios and "gas often referred to as evaporation technique"i) the crystallographic aspccts ofthe particles are consid- ered in some detail. Emphasis is laid on the crystal (GET). When the pressure of an inert is in the inorphology and the related crystal structures ef the gas range from about one to several tens of Torr, the particles efsome 24 elements. size of the particles produced by GET is in thc range from several to several thousand nm, de- pending on the materials evaporated, the naturc of the inert gas and various other evaporation conditions. One of the most characteristic fea- tures of the particles thus produced is that the particles have, generally speaking, very well- defined crystal habits when the particle size is in the range from about ten to several hundred nm. The crystal morphology and the relevant crystal structures of these particles greatly interested sDme invcstigators in Japan, 4nd encouraged them to study these propenies by means of elec- tron microscopy and electron difliraction. -
Uniform Polychora
BRIDGES Mathematical Connections in Art, Music, and Science Uniform Polychora Jonathan Bowers 11448 Lori Ln Tyler, TX 75709 E-mail: [email protected] Abstract Like polyhedra, polychora are beautiful aesthetic structures - with one difference - polychora are four dimensional. Although they are beyond human comprehension to visualize, one can look at various projections or cross sections which are three dimensional and usually very intricate, these make outstanding pieces of art both in model form or in computer graphics. Polygons and polyhedra have been known since ancient times, but little study has gone into the next dimension - until recently. Definitions A polychoron is basically a four dimensional "polyhedron" in the same since that a polyhedron is a three dimensional "polygon". To be more precise - a polychoron is a 4-dimensional "solid" bounded by cells with the following criteria: 1) each cell is adjacent to only one other cell for each face, 2) no subset of cells fits criteria 1, 3) no two adjacent cells are corealmic. If criteria 1 fails, then the figure is degenerate. The word "polychoron" was invented by George Olshevsky with the following construction: poly = many and choron = rooms or cells. A polytope (polyhedron, polychoron, etc.) is uniform if it is vertex transitive and it's facets are uniform (a uniform polygon is a regular polygon). Degenerate figures can also be uniform under the same conditions. A vertex figure is the figure representing the shape and "solid" angle of the vertices, ex: the vertex figure of a cube is a triangle with edge length of the square root of 2. -
The View from Six Dimensions
Walls and Bridges The view from Six Dimensiosn Wendy Y. Krieger [email protected] ∗ January, Abstract Walls divide, bridges unite. This idea is applied to devising a vocabulary suited for the study of higher dimensions. Points are connected, solids divided. In higher dimensions, there are many more products and concepts visible. The four polytope products (prism, tegum, pyramid and comb), lacing and semiate figures, laminates are all discussed. Many of these become distinct in four to six dimensions. Walls and Bridges Consider a knife. Its main action is to divide solids into pieces. This is done by a sweeping action, although the presence of solid materials might make the sweep a little less graceful. What might a knife look like in four dimensions. A knife would sweep a three-dimensional space, and thus the blade is two-dimensional. The purpose of the knife is to divide, and therefore its dimension is fixed by what it divides. Walls divide, bridges unite. When things are thought about in the higher dimensions, the dividing or uniting nature of it is more important than its innate dimensionality. A six-dimensional blade has four dimensions, since its sweep must make five dimensions. There are many idioms that suggest the role of an edge or line is to divide. This most often hap pens when the referent dimension is the two-dimensional ground, but the edge of a knife makes for a three-dimensional referent. A line in the sand, a deadline, and to the edge, all suggest boundaries of two-dimensional areas, where the line or edge divides. -
Crystallography Ll Lattice N-Dimensional, Infinite, Periodic Array of Points, Each of Which Has Identical Surroundings
