Polyominoes with Maximum Convex Hull Sascha Kurz
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Snakes in the Plane
Snakes in the Plane by Paul Church A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Computer Science Waterloo, Ontario, Canada, 2008 c 2008 Paul Church I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract Recent developments in tiling theory, primarily in the study of anisohedral shapes, have been the product of exhaustive computer searches through various classes of poly- gons. I present a brief background of tiling theory and past work, with particular empha- sis on isohedral numbers, aperiodicity, Heesch numbers, criteria to characterize isohedral tilings, and various details that have arisen in past computer searches. I then develop and implement a new “boundary-based” technique, characterizing shapes as a sequence of characters representing unit length steps taken from a finite lan- guage of directions, to replace the “area-based” approaches of past work, which treated the Euclidean plane as a regular lattice of cells manipulated like a bitmap. The new technique allows me to reproduce and verify past results on polyforms (edge-to-edge as- semblies of unit squares, regular hexagons, or equilateral triangles) and then generalize to a new class of shapes dubbed polysnakes, which past approaches could not describe. My implementation enumerates polyforms using Redelmeier’s recursive generation algo- rithm, and enumerates polysnakes using a novel approach. -
A Flowering of Mathematical Art
A Flowering of Mathematical Art Jim Henle & Craig Kasper The Mathematical Intelligencer ISSN 0343-6993 Volume 42 Number 1 Math Intelligencer (2020) 42:36-40 DOI 10.1007/s00283-019-09945-0 1 23 Your article is protected by copyright and all rights are held exclusively by Springer Science+Business Media, LLC, part of Springer Nature. This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self- archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy For Our Mathematical Pleasure (Jim Henle, Editor) 1 have argued that the creation of mathematical A Flowering of structures is an art. The previous column discussed a II tiny genre of that art: numeration systems. You can’t describe that genre as ‘‘flowering.’’ But activity is most Mathematical Art definitely blossoming in another genre. Around the world, hundreds of artists are right now creating puzzles of JIM HENLE , AND CRAIG KASPER subtlety, depth, and charm. We are in the midst of a renaissance of logic puzzles. A Renaissance The flowering began with the discovery in 2004 in England, of the discovery in 1980 in Japan, of the invention in 1979 in the United States, of the puzzle type known today as sudoku. -
A New Mathematical Model for Tiling Finite Regions of the Plane with Polyominoes
Volume 15, Number 2, Pages 95{131 ISSN 1715-0868 A NEW MATHEMATICAL MODEL FOR TILING FINITE REGIONS OF THE PLANE WITH POLYOMINOES MARCUS R. GARVIE AND JOHN BURKARDT Abstract. We present a new mathematical model for tiling finite sub- 2 sets of Z using an arbitrary, but finite, collection of polyominoes. Unlike previous approaches that employ backtracking and other refinements of `brute-force' techniques, our method is based on a systematic algebraic approach, leading in most cases to an underdetermined system of linear equations to solve. The resulting linear system is a binary linear pro- gramming problem, which can be solved via direct solution techniques, or using well-known optimization routines. We illustrate our model with some numerical examples computed in MATLAB. Users can download, edit, and run the codes from http://people.sc.fsu.edu/~jburkardt/ m_src/polyominoes/polyominoes.html. For larger problems we solve the resulting binary linear programming problem with an optimization package such as CPLEX, GUROBI, or SCIP, before plotting solutions in MATLAB. 1. Introduction and motivation 2 Consider a planar square lattice Z . We refer to each unit square in the lattice, namely [~j − 1; ~j] × [~i − 1;~i], as a cell.A polyomino is a union of 2 a finite number of edge-connected cells in the lattice Z . We assume that the polyominoes are simply-connected. The order (or area) of a polyomino is the number of cells forming it. The polyominoes of order n are called n-ominoes and the cases for n = 1; 2; 3; 4; 5; 6; 7; 8 are named monominoes, dominoes, triominoes, tetrominoes, pentominoes, hexominoes, heptominoes, and octominoes, respectively. -
Coverage Path Planning Using Reinforcement Learning-Based TSP for Htetran—A Polyabolo-Inspired Self-Reconfigurable Tiling Robot
