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Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity Eppur Si Muove: Biomimetic Embedding of N-Tuple Helices in Spherical Polyhedra - /
Alternative view of segmented documents via Kairos 23 October 2017 | Draft Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity Eppur si muove: Biomimetic embedding of N-tuple helices in spherical polyhedra - / - Introduction Symbolic stars vs Strategic pillars; Polyhedra vs Helices; Logic vs Comprehension? Dynamic bonding patterns in n-tuple helices engendering n-fold rotating symbols Embedding the triple helix in a spherical octahedron Embedding the quadruple helix in a spherical cube Embedding the quintuple helix in a spherical dodecahedron and a Pentagramma Mirificum Embedding six-fold, eight-fold and ten-fold helices in appropriately encircled polyhedra Embedding twelve-fold, eleven-fold, nine-fold and seven-fold helices in appropriately encircled polyhedra Neglected recognition of logical patterns -- especially of opposition Dynamic relationship between polyhedra engendered by circles -- variously implying forms of unity Symbol rotation as dynamic essential to engaging with value-inversion References Introduction The contrast to the geocentric model of the solar system was framed by the Italian mathematician, physicist and philosopher Galileo Galilei (1564-1642). His much-cited phrase, " And yet it moves" (E pur si muove or Eppur si muove) was allegedly pronounced in 1633 when he was forced to recant his claims that the Earth moves around the immovable Sun rather than the converse -- known as the Galileo affair. Such a shift in perspective might usefully inspire the recognition that the stasis attributed so widely to logos and other much-valued cultural and heraldic symbols obscures the manner in which they imply a fundamental cognitive dynamic. Cultural symbols fundamental to the identity of a group might then be understood as variously moving and transforming in ways which currently elude comprehension. -
The Geometry of the Simplex Method and Applications to the Assignment Problems
The Geometry of the Simplex Method and Applications to the Assignment Problems By Rex Cheung SENIOR THESIS BACHELOR OF SCIENCE in MATHEMATICS in the COLLEGE OF LETTERS AND SCIENCE of the UNIVERSITY OF CALIFORNIA, DAVIS Approved: Jes´usA. De Loera March 13, 2011 i ii ABSTRACT. In this senior thesis, we study different applications of linear programming. We first look at different concepts in linear programming. Then we give a study on some properties of polytopes. Afterwards we investigate in the Hirsch Conjecture and explore the properties of Santos' counterexample as well as a perturbation of this prismatoid. We also give a short survey on some attempts in finding more counterexamples using different mathematical tools. Lastly we present an application of linear programming: the assignment problem. We attempt to solve this assignment problem using the Hungarian Algorithm and the Gale-Shapley Algorithm. Along with the algorithms we also give some properties of the Birkhoff Polytope. The results in this thesis are obtained collaboratively with Jonathan Gonzalez, Karla Lanzas, Irene Ramirez, Jing Lin, and Nancy Tafolla. Contents Chapter 1. Introduction to Linear Programming 1 1.1. Definitions and Motivations 1 1.2. Methods for Solving Linear Programs 2 1.3. Diameter and Pivot Bound 8 1.4. Polytopes 9 Chapter 2. Hirsch Conjecture and Santos' Prismatoid 11 2.1. Introduction 11 2.2. Prismatoid and Santos' Counterexample 13 2.3. Properties of Santos' Prismatoid 15 2.4. A Perturbed Version of Santos' Prismatoid 18 2.5. A Search for Counterexamples 22 Chapter 3. An Application of Linear Programming: The Assignment Problem 25 3.1. -
Plane and Solid Geometry