crystallography ll Lattice n-dimensional, infinite, periodic array of points, each of which has identical surroundings. use this as test for lattice points A2 ("bcc") structure lattice points Lattice n-dimensional, infinite, periodic array of points, each of which has identical surroundings. use this as test for lattice points CsCl structure lattice points Choosing unit cells in a lattice Want very small unit cell - least complicated, fewer atoms Prefer cell with 90° or 120°angles - visualization & geometrical calculations easier Choose cell which reflects symmetry of lattice & crystal structure Choosing unit cells in a lattice Sometimes, a good unit cell has more than one lattice point 2-D example: Primitive cell (one lattice pt./cell) has End-centered cell (two strange angle lattice pts./cell) has 90° angle Choosing unit cells in a lattice Sometimes, a good unit cell has more than one lattice point 3-D example: body-centered cubic (bcc, or I cubic) (two lattice pts./cell) The primitive unit cell is not a cube 14 Bravais lattices Allowed centering types: P I F C primitive body-centered face-centered C end-centered Primitive R - rhombohedral rhombohedral cell (trigonal) centering of trigonal cell 14 Bravais lattices Combine P, I, F, C (A, B), R centering with 7 crystal systems Some combinations don't work, some don't give new lattices - C tetragonal C-centering destroys cubic = P tetragonal symmetry 14 Bravais lattices Only 14 possible (Bravais, 1848) System Allowed centering Triclinic P (primitive) Monoclinic P, I (innerzentiert) Orthorhombic P, I, F (flächenzentiert), A (end centered) Tetragonal P, I Cubic P, I, F Hexagonal P Trigonal P, R (rhombohedral centered) Choosing unit cells in a lattice Unit cell shape must be: 2-D - parallelogram (4 sides) 3-D - parallelepiped (6 faces) Not a unit cell: Choosing unit cells in a lattice Unit cell shape must be: 2-D - parallelogram (4 sides) 3-D - parallelepiped (6 faces) Not a unit cell: correct cell Stereographic projections Show or represent 3-D object in 2-D Procedure: 1. -
Systematics of Atomic Orbital Hybridization of Coordination Polyhedra: Role of F Orbitals
molecules Article Systematics of Atomic Orbital Hybridization of Coordination Polyhedra: Role of f Orbitals R. Bruce King Department of Chemistry, University of Georgia, Athens, GA 30602, USA; [email protected] Academic Editor: Vito Lippolis Received: 4 June 2020; Accepted: 29 June 2020; Published: 8 July 2020 Abstract: The combination of atomic orbitals to form hybrid orbitals of special symmetries can be related to the individual orbital polynomials. Using this approach, 8-orbital cubic hybridization can be shown to be sp3d3f requiring an f orbital, and 12-orbital hexagonal prismatic hybridization can be shown to be sp3d5f2g requiring a g orbital. The twists to convert a cube to a square antiprism and a hexagonal prism to a hexagonal antiprism eliminate the need for the highest nodality orbitals in the resulting hybrids. A trigonal twist of an Oh octahedron into a D3h trigonal prism can involve a gradual change of the pair of d orbitals in the corresponding sp3d2 hybrids. A similar trigonal twist of an Oh cuboctahedron into a D3h anticuboctahedron can likewise involve a gradual change in the three f orbitals in the corresponding sp3d5f3 hybrids. Keywords: coordination polyhedra; hybridization; atomic orbitals; f-block elements 1. Introduction In a series of papers in the 1990s, the author focused on the most favorable coordination polyhedra for sp3dn hybrids, such as those found in transition metal complexes. Such studies included an investigation of distortions from ideal symmetries in relatively symmetrical systems with molecular orbital degeneracies [1] In the ensuing quarter century, interest in actinide chemistry has generated an increasing interest in the involvement of f orbitals in coordination chemistry [2–7]. -
Chains of Antiprisms
Chains of antiprisms Citation for published version (APA): Verhoeff, T., & Stoel, M. (2015). Chains of antiprisms. In Bridges Baltimore 2015 : Mathematics, Music, Art, Architecture, Culture, Baltimore, MD, USA, July 29 - August 1, 2015 (pp. 347-350). The Bridges Organization. Document status and date: Published: 01/01/2015 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.