sensors Article Coverage Path Planning Using Reinforcement Learning-Based TSP for hTetran—A Polyabolo-Inspired Self-Reconfigurable Tiling Robot Anh Vu Le 1,2 , Prabakaran Veerajagadheswar 1, Phone Thiha Kyaw 3 , Mohan Rajesh Elara 1 and Nguyen Huu Khanh Nhan 2,∗ 1 ROAR Lab, Engineering Product Development, Singapore University of Technology and Design, Singapore 487372, Singapore; [email protected] (A.V.L); [email protected] (P.V); [email protected] (M.R.E.) 2 Optoelectronics Research Group, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam 3 Department of Mechatronic Engineering, Yangon Technological University, Insein 11101, Myanmar; [email protected] * Correspondence: [email protected] Abstract: One of the critical challenges in deploying the cleaning robots is the completion of covering the entire area. Current tiling robots for area coverage have fixed forms and are limited to cleaning only certain areas. The reconfigurable system is the creative answer to such an optimal coverage problem. The tiling robot’s goal enables the complete coverage of the entire area by reconfigur- ing to different shapes according to the area’s needs. In the particular sequencing of navigation, it is essential to have a structure that allows the robot to extend the coverage range while saving Citation: Le, A.V.; energy usage during navigation. This implies that the robot is able to cover larger areas entirely Veerajagadheswar, P.; Thiha Kyaw, P.; with the least required actions. This paper presents a complete path planning (CPP) for hTetran, Elara, M.R.; Nhan, N.H.K. -
SUDOKU SOLVER Study and Implementation
SUDOKU SOLVER Study and Implementation Abstract In this project we studied two algorithms to solve a Sudoku game and implemented a playable application for all major operating systems. Qizhong Mao, Baiyi Tao, Arif Aziz {qzmao, tud46571, arif.aziz}@temple.edu CIS 5603 Project Report Qizhong Mao, Baiyi Tao, Arif Aziz Table of Contents Introduction .......................................................................................................................... 3 Variants ................................................................................................................................ 3 Standard Sudoku ...........................................................................................................................3 Small Sudoku .................................................................................................................................4 Mini Sudoku ..................................................................................................................................4 Jigsaw Sudoku ...............................................................................................................................4 Hyper Sudoku ................................................................................................................................5 Dodeka Sudoku ..............................................................................................................................5 Number Place Challenger ...............................................................................................................5 -
Applying Parallel Programming Polyomino Problem
______________________________________________________ Case Study Applying Parallel Programming to the Polyomino Problem Kevin Gong UC Berkeley CS 199 Fall 1991 Faculty Advisor: Professor Katherine Yelick ______________________________________________________ INTRODUCTION - 1 ______________________________________________________ ______________________________________________________ INTRODUCTION - 2 ______________________________________________________ Acknowledgements Thanks to Martin Gardner, for introducing me to polyominoes (and a host of other problems). Thanks to my faculty advisor, Professor Katherine Yelick. Parallel runs for this project were performed on a 12-processor Sequent machine. Sequential runs were performed on Mammoth. This machine was purchased under an NSF CER infrastructure grant, as part of the Mammoth project. Format This paper was printed on a Laserwriter II using an Apple Macintosh SE running Microsoft Word. Section headings and tables are printed in Times font. The text itself is printed in Palatino font. Headers and footers are in Helvetica. ______________________________________________________ INTRODUCTION - 3 ______________________________________________________ ______________________________________________________ INTRODUCTION - 4 ______________________________________________________ Contents 1 Introduction 2 Problem 3 Existing Research 4 Sequential Enumeration 4.1 Extension Method 4.1.1 Generating Polyominoes 4.1.2 Hashing 4.1.3 Storing Polyominoes For Uniqueness Check 4.1.4 Data Structures 4.2 Rooted -
Rebuilding Objects with Linear Algebra