DUKE UNIVERSITY LIBRARY vv i\\S FOUR DISCOVERERS AND FOUNDERS OF GEOMETRY (See pages 4-90-498) PLANE AND SOLID GEOMETRY BY FLETCHER DURELL, Ph.D. HEAD OP THE MATHEMATICAL DEPARTMENT, THE LAWRENCEVILLE SCHOOL NEW YORK CHARLES E. MERRILL CO, wm r DURELL’S MATHEMATICAL SERIES ARITHMETIC Two Book Series Elementary Arithmetic Teachers’ Edition Advanced Arithmetic Three Book Series Book One Book Two Book Three ALGEBRA Two Book Course Book One Book Two Book Two with Advanced Work Introductory Algebra School Algebra GEOMETRY Plane Geometry Solid Geometry Plane and Solid Geometry, TRIGONOMETRY Plane Trigonometry and Tables Plane and Spherical Trigonometry and Tables. • Plane and Spherical Trigonometry with Surveying and Tables Logarithmic and Trigonometric Tables Copyright. 1904, by Charles E. Merrill Cft It o o UV 5 / 3 .^ PREFACE One of the main purposes in writing this book has been to try to present the subject of Geometry so that the pupil shall understand it not merely as a series of correct deductions, but shall realize the value and meaning of its principles as well. This aspect of the subject has been directly presented in some places, and it is hoped that it per- vades and shapes the presentation in all places. Again, teachers of Geometry generally agree that the most difficult part of their work lies in developing in pupils the power to work original exercises. The second main purpose of the book is to aid in the solution of this difficulty by arranging original exercises in groups, each of the earlier groups to be worked by a distinct method. -
Covering Properties of Lattice Polytopes
Covering properties of lattice polytopes Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) eingereicht am Fachbereich Mathematik und Informatik der Freien Universit¨atBerlin von Giulia Codenotti Berlin 24. Oktober 2019 Supervisor and first reviewer: Prof. Dr. Christian Haase Institut f¨urMathematik Freie Universit¨atBerlin Co-supervisor and second reviewer: Prof. Francisco Santos Departamento de Matem´aticas,Estad´ısticay Computaci´on Universidad de Cantabria Date of defense: 17.01.2020 Institut f¨urMathematik Freie Universit¨atBerlin Summary This thesis studies problems concerning the interaction between polytopes and lattices. Motivation for the study of lattice polytopes comes from two very different fields: dis- crete optimization, in particular integer linear programming, and algebraic geometry, specifically the study of toric varieties. The first topic we study is the existence of unimodular covers for certain interesting families of 3-dimensional lattice polytopes. A unimodular cover of a lattice polytope is a collection of unimodular simplices whose union equals the polytope. Admitting a unimodular cover is a weaker property than admitting a unimodular triangulation, and stronger than having the integer decomposition property (IDP). This last property is particularly interesting in the algebraic context, and there are various conjectures relating it to smoothness ([Oda97]). We show that unimodular covers exist for all 3- dimensional parallelepipeds and for all Cayley sums of polygons where one polygon is a weak Minkowski summand of the other. For both classes of polytopes only the IDP property was previously known. We then explore questions related to the so-called flatness constant, the largest width that a hollow convex body can have in a given dimension. -
Polyhedral Models of the Projective Plane
Bridges 2018 Conference Proceedings Polyhedral Models of the Projective Plane Paul Gailiunas 25 Hedley Terrace, Gosforth, Newcastle, NE3 1DP, England; [email protected] Abstract The tetrahemihexahedron is the only uniform polyhedron that is topologically equivalent to the projective plane. Many other polyhedra having the same topology can be constructed by relaxing the conditions, for example those with faces that are regular polygons but not transitive on the vertices, those with planar faces that are not regular, those with faces that are congruent but non-planar, and so on. Various techniques for generating physical realisations of such polyhedra are discussed, and several examples of different types described. Artists such as Max Bill have explored non-orientable surfaces, in particular the Möbius strip, and Carlo Séquin has considered what is possible with some models of the projective plane. The models described here extend the range of possibilities. The Tetrahemihexahedron A two-dimensional projective geometry is characterised by the pair of axioms: any two distinct points determine a unique line; any two distinct lines determine a unique point. There are projective geometries with a finite number of points/lines but more usually the projective plane is considered as an ordinary Euclidean plane plus a line “at infinity”. By the second axiom any line intersects the line “at infinity” in a single point, so a model of the projective plane can be constructed by taking a topological disc (it is convenient to think of it as a hemisphere) and identifying opposite points. The first known analytical surface matching this construction was discovered by Jakob Steiner in 1844 when he was in Rome, and it is known as the Steiner Roman surface. -
