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ON a PROOF of the TREE CONJECTURE for TRIANGLE TILING BILLIARDS Olga Paris-Romaskevich
ON A PROOF OF THE TREE CONJECTURE FOR TRIANGLE TILING BILLIARDS Olga Paris-Romaskevich To cite this version: Olga Paris-Romaskevich. ON A PROOF OF THE TREE CONJECTURE FOR TRIANGLE TILING BILLIARDS. 2019. hal-02169195v1 HAL Id: hal-02169195 https://hal.archives-ouvertes.fr/hal-02169195v1 Preprint submitted on 30 Jun 2019 (v1), last revised 8 Oct 2019 (v3) 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. ON A PROOF OF THE TREE CONJECTURE FOR TRIANGLE TILING BILLIARDS. OLGA PARIS-ROMASKEVICH To Manya and Katya, to the moments we shared around mathematics on one winter day in Moscow. Abstract. Tiling billiards model a movement of light in heterogeneous medium consisting of homogeneous cells in which the coefficient of refraction between two cells is equal to −1. The dynamics of such billiards depends strongly on the form of an underlying tiling. In this work we consider periodic tilings by triangles (and cyclic quadrilaterals), and define natural foliations associated to tiling billiards in these tilings. By studying these foliations we manage to prove the Tree Conjecture for triangle tiling billiards that was stated in the work by Baird-Smith, Davis, Fromm and Iyer, as well as its generalization that we call Density property. -
Tiles of the Plane: 6.891 Final Project
Tiles of the Plane: 6.891 Final Project Luis Blackaller Kyle Buza Justin Mazzola Paluska May 14, 2007 1 Introduction Our 6.891 project explores ways of generating shapes that tile the plane. A plane tiling is a covering of the plane without holes or overlaps. Figure 1 shows a common tiling from a building in a small town in Mexico. Traditionally, mathematicians approach the tiling problem by starting with a known group of symme- tries that will generate a set of tiles that maintain with them. It turns out that all periodic 2D tilings must have a fundamental region determined by 2 translations as described by one of the 17 Crystallographic Symmetry Groups [1]. As shown in Figure 2, to generate a set of tiles, one merely needs to choose which of the 17 groups is needed and create the desired tiles using the symmetries in that group. On the other hand, there are aperiodic tilings, such as Penrose Tiles [2], that use different sets of generation rules, and a completely different approach that relies on heavy math like optimization theory and probability theory. We are interested in the inverse problem: given a set of shapes, we would like to see what kinds of tilings can we create. This problem is akin to what a layperson might encounter when tiling his bathroom — the set of tiles is fixed, but the ways in which the tiles can be combined is unconstrained. In general, this is an impossible problem: Robert Berger proved that the problem of deciding whether or not an arbitrary finite set of simply connected tiles admits a tiling of the plane is undecidable [3]. -
Simple Rules for Incorporating Design Art Into Penrose and Fractal Tiles
Bridges 2012: Mathematics, Music, Art, Architecture, Culture Simple Rules for Incorporating Design Art into Penrose and Fractal Tiles San Le SLFFEA.com [email protected] Abstract Incorporating designs into the tiles that form tessellations presents an interesting challenge for artists. Creating a viable M.C. Escher-like image that works esthetically as well as functionally requires resolving incongruencies at a tile’s edge while constrained by its shape. Escher was the most well known practitioner in this style of mathematical visualization, but there are significant mathematical objects to which he never applied his artistry including Penrose Tilings and fractals. In this paper, we show that the rules of creating a traditional tile extend to these objects as well. To illustrate the versatility of tiling art, images were created with multiple figures and negative space leading to patterns distinct from the work of others. 1 1 Introduction M.C. Escher was the most prominent artist working with tessellations and space filling. Forty years after his death, his creations are still foremost in people’s minds in the field of tiling art. One of the reasons Escher continues to hold such a monopoly in this specialty are the unique challenges that come with creating Escher type designs inside a tessellation[15]. When an image is drawn into a tile and extends to the tile’s edge, it introduces incongruencies which are resolved by continuously aligning and refining the image. This is particularly true when the image consists of the lizards, fish, angels, etc. which populated Escher’s tilings because they do not have the 4-fold rotational symmetry that would make it possible to arbitrarily rotate the image ± 90, 180 degrees and have all the pieces fit[9]. -
Fundamental Principles Governing the Patterning of Polyhedra
