Tightening Curves on Surfaces Monotonically with Applications
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An FPT Algorithm for the Embeddability of Graphs Into Two-Dimensional Simplicial Complexes∗
An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes∗ Éric Colin de Verdière† Thomas Magnard‡ Abstract We consider the embeddability problem of a graph G into a two-dimensional simplicial com- plex C: Given G and C, decide whether G admits a topological embedding into C. The problem is NP-hard, even in the restricted case where C is homeomorphic to a surface. It is known that the problem admits an algorithm with running time f(c)nO(c), where n is the size of the graph G and c is the size of the two-dimensional complex C. In other words, that algorithm is polynomial when C is fixed, but the degree of the polynomial depends on C. We prove that the problem is fixed-parameter tractable in the size of the two-dimensional complex, by providing a deterministic f(c)n3-time algorithm. We also provide a randomized algorithm with O(1) expected running time 2c nO(1). Our approach is to reduce to the case where G has bounded branchwidth via an irrelevant vertex method, and to apply dynamic programming. We do not rely on any component of the existing linear-time algorithms for embedding graphs on a fixed surface; the only elaborated tool that we use is an algorithm to compute grid minors. 1 Introduction An embedding of a graph G into a host topological space X is a crossing-free topological drawing of G into X. The use and computation of graph embeddings is central in the communities of computational topology, topological graph theory, and graph drawing. -
Downloaded from Bookstore.Ams.Org 30-60-90 Triangle, 190, 233 36-72
Index 30-60-90 triangle, 190, 233 intersects interior of a side, 144 36-72-72 triangle, 226 to the base of an isosceles triangle, 145 360 theorem, 96, 97 to the hypotenuse, 144 45-45-90 triangle, 190, 233 to the longest side, 144 60-60-60 triangle, 189 Amtrak model, 29 and (logical conjunction), 385 AA congruence theorem for asymptotic angle, 83 triangles, 353 acute, 88 AA similarity theorem, 216 included between two sides, 104 AAA congruence theorem in hyperbolic inscribed in a semicircle, 257 geometry, 338 inscribed in an arc, 257 AAA construction theorem, 191 obtuse, 88 AAASA congruence, 197, 354 of a polygon, 156 AAS congruence theorem, 119 of a triangle, 103 AASAS congruence, 179 of an asymptotic triangle, 351 ABCD property of rigid motions, 441 on a side of a line, 149 absolute value, 434 opposite a side, 104 acute angle, 88 proper, 84 acute triangle, 105 right, 88 adapted coordinate function, 72 straight, 84 adjacency lemma, 98 zero, 84 adjacent angles, 90, 91 angle addition theorem, 90 adjacent edges of a polygon, 156 angle bisector, 100, 147 adjacent interior angle, 113 angle bisector concurrence theorem, 268 admissible decomposition, 201 angle bisector proportion theorem, 219 algebraic number, 317 angle bisector theorem, 147 all-or-nothing theorem, 333 converse, 149 alternate interior angles, 150 angle construction theorem, 88 alternate interior angles postulate, 323 angle criterion for convexity, 160 alternate interior angles theorem, 150 angle measure, 54, 85 converse, 185, 323 between two lines, 357 altitude concurrence theorem, -
The Octagonal Pets by Richard Evan Schwartz
The Octagonal PETs by Richard Evan Schwartz 1 Contents 1 Introduction 11 1.1 WhatisaPET?.......................... 11 1.2 SomeExamples .......................... 12 1.3 GoalsoftheMonograph . 13 1.4 TheOctagonalPETs . 14 1.5 TheMainTheorem: Renormalization . 15 1.6 CorollariesofTheMainTheorem . 17 1.6.1 StructureoftheTiling . 17 1.6.2 StructureoftheLimitSet . 18 1.6.3 HyperbolicSymmetry . 21 1.7 PolygonalOuterBilliards. 22 1.8 TheAlternatingGridSystem . 24 1.9 ComputerAssists ......................... 26 1.10Organization ........................... 28 2 Background 29 2.1 LatticesandFundamentalDomains . 29 2.2 Hyperplanes............................ 30 2.3 ThePETCategory ........................ 31 2.4 PeriodicTilesforPETs. 32 2.5 TheLimitSet........................... 34 2.6 SomeHyperbolicGeometry . 35 2.7 ContinuedFractions . 