The Shattered Urn: Rebuilding Objects with Linear Algebra John Burkardt Department of Mathematics University of South Carolina 20 November 2018 1 / 94 The Shattered Urn: Rebuilding Objects with Linear Algebra Computational Mathematics Seminar sponsored by: Mathematics Department University of Pittsburgh ... 10:00-11:00, 20 November 2018 427 Thackeray Hall References: http://people.math.sc.edu/burkardt/presentations/polyominoes 2018 pitt.pdf Marcus Garvie, John Burkardt, A New Mathematical Model for Tiling Finite Regions of the Plane with Polyominoes, submitted. 2 / 94 The Portland Vase 25AD - A Reconstruction Problem 3 / 94 Ostomachion - Archimedes, 287-212 BC 4 / 94 Tangrams - Song Dynasty, 960-1279 5 / 94 Dudeney - The Broken Chessboard (1919) 6 / 94 Golomb - Polyominoes (1965) 7 / 94 Puzzle Statement 8 / 94 Tile Variations: Reflections, Rotations, All Orientations 9 / 94 Puzzle Variations: Holes or Infinite Regions 10 / 94 Is It Math? \One can guess that there are several tilings of a 6 × 10 rectangle using the twelve pentominoes. However, one might not predict just how many there are. An exhaustive computer search has found that there are 2339 such tilings. These questions make nice puzzles, but are not the kind of interesting mathematical problem that we are looking for." \Tilings" - Federico Ardila, Richard Stanley 11 / 94 Is it \Simple" Computer Science? 12 / 94 Sequential Solution: Backtracking 13 / 94 Is it \Clever" Computer Science? 14 / 94 Simultaneous Solution: Linear Algebra? 15 / 94 Exact Cover Knuth: \Tiling is a version -
Some Open Problems in Polyomino Tilings
Some Open Problems in Polyomino Tilings Andrew Winslow1 University of Texas Rio Grande Valley, Edinburg, TX, USA [email protected] Abstract. The author surveys 15 open problems regarding the algorith- mic, structural, and existential properties of polyomino tilings. 1 Introduction In this work, we consider a variety of open problems related to polyomino tilings. For further reference on polyominoes and tilings, the original book on the topic by Golomb [15] (on polyominoes) and more recent book of Gr¨unbaum and Shep- hard [23] (on tilings) are essential. Also see [2,26] and [19,21] for open problems in polyominoes and tiling more broadly, respectively. We begin by introducing the definitions used throughout; other definitions are introduced as necessary in later sections. Definitions. A polyomino is a subset of R2 formed by a strongly connected union of axis-aligned unit squares centered at locations on the square lattice Z2. Let T = fT1;T2;::: g be an infinite set of finite simply connected closed sets of R2. Provided the elements of T have pairwise disjoint interiors and cover the Euclidean plane, then T is a tiling and the elements of T are called tiles. Provided every Ti 2 T is congruent to a common shape T , then T is mono- hedral, T is the prototile of T , and the elements of T are called copies of T . In this case, T is said to have a tiling. 2 Isohedral Tilings We begin with monohedral tilings of the plane where the prototile is a polyomino. If a tiling T has the property that for all Ti;Tj 2 T , there exists a symmetry of T mapping Ti to Tj, then T is isohedral; otherwise the tiling is non-isohedral (see Figure 1). -
Enumeration of Skew Ferrers Diagrams*
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 112 (1993) 65-79 65 North-Holland Enumeration of skew Ferrers diagrams* M.P. Delest and J.M. Fedou Dhpartement d’lnformatique. Unirersith de Bordeaux I. LaBRI**. 351 Course de la Liberation, 33405 Talence Cedex, France Received 7 November 1989 Revised 5 December 1990 Abstract Delest, M.P. and J.M. Fedou, Enumeration of skew Ferrers diagrams, Discrete Mathematics 112 (1993) 65-79. In this paper, we show that the generating function for skew Ferrers diagrams according to their width and area is the quotient of new basic Bessel functions. R&sum& Delest, M.P. and J.M. Fedou, Enumeration of skew Ferrers diagrams, Discrete Mathematics 112 (1993) 65-79. Nous montrons dans cet article que la fonction g&ntratrice des diagrammes de Ferrers gauches selon les paramktres ptrimttre et aire s’exprime en fonction du quotient des q analogues de deux fonctions de Bessel. Introduction Ferrers diagrams, related to the well-known partitions of an integer, have been extensively studied. See for instance Andrews’ book [3]. A partition of an integer n is a decreasing sequence of integers, nl, n2,. , nk, such that n, + n2 + ... + nk = n. The geometric figure formed by the k rows having respectively n, , n2,. , nk cells (see Fig. 1) is called the Ferrers diagram associated to the partition (n, , n2,. nk) of n. Filling Ferrers diagrams with numbers gives plane partitions, which are related to repres- entations of the symmetric group [13]. -