Transformations (Part 2) Composition of Isometries Solids
Study Guide 3/ Page 1 Transformations (part 2) Congruence Transformations (Isometries) 1. You should be able to list all isometries and all that is required to unambiguously determine each of them. 2. Discuss point symmetry and identity. Why do we often leave them out from the list of isometries? 3. Discuss glide reflection. Why do we want the vector to be parallel to the line of reflection? Tools of transformations 4. You should be able to solve elementary problems (see examples at https://www.geogebra.org/m/UL59oGXJ ) and provide useful instructional tips for working with typical tools such as transparency, square dot paper or technology. 5. Given a pre-image and its image, find the corresponding isometry. https://www.geogebra.org/m/jpDtBfw2 . Composition of Isometries 6. Explain why the composition of three reflections cannot yield a translation or rotation. Try to explain it without using technology, just by referring to some properties of isometries. 7. Explain why reflection has a special position among the isometries and illustrate it on a few examples. 8. Prove that the angle of rotation generated by two reflections with intersecting lines is 2x the angle between the lines. 9. Show that the vector of translation generated by two reflections with parallel lines is twice as long as the distance between the two lines. In other words, describe the translation vector that maps E onto E’. Solids Classification of solids You should be familiar with the basic terminology and names: Vertex, Edge, Face; Solid, Polyhedron, Prism (right and oblique), Antiprism, Pyramid (right and oblique), Platonic Solid, Regular Polyhedron, Archimedean Solid, Semiregular Polyhedron, Sphere, Cone, and Cylinder. -
Star and Convex Regular Polyhedra by Origami. Build Polyhedra by Origami.]
Star and convex regular polyhedra by Origami. Build polyhedra by Origami.] Marcel Morales Alice Morales 2009 E D I T I O N M ORALES Polyhedron by Origami I) Table of convex regular Polyhedra ................................................................................................. 4 II) Table of star regular Polyhedra (non convex) .............................................................................. 5 III) Some Polyhedra that should be named regular done with equilateral triangles .......................... 6 IV) Some Polyhedra that should be named regular done with rectangle triangles ............................ 7 V) Some geometry of polygons ........................................................................................................ 9 VI) The convex regular polygons ..................................................................................................... 12 VII) The star regular polygons .......................................................................................................... 12 VIII) Regular convex polygons by folding paper ............................................................................. 14 1) The square .......................................................................................................................................... 14 2) The equilateral triangle ....................................................................................................................... 14 3) The regular pentagon ......................................................................................................................... -
The Width of 5-Dimensional Prismatoids, As We Do in This Paper