37 2.8 SomeAnalysis........................... 39 I FriendsoftheOctagonalPETs 40 3 Multigraph PETs 41 3.1 TheAbstractConstruction. 41 3.2 TheReflectionLemma . 43 3.3 ConstructingMultigraphPETs . 44 3.4 PlanarExamples ......................... 45 3.5 ThreeDimensionalExamples . 46 3.6 HigherDimensionalGeneralizations . 47 2 4 TheAlternatingGridSystem 48 4.1 BasicDefinitions ......................... 48 4.2 CompactifyingtheGenerators . 50 4.3 ThePETStructure........................ 52 4.4 CharacterizingthePET . 54 4.5 AMoreSymmetricPicture. 55 4.5.1 CanonicalCoordinates . 55 4.5.2 TheDoubleFoliation . 56 4.5.3 TheOctagonalPETs . 57 4.6 UnboundedOrbits ........................ 57 4.7 TheComplexOctagonalPETs. 58 4.7.1 ComplexCoordinates. -
Geometry: Neutral MATH 3120, Spring 2016 Many Theorems of Geometry Are True Regardless of Which Parallel Postulate Is Used
Geometry: Neutral MATH 3120, Spring 2016 Many theorems of geometry are true regardless of which parallel postulate is used. A neutral geom- etry is one in which no parallel postulate exists, and the theorems of a netural geometry are true for Euclidean and (most) non-Euclidean geomteries. Spherical geometry is a special case of Non-Euclidean geometries where the great circles on the sphere are lines. This leads to spherical trigonometry where triangles have angle measure sums greater than 180◦. While this is a non-Euclidean geometry, spherical geometry develops along a separate path where the axioms and theorems of neutral geometry do not typically apply. The axioms and theorems of netural geometry apply to Euclidean and hyperbolic geometries. The theorems below can be proven using the SMSG axioms 1 through 15. In the SMSG axiom list, Axiom 16 is the Euclidean parallel postulate. A neutral geometry assumes only the first 15 axioms of the SMSG set. Notes on notation: The SMSG axioms refer to the length or measure of line segments and the measure of angles. Thus, we will use the notation AB to describe a line segment and AB to denote its length −−! −! or measure. We refer to the angle formed by AB and AC as \BAC (with vertex A) and denote its measure as m\BAC. 1 Lines and Angles Definitions: Congruence • Segments and Angles. Two segments (or angles) are congruent if and only if their measures are equal. • Polygons. Two polygons are congruent if and only if there exists a one-to-one correspondence between their vertices such that all their corresponding sides (line sgements) and all their corre- sponding angles are congruent. -
Arxiv:1812.02806V3 [Math.GT] 27 Jun 2019 Sphere Formed by the Triangles Bounds a Ball and the Vertex in the Identification Space Looks Like the Centre of a Ball
TRAVERSING THREE-MANIFOLD TRIANGULATIONS AND SPINES J. HYAM RUBINSTEIN, HENRY SEGERMAN AND STEPHAN TILLMANN Abstract. A celebrated result concerning triangulations of a given closed 3–manifold is that any two triangulations with the same number of vertices are connected by a sequence of so-called 2-3 and 3-2 moves. A similar result is known for ideal triangulations of topologically finite non-compact 3–manifolds. These results build on classical work that goes back to Alexander, Newman, Moise, and Pachner. The key special case of 1–vertex triangulations of closed 3–manifolds was independently proven by Matveev and Piergallini. The general result for closed 3–manifolds can be found in work of Benedetti and Petronio, and Amendola gives a proof for topologically finite non-compact 3–manifolds. These results (and their proofs) are phrased in the dual language of spines. The purpose of this note is threefold. We wish to popularise Amendola’s result; we give a combined proof for both closed and non-compact manifolds that emphasises the dual viewpoints of triangulations and spines; and we give a proof replacing a key general position argument due to Matveev with a more combinatorial argument inspired by the theory of subdivisions. 1. Introduction Suppose you have a finite number of triangles. If you identify edges in pairs such that no edge remains unglued, then the resulting identification space looks locally like a plane and one obtains a closed surface, a two-dimensional manifold without boundary. The classification theorem for surfaces, which has its roots in work of Camille Jordan and August Möbius in the 1860s, states that each closed surface is topologically equivalent to a sphere with some number of handles or crosscaps. -