Parity and Tiling by Trominoes
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 40 (2008), Pages 115–136 Parity and tiling by trominoes Michael Reid Department of Mathematics University of Central Florida Orlando, FL 32816 U.S.A. [email protected] Abstract The problem of counting tilings by dominoes and other dimers and finding arithmetic significance in these numbers has received considerable atten- tion. In contrast, little attention has been paid to the number of tilings by more complex shapes. In this paper, we consider tilings by trominoes and the parity of the number of tilings. We mostly consider reptilings and tilings of related shapes by the L tromino. We were led to this by revisiting a theorem of Hochberg and Reid (Discrete Math. 214 (2000), 255–261) about tiling with d-dimensional notched cubes, for d 3; the L tromino is the 2-dimensional notched cube. We conjecture≥ that the number of tilings of a region shaped like an L tromino, but scaled by a factor of n, is odd if and only if n is a power of 2. More generally, we conjecture the the number of tilings of a region obtained by scaling an L tromino by a factor of m in the x direction and a factor of n in the y direction, is odd if and only if m = n and the common value is a power of 2. The conjecture is proved for odd values of m and n, and also for several small even values. In the final section, we briefly consider tilings by other shapes. 1 Introduction In this paper, we consider the number of tilings of certain regions by L trominoes, and try to understand the parity of this number. -
SOMA CUBE Thorleif's SOMA Page
SOMA CUBE Thorleif's SOMA page : http://fam-bundgaard.dk/SOMA/SOMA.HTM Standard SOMA figures Stran 1 Stran 2 Stran 3 Stran 4 Stran 5 Stran 6 Stran 7 Stran 8 Stran 9 Stran 10 Stran 11 Stran 12 Stran 13 Stran 14 Stran 15 Stran 16 Stran 17 Stran 18 Stran 19 Stran 20 Stran 21 Stran 22 Stran 23 Stran 24 Stran 25 Stran 26 Stran 27 Stran 28 Stran 29 Stran 30 Special SOMA collection Stran 31 . Pairs of figures Pairs Stran 32 Figures NOT using using 7 pieces. Figures NOT Stran 33 Double set Double figures. Stran 34 Double set Double figures. Stran 35 Double set Double figures. Stran 36 Special Double/Tripple etc. set figures. Double/Tripple Special Stran 37 Paulo Brina 'Quad' collection Stran 38 Quad set figures. Quad set Stran 39 SOMA NEWS A 4 set SOMA figure is large. March 25 2010 Just as I thought SOMA was fading, I heard from Paulo Brina of Belo Horizonte, Brasil. The story he told was BIG SOMA Joining the rest of us, SOMA maniacs, Paulo was tired of the old 7 piece soma, so he began to play Tetra-SOMA, using 4 sets = 28 pieces = 108 cubes. The tetra set, home made, stainless steel (it's polished, so, it's hard to take photos) Notice the extreme beauty this polished set exhibits. :o) As Paulo wrote "It's funny. More possibilities, more complexity." Stran 40 Lets look at the pictures. 001 bloc 2x7x9 = 126, so we have 18 holes/gaps 002 A 5x5x5 cube would make 125, . -
Isohedral Polyomino Tiling of the Plane
Discrete Comput Geom 21:615–630 (1999) GeometryDiscrete & Computational © 1999 Springer-Verlag New York Inc. Isohedral Polyomino Tiling of the Plane¤ K. Keating and A. Vince Department of Mathematics, University of Florida, Gainesville, FL 32611, USA keating,vince @math.ufl.edu f g Abstract. A polynomial time algorithm is given for deciding, for a given polyomino P, whether there exists an isohedral tiling of the Euclidean plane by isometric copies of P. The decidability question for general tilings by copies of a single polyomino, or even periodic tilings by copies of a single polyomino, remains open. 1. Introduction Besides its place among topics in recreational mathematics, polyomino tiling of the plane is relevant to decidability questions in logic and data storage questions in parallel processing. A polyomino is a rookwise connected tile formed by joining unit squares at their edges. We always assume that our polyominoes are made up of unit squares in the Cartesian plane whose centers are lattice points and whose sides are parallel to the coordinate axes. A polyomino consisting of n unit squares is referred to as an n-omino. Polyominoes were introduced in 1953 by Golomb; his book Polyominoes, originally published in 1965 [3], has recently been revised and reissued [5]. See also his survey article [6] for an overview of polyomino tiling problems. One of the first questions to arise in the subject was the following: Given a single polyomino P, can isometric copies of P tile the plane? It is known that every n-omino for n 6 tiles the plane. Of the 108 7-ominoes, only 4 of them, shown in Fig.