The width of 5-dimensional prismatoids Benjamin Matschke∗ Francisco Santos∗∗ Christophe Weibel∗∗∗ June 23, 2018 Abstract Santos’ construction of counter-examples to the Hirsch Conjecture (2012) is based on the existence of prismatoids of dimension d of width greater than d. Santos, Stephen and Thomas (2012) have shown that this cannot occur in d 4. Motivated by this we here study the width of ≤ 5-dimensional prismatoids, obtaining the following results: There are 5-prismatoids of width six with only 25 vertices, versus the • 48 vertices in Santos’ original construction. This leads to non-Hirsch polytopes of dimension 20, rather than the original dimension 43. There are 5-prismatoids with n vertices and width Ω(√n) for arbi- • trarily large n. Hence, the width of 5-prismatoids is unbounded. Mathematics Subject Classification: 52B05, 52B55, 90C05 1 Introduction The Hirsch Conjecture. In 1957, Warren Hirsch conjectured that the (com- binatorial) diameter of any d-dimensional convex polytope or polyhedron with n facets is at most n d. Here the diameter of a polytope is the diameter of its graph, that is, the maximal− number of edges that one needs to pass in order to go from one vertex to another. arXiv:1202.4701v2 [math.CO] 6 May 2013 Hirsch’s motivation was that the simplex method, devised by Dantzig ten years earlier, finds the optimal solution of a linear program by walking along the graph of the feasibility polyhedron. In particular, the maximum diame- ter among polyhedra with a given number of facets is a lower bound for the computational complexity of the simplex algorithm for any pivoting rule. -
Continuous Flattening of All Polyhedral Manifolds Using Countably Infinite Creases
Continuous Flattening of All Polyhedral Manifolds using Countably Infinite Creases Zachary Abel∗ Erik D. Demaine† Martin L. Demaine† Jason S. Ku∗ Jayson Lynch† Jin-ichi Itoh‡ Chie Nara§ Abstract We prove that any finite polyhedral manifold in 3D can be continuously flattened into 2D while preserving intrinsic distances and avoiding crossings, answering a 19-year-old open problem, if we extend standard folding models to allow for countably infinite creases. The most general cases previously known to be continuously flattenable were convex polyhedra and semi-orthogonal polyhedra. For non-orientable manifolds, even the existence of an instantaneous flattening (flat folded state) is a new result. Our solution extends a method for flattening semi-orthogonal polyhedra: slice the polyhedron along parallel planes and flatten the polyhedral strips between consecutive planes. We adapt this approach to arbitrary nonconvex polyhedra by generalizing strip flattening to nonorthogonal corners and slicing along a countably infinite number of parallel planes, with slices densely approaching every vertex of the manifold. We also show that the area of the polyhedron that needs to support moving creases (which are necessary for closed polyhedra by the Bellows Theorem) can be made arbitrarily small. 1 Introduction We crush polyhedra flat all the time, such as when we recycle cereal boxes or store airbags in a steering wheel. But is this actually possible without tearing or stretching the material? This problem was first posed in 2001 [DDL01] (see [DO07, Chapter 18]): does every polyhedron have a continuous motion that preserves the metric (intrinsic shortest paths), avoids crossings, and ends in a flat folded state? This problem is Open Problem 18.1 of the book Geometric Folding Algorithms [DO07]. -
On the Shape Simulation of Aggregate and Cement Particles in a DEM System
Hindawi Publishing Corporation Advances in Materials Science and Engineering Volume 2015, Article ID 692768, 7 pages http://dx.doi.org/10.1155/2015/692768 Research Article On the Shape Simulation of Aggregate and Cement Particles in a DEM System Huan He,1,2 Piet Stroeven,2 Eric Pirard,3 and Luc Courard3 1 College of Architecture and Civil Engineering, Beijing University of Technology, Beijing 100122, China 2Faculty of Civil Engineering and Geosciences, Delft University of Technology, Stevinweg 1, 2628 CN Delft, Netherlands 3GeMMe (Minerals Engineering, Materials and Environment), University of Liege,` Chemin des Chevreuils 1, 4000 Liege,` Belgium Correspondence should be addressed to Huan He; [email protected] Received 30 April 2015; Revised 6 July 2015; Accepted 7 July 2015 Academic Editor: Ana S. Guimaraes˜ Copyright © 2015 Huan He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Aggregate occupies at least three-quarters of the volume of concrete, so its impact on concrete’s properties is significant. Both size and shape of aggregate influence workability, mechanical properties, and durability of concrete. On the other hand, the shape of cement particles plays also an important role in the hydration process due to surface dissolution in the hardening process. Additionally, grain dispersion, shape, and size govern the pore percolation process that is of crucial importance for concrete durability. Discrete element modeling (DEM) is commonly employed for simulation of concrete structure. To be able to do so, the assessed grain shape should be implemented. -