The Algebra of Projective Spheres on Plane, Sphere and Hemisphere
Journal of Applied Mathematics and Physics, 2020, 8, 2286-2333 https://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352 The Algebra of Projective Spheres on Plane, Sphere and Hemisphere István Lénárt Eötvös Loránd University, Budapest, Hungary How to cite this paper: Lénárt, I. (2020) Abstract The Algebra of Projective Spheres on Plane, Sphere and Hemisphere. Journal of Applied Numerous authors studied polarities in incidence structures or algebrization Mathematics and Physics, 8, 2286-2333. of projective geometry [1] [2]. The purpose of the present work is to establish https://doi.org/10.4236/jamp.2020.810171 an algebraic system based on elementary concepts of spherical geometry, ex- tended to hyperbolic and plane geometry. The guiding principle is: “The Received: July 17, 2020 Accepted: October 27, 2020 point and the straight line are one and the same”. Points and straight lines are Published: October 30, 2020 not treated as dual elements in two separate sets, but identical elements with- in a single set endowed with a binary operation and appropriate axioms. It Copyright © 2020 by author(s) and consists of three sections. In Section 1 I build an algebraic system based on Scientific Research Publishing Inc. This work is licensed under the Creative spherical constructions with two axioms: ab= ba and (ab)( ac) = a , pro- Commons Attribution International viding finite and infinite models and proving classical theorems that are License (CC BY 4.0). adapted to the new system. In Section Two I arrange hyperbolic points and http://creativecommons.org/licenses/by/4.0/ straight lines into a model of a projective sphere, show the connection be- Open Access tween the spherical Napier pentagram and the hyperbolic Napier pentagon, and describe new synthetic and trigonometric findings between spherical and hyperbolic geometry. -
Saccheri and Lambert Quadrilateral in Hyperbolic Geometry
MA 408, Computer Lab Three Saccheri and Lambert Quadrilaterals in Hyperbolic Geometry Name: Score: Instructions: For this lab you will be using the applet, NonEuclid, developed by Castel- lanos, Austin, Darnell, & Estrada. You can either download it from Vista (Go to Labs, click on NonEuclid), or from the following web address: http://www.cs.unm.edu/∼joel/NonEuclid/NonEuclid.html. (Click on Download NonEuclid.Jar). Work through this lab independently or in pairs. You will have the opportunity to finish anything you don't get done in lab at home. Please, write the solutions of homework problems on a separate sheets of paper and staple them to the lab. 1 Introduction In the previous lab you observed that it is impossible to construct a rectangle in the Poincar´e disk model of hyperbolic geometry. In this lab you will study two types of quadrilaterals that exist in the hyperbolic geometry and have many properties of a rectangle. A Saccheri quadrilateral has two right angles adjacent to one of the sides, called the base. Two sides that are perpendicular to the base are of equal length. A Lambert quadrilateral is a quadrilateral with three right angles. In the Euclidean geometry a Saccheri or a Lambert quadrilateral has to be a rectangle, but the hyperbolic world is different ... 2 Saccheri Quadrilaterals Definition: A Saccheri quadrilateral is a quadrilateral ABCD such that \ABC and \DAB are right angles and AD ∼= BC. The segment AB is called the base of the Saccheri quadrilateral and the segment CD is called the summit. The two right angles are called the base angles of the Saccheri quadrilateral, and the angles \CDA and \BCD are called the summit angles of the Saccheri quadrilateral. -
The Role of Definitions in Geometry for Prospective Middle and High School Teachers