FUNDAMENTAL PRINCIPLES GOVERNING THE PATTERNING OF POLYHEDRA B.G. Thomas and M.A. Hann School of Design, University of Leeds, Leeds LS2 9JT, UK. [email protected] ABSTRACT: This paper is concerned with the regular patterning (or tiling) of the five regular polyhedra (known as the Platonic solids). The symmetries of the seventeen classes of regularly repeating patterns are considered, and those pattern classes that are capable of tiling each solid are identified. Based largely on considering the symmetry characteristics of both the pattern and the solid, a first step is made towards generating a series of rules governing the regular tiling of three-dimensional objects. Key words: symmetry, tilings, polyhedra 1. INTRODUCTION A polyhedron has been defined by Coxeter as “a finite, connected set of plane polygons, such that every side of each polygon belongs also to just one other polygon, with the provision that the polygons surrounding each vertex form a single circuit” (Coxeter, 1948, p.4). The polygons that join to form polyhedra are called faces, 1 these faces meet at edges, and edges come together at vertices. The polyhedron forms a single closed surface, dissecting space into two regions, the interior, which is finite, and the exterior that is infinite (Coxeter, 1948, p.5). The regularity of polyhedra involves regular faces, equally surrounded vertices and equal solid angles (Coxeter, 1948, p.16). Under these conditions, there are nine regular polyhedra, five being the convex Platonic solids and four being the concave Kepler-Poinsot solids. The term regular polyhedron is often used to refer only to the Platonic solids (Cromwell, 1997, p.53). -
Tiling the Plane with Equilateral Convex Pentagons Maria Fischer1
Parabola Volume 52, Issue 3 (2016) Tiling the plane with equilateral convex pentagons Maria Fischer1 Mathematicians and non-mathematicians have been concerned with finding pentago- nal tilings for almost 100 years, yet tiling the plane with convex pentagons remains the only unsolved problem when it comes to monohedral polygonal tiling of the plane. One of the oldest and most well known pentagonal tilings is the Cairo tiling shown below. It can be found in the streets of Cairo, hence the name, and in many Islamic decorations. Figure 1: Cairo tiling There have been 15 types of such pentagons found so far, but it is not clear whether this is the complete list. We will look at the properties of these 15 types and then find a complete list of equilateral convex pentagons which tile the plane – a problem which has been solved by Hirschhorn in 1983. 1Maria Fischer completed her Master of Mathematics at UNSW Australia in 2016. 1 Archimedean/Semi-regular tessellation An Archimedean or semi-regular tessellation is a regular tessellation of the plane by two or more convex regular polygons such that the same polygons in the same order surround each polygon. All of these polygons have the same side length. There are eight such tessellations in the plane: # 1 # 2 # 3 # 4 # 5 # 6 # 7 # 8 Number 5 and number 7 involve triangles and squares, number 1 and number 8 in- volve triangles and hexagons, number 2 involves squares and octagons, number 3 tri- angles and dodecagons, number 4 involves triangles, squares and hexagons and num- ber 6 squares, hexagons and dodecagons. -
Five-Vertex Archimedean Surface Tessellation by Lanthanide-Directed Molecular Self-Assembly
Five-vertex Archimedean surface tessellation by lanthanide-directed molecular self-assembly David Écijaa,1, José I. Urgela, Anthoula C. Papageorgioua, Sushobhan Joshia, Willi Auwärtera, Ari P. Seitsonenb, Svetlana Klyatskayac, Mario Rubenc,d, Sybille Fischera, Saranyan Vijayaraghavana, Joachim Reicherta, and Johannes V. Bartha,1 aPhysik Department E20, Technische Universität München, D-85478 Garching, Germany; bPhysikalisch-Chemisches Institut, Universität Zürich, CH-8057 Zürich, Switzerland; cInstitute of Nanotechnology, Karlsruhe Institute of Technology, D-76344 Eggenstein-Leopoldshafen, Germany; and dInstitut de Physique et Chimie des Matériaux de Strasbourg (IPCMS), CNRS-Université de Strasbourg, F-67034 Strasbourg, France Edited by Kenneth N. Raymond, University of California, Berkeley, CA, and approved March 8, 2013 (received for review December 28, 2012) The tessellation of the Euclidean plane by regular polygons has by five interfering laser beams, conceived to specifically address been contemplated since ancient times and presents intriguing the surface tiling problem, yielded a distorted, 2D Archimedean- aspects embracing mathematics, art, and crystallography. Sig- like architecture (24). nificant efforts were devoted to engineer specific 2D interfacial In the last decade, the tools of supramolecular chemistry on tessellations at the molecular level, but periodic patterns with surfaces have provided new ways to engineer a diversity of sur- distinct five-vertex motifs remained elusive. Here, we report a face-confined molecular architectures, mainly exploiting molecular direct scanning tunneling microscopy investigation on the cerium- recognition of functional organic species or the metal-directed directed assembly of linear polyphenyl molecular linkers with assembly of molecular linkers (5). Self-assembly protocols have terminal carbonitrile groups on a smooth Ag(111) noble-metal sur- been developed to achieve regular surface tessellations, includ- face. -
Grade 6 Math Circles Tessellations Tiling the Plane
Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 6 Math Circles October 13/14, 2015 Tessellations Tiling the Plane Do the following activity on a piece of graph paper. Build a pattern that you can repeat all over the page. Your pattern should use one, two, or three different `tiles' but no more than that. It will need to cover the page with no holes or overlapping shapes. Think of this exercises as if you were using tiles to create a pattern for your kitchen counter or a floor. Your pattern does not have to fill the page with straight edges; it can be a pattern with bumpy edges that does not fit the page perfectly. The only rule here is that we have no holes or overlapping between our tiles. Here are two examples, one a square tiling and another that we will call the up-down arrow tiling: Another word for tiling is tessellation. After you have created a tessellation, study it: did you use a weird shape or shapes? Or is your tiling simple and only uses straight lines and 1 polygons? Recall that a polygon is a many sided shape, with each side being a straight line. For example, triangles, trapezoids, and tetragons (quadrilaterals) are all polygons but circles or any shapes with curved sides are not polygons. As polygons grow more sides or become more irregular, you may find it difficult to use them as tiles. When given the previous activity, many will come up with the following tessellations. -
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. -
Robotic Fabrication of Modular Formwork for Non-Standard Concrete Structures
Robotic Fabrication of Modular Formwork for Non-Standard Concrete Structures Milena Stavric1, Martin Kaftan2 1,2TU Graz, Austria, 2CTU, Czech Republic 1www. iam2.tugraz.at, 2www. iam2.tugraz.at [email protected], [email protected] Abstract. In this work we address the fast and economical realization of complex formwork for concrete with the advantage of industrial robot arm. Under economical realization we mean reduction of production time and material efficiency. The complex form of individual formwork parts can be in our case double curved surface of complex mesh geometry. We propose the fabrication of the formwork by straight or shaped hot wire. We illustrate on several projects different approaches to mould production, where the proposed process demonstrates itself effective. In our approach we deal with the special kinds of modularity and specific symmetry of the formwork Keywords. Robotic fabrication; formwork; non-standard structures. INTRODUCTION Figure 1 Designing free-form has been more possible for Robot ABB I140. architects with the help of rapidly advancing CAD programs. However, the skin, the facade of these building in most cases needs to be produced by us- ing costly formwork techniques whether it’s made from glass, concrete or other material. The problem of these techniques is that for every part of the fa- cade must be created an unique mould. The most common type of formwork is the polystyrene (EPS or XPS) mould which is 3D formed using a CNC mill- ing machine. This technique is very accurate, but it requires time and produces a lot of waste material. On the other hand, the polystyrene can be cut fast and with minimum waste by heated wire, especially systems still pertinent in the production of archi- when both sides of the cut foam can be used. -
Pentaplexity: Comparing Fractal Tilings and Penrose Tilings
Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Pentaplexity: Comparing fractal tilings and Penrose tilings Adam Brunell and Daniel Sherwood Vassar College [email protected] [email protected] | April 6, 2013 Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Overview 1 Introducing Fractal Penrose Tilings 2 Introducing Penrose’s Original Pentaplexity Tiling 3 Comparing Fractal Tiling and Pentaplexity Tiling Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Penrose’s Kites and Darts Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Drawing the Aorta in Kites and Darts Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Fractal Penrose Tiling Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Fractal Tile Set Adam Brunell and Daniel Sherwood Penrose -
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. -
Regular Tilings with Heptagons
Tilings of the plane Terminology There appears to be some confusion as to the meanings of the terms semi-regular, demi-regular, quasi-regular etc. I shall use the following scheme instead. Monomorphic – consisting of one shape of tile only Bimorphic – consisting of two shapes of tile Trimorphic - consisting of three shapes etc. Regular – consisting of regular polygons only Irregular – containing at least one irregular polygon First order – containing vertices of one type only (ie all verticese are identical) Second order – having two kinds of vertex Third order – having three kinds of vertex etc. Conformal – in which all the vertices coincide with vertices of another polygon Non-conformal – in which vertices of one polygon lie on the edge of anotherRegular Monomorphic Tilings There are three conformal regular monomorphic tilings Square tiling 1S It has 4-fold rotational symmetry, 4 reflection lines and 2 orthogonal translations There are an infinite number of non-conformal variations with lesser symmetry. Triangular tiling 1T It has 6-fold rotational symmetry, 3 reflection lines and 3 translations. Again, there are an infinite number of non-conformal variations with lesser symmetry Hexagonal tiling 1H It has both 6-fold and 3-fold rotational symmetry Regular Bimorphic Tilings Squares and triangles As far as I know there are only two first order regular bimorphic tilings containing squares and triangles This is a first order regular tiling. Each vertex of this tiling has the order SSTTT. 1S 2T The order of this tiling is STSTT. 4S 8T There appear to be a large number of regular bimorphic (ST) tilings of higher orders.