Face-Based Volume-Of-Fluid Interface Positioning in Arbitrary Polyhedra
Face-based Volume-of-Fluid interface positioning in arbitrary polyhedra Johannes Kromer and Dieter Bothey Mathematical Modeling and Analysis, Technische Universit¨atDarmstadt Alarich-Weiss-Strasse 10, 64287 Darmstadt, Germany yEmail for correspondence: [email protected] Abstract We introduce a fast and robust algorithm for finding a plane Γ with given normal nΓ, which truncates an arbitrary polyhedron such that the remaining sub-polyhedron admits a given volume α∗ . In the literature, this P jPj is commonly referred to as Volume-of-Fluid (VoF) interface positioning problem. The novelty of our work is twofold: firstly, by recursive application of the Gaussian divergence theorem, the volume of a truncated polyhedron can be computed at high efficiency, based on summation over quantities associated to the faces of the polyhedron. One obtains a very convenient piecewise parametrization (within so-called brackets) in terms of the signed distance s to the plane Γ. As an implication, one can restrain from the costly necessity to establish topological connectivity, rendering the present approach most suitable for the application to unstructured computational meshes. Secondly, in the vicinity of the truncation position s, the volume can be expressed exactly, i.e. in terms of a cubic polynomial of the normal distance to the PLIC plane. The local knowledge of derivatives enables to construct a root-finding algorithm that pairs bracketing and higher-order approximation. The performance is assessed by conducting an extensive set of numerical experiments, considering convex and non-convex polyhedra of genus (i.e., number of holes) zero and one in combination with carefully selected volume fractions α∗ (including α∗ 0 and α∗ 1) and ≈ ≈ normal orientations nΓ. -
Refort Resumes
REFORT RESUMES ED 013 517 EC 000 574 NINTH GRADE PLANE AND SOLID GEOMETRY FOR THEACADEMICALLY TALENTED, TEACHERS GUIDE. OHIO STATE DEFT. Cf EDUCATION, COLUMBUS CLEVELAND FUBLIC SCHOOLS, OHIO FUD DATE 63 EDRS PRICE MF--$1.00 HC-$10.46 262F. DESCRIPTORS- *GIFTED, =PLANE GEOMETRY, *SOLID GEOMETRY, *CURRICULUM GUIDES, UNITS Cf STUDY (SUBJECTFIELDS), SPECIAL EDUCATION, GRADE 9, COLUMBUS A UNIFIED TWO--SEMESTER COURSE INPLANE AND SOLID GEOMETRY FOR THE GIFTED IS PRESENTED IN 15 UNITS,EACH SPECIFYING THE NUMBER OF INSTRUCTIONALSESSIONS REQUIRED. UNITS' APE SUBDIVIDED BY THE TOPIC ANDITS CONCEPTS, VOCABULARY, SYMBOLISM, REFERENCES (TO SEVENTEXTBCOKS LISTED IN THE GUIDE), AND SUGGESTIONS. THE APPENDIXCCNTAINS A FALLACIOUS PROOF, A TABLE COMPARING EUCLIDEANAND NON-EUCLIDEAN GEOMETRY, PROJECTS FORINDIVIDUAL ENRICHMENT, A GLOSSARY, AND A 64 -ITEM BIBLIOGRAPHY.RESULTS OF THE STANDARDIZED TESTS SHOWED THAT THE ACCELERATESSCORED AS WELL OR BETTER IN ALMOST ALL CASES THANTHE REGULAR CLASS PUPILS, EVEN THOUGH THE ACCELERATES WEREYOUNGER. SUBJECTIVE EVALUATION Cf ADMINISTRATION; COUNSELORS,TEACHERS, AND PUPILS SHOWED THE PROGRAM WAS HIGHLYSUCCESSFUL. (RM) AY, TEACHERS' GUIDE Ninth Grade Plane and Solid Geometry foi the Academically Talented Issued by E. E. DOLT Superintendent of Public Instruction Columbus, Ohio 1963 U.S. DEPARTMENT OF HEALTH, EDUCATION & WELFARE OFFICE OF EDUCATION THIS DOCUMENT HAS BEEN REPRODUCED EXACTLY AS RECEIVED FROM THE PERSON OR ORGANIZATION ORIGINATING IT.POINTS OF VIEW OR OPINIONS STATED DO NOT NECESSARILY REPRESENT OFFICIAL OFFICE OF EDUCATION POSITION OR POLICY. TEACHERS' GUIDE Ninth Grade Plane andSolid Geometry for the Academically Talented Prepared by CLEVELAND PUBLIC SCHOOLS Division of Mathematics In Cooperation With THE OHIO DEPARTMENT OFEDUCATION Under th,.. Direction of R.