2018 HAWAII UNIVERSITY INTERNATIONAL CONFERENCES STEAM - SCIENCE, TECHNOLOGY & ENGINEERING, ARTS, MATHEMATICS & EDUCATION JUNE 6 - 8, 2018 PRINCE WAIKIKI, HONOLULU, HAWAII THE ROLE OF DEFINITIONS IN GEOMETRY FOR PROSPECTIVE MIDDLE AND HIGH SCHOOL TEACHERS SZABO, TAMAS DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN WHITEWATER, WISCONSIN Dr. Tamas Szabo Department of Mathematics University of Wisconsin Whitewater, Wisconsin THE ROLE OF DEFINITIONS IN GEOMETRY FOR PROSPECTIVE MIDDLE AND HIGH SCHOOL TEACHERS ABSTRACT. Does verbalizing a key definition before solving a related problem help students solving the problem? Perhaps surprisingly, the answers turns out to be NO, based on this yearlong experiment conducted with mathematics education majors and minors in two geometry classes. This article describes the experiment, the results, suggests possible explanations, and derives some peda-gogical conclusions that are useful for teachers of any mathematics or science course. Keywords: concept definition, concept image, proof writing, problem solving, secondary mathematics teachers. 1. INTRODUCTION Definitions play a very important role in mathematics, the learning of mathematics, and the teaching of mathematics alike. However, knowing a definition is not equivalent with knowing a concept. To become effective problem solvers, students need to develop an accurate concept image. The formal definition is only a small part of the concept image, which also includes examples and non-examples, properties and connections to other concepts. The distinction between concept definition and concept image has been examined by many research studies (e.g., Edwards, 1997; Vinner and Dreyfus, 1989; Tall, 1992; Vinner, 1991). Another group of articles analyze, argue for, and provide examples of engaging students in the construction of definitions (e.g., Zandieh and Rasmussen, 2010; Herbst et al., 2005; Johnson et al., 2014; Zaslavski and Shir, 2005). -
Arxiv:Math/0405482V2 [Math.CO] 12 Sep 2016 (A) Lozenge Tiling (B) Lozenge Dual (C) Rhombus Tiling (D) Rhombus Dual
FROM DOMINOES TO HEXAGONS DYLAN P. THURSTON Abstract. There is a natural generalization of domino tilings to tilings of a polygon by hexagons, or, dually, configurations of oriented curves that meet in triples. We show exactly when two such tilings can be connected by a series of moves analogous to the domino flip move. The triple diagrams that result have connections to Legendrian knots, cluster algebras, and planar algebras. 1. Introduction The study of tilings of a planar region by lozenges, as in Figure 1(a), or more generally by rhombuses, as in Figure 1(c), has a long history in combinatorics. Interesting questions include deciding when a region can be tiled, connectivity of the space of tilings under the basic move $ , counting the number of tilings, and behavior of a random tiling. One useful tool for studying these tilings is the dual picture, as in Figures 1(b). It consists of replacing each rhombus in the tiling by a cross of two strands connecting opposite, parallel sides. If we trace a strand through the tiling it comes out on a parallel face on the opposite side of the region, independently of the tiling of the region. Lozenge tilings give patterns of strands with three families of parallel (non-intersecting strands). If we drop the restriction to three families of parallel strands, as in Figure 1(d), and take any set of strands connecting boundary points which are both generic (only two strands intersect at a time) and minimal (no strand intersects itself and no two strands intersect more than once), then there is a corresponding dual tiling by rhombuses, as in Figure 1(c). -
Annales Sciennifiques Supérieuke D L ÉCOLE Hormale
ISSN 0012-9593 ASENAH quatrième série - tome 46 fascicule 5 septembre-octobre 2013 aNNALES SCIENnIFIQUES d L ÉCOLE hORMALE SUPÉRIEUkE Alexander B. GONCHAROV & Richard KENYON Dimers and cluster integrable systems SOCIÉTÉ MATHÉMATIQUE DE FRANCE Ann. Scient. Éc. Norm. Sup. 4 e série, t. 46, 2013, p. 747 à 813 DIMERS AND CLUSTER INTEGRABLE SYSTEMS ʙʏ Aʟɴʀ B. GONCHAROV ɴ Rɪʜʀ KENYON Aʙʀ. – We show that the dimer model on a bipartite graph Γ on a torus gives rise to a quantum integrable system of special type, which we call a cluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space LΓ of line bundles with connections on the graph Γ. The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs Γ1 and Γ2 are equivalent if the Newton polygons of the correspond- ing partition functions coincide up to translation. We define elementary transformations of bipartite surface graphs, and show that two equivalent minimal bipartite graphs are related by a sequence of elementary transformations. For each elementary transformation we define a birational Poisson iso- morphism LΓ LΓ providing an equivalence of the integrable systems. We show that it is a cluster 1 → 2 Poisson transformation, as defined in [10]. We show that for any convex integral polygon N there is a non-empty finite set of minimal graphs Γ for which N is the Newton polygon of the partition function related to Γ. Gluing the varieties LΓ for graphs Γ related by elementary transformations via the corresponding cluster Poisson transformations, we get a Poisson space X N . -
Volume 6 (2006) 1–16
FORUM GEOMETRICORUM A Journal on Classical Euclidean Geometry and Related Areas published by Department of Mathematical Sciences Florida Atlantic University b bbb FORUM GEOM Volume 6 2006 http://forumgeom.fau.edu ISSN 1534-1178 Editorial Board Advisors: John H. Conway Princeton, New Jersey, USA Julio Gonzalez Cabillon Montevideo, Uruguay Richard Guy Calgary, Alberta, Canada Clark Kimberling Evansville, Indiana, USA Kee Yuen Lam Vancouver, British Columbia, Canada Tsit Yuen Lam Berkeley, California, USA Fred Richman Boca Raton, Florida, USA Editor-in-chief: Paul Yiu Boca Raton, Florida, USA Editors: Clayton Dodge Orono, Maine, USA Roland Eddy St. John’s, Newfoundland, Canada Jean-Pierre Ehrmann Paris, France Chris Fisher Regina, Saskatchewan, Canada Rudolf Fritsch Munich, Germany Bernard Gibert St Etiene, France Antreas P. Hatzipolakis Athens, Greece Michael Lambrou Crete, Greece Floor van Lamoen Goes, Netherlands Fred Pui Fai Leung Singapore, Singapore Daniel B. Shapiro Columbus, Ohio, USA Steve Sigur Atlanta, Georgia, USA Man Keung Siu Hong Kong, China Peter Woo La Mirada, California, USA Technical Editors: Yuandan Lin Boca Raton, Florida, USA Aaron Meyerowitz Boca Raton, Florida, USA Xiao-Dong Zhang Boca Raton, Florida, USA Consultants: Frederick Hoffman Boca Raton, Floirda, USA Stephen Locke Boca Raton, Florida, USA Heinrich Niederhausen Boca Raton, Florida, USA Table of Contents Khoa Lu Nguyen and Juan Carlos Salazar, On the mixtilinear incircles and excircles,1 Juan Rodr´ıguez, Paula Manuel and Paulo Semi˜ao, A conic associated with the Euler line,17 Charles Thas, A note on the Droz-Farny theorem,25 Paris Pamfilos, The cyclic complex of a cyclic quadrilateral,29 Bernard Gibert, Isocubics with concurrent normals,47 Mowaffaq Hajja and Margarita Spirova, A characterization of the centroid using June Lester’s shape function,53 Christopher J. -
Saccheri Quadrilaterals Definition: Let Be Any Line Segment, and Erect Two
Saccheri Quadrilaterals Definition: Let be any line segment, and erect two perpendiculars at the endpoints A and B. Mark off points C and D on these perpendiculars so that C and D lie on the same side of the line , and BC = AD. Join C and D. The resulting quadrilateral is a Saccheri Quadrilateral. Side is called the base, and the legs, and side the summit. The angles at C and D are called the summit angles. Lemma: A Saccheri Quadrilateral is convex. ~ By construction, D and C are on the same side of the line , and by PSP, will be as well. If intersected at a point F, one of the triangles ªADF or ªBFC would have two angles of at least measure 90, a contradiction (one of the linear pair of angles at F must be obtuse or right). Finally, if and met at a point E then ªABE would be a triangle with two right angles. Thus and must lie entirely on one side of each other. So, GABCD is convex. Theorem: The summit angles of a Saccheri Quadrilateral are congruent. ~ The SASAS version of using SAS to prove the base angles of an isosceles triangle are congruent. GDABC GCBAD, by SASAS, so pD pC. Corollaries: • The diagonals of a Saccheri Quadrilateral are congruent. (Proof: ªABC ªBAD by SAS; CPCF gives AC = BD.) • The line joining the midpoints of the base and summit of a quadrilateral is the perpendicular bisector of both the base and summit. (Proof: Let N and M be the midpoints of summit and base, respectively.