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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. -
Edgy Puzzles Karl Schaffer Karl [email protected]
Edgy Puzzles Karl Schaffer [email protected] Countless puzzles involve decomposing areas or volumes of two or three-dimensional figures into smaller figures. “Polyform” puzzles include such well-known examples as pentominoes, tangrams, and soma cubes. This paper will examine puzzles in which the edge sets, or "skeletons," of various symmetric figures like polyhedra are decomposed into multiple copies of smaller graphs, and note their relationship to representations by props or body parts in dance performance. The edges of the tetrahedron in Figure 1 are composed of a folded 9-gon, while the cube and octahedron are each composed of six folded paths of length 2. These constructions have been used in dances created by the author and his collaborators. The photo from the author’s 1997 dance “Pipe Dreams” shows an octahedron in which each dancer wields three lengths of PVC pipe held together by cord at the two internal vertices labeled a in the diagram on the right. The shapes created by the dancers, which might include whimsical designs reminiscent of animals or other objects as well as mathematical forms, seem to appear and dissolve in fluid patterns, usually in time to a musical score. L R a R L a L R Figure 1. PVC pipe polyhedral skeletons used in dances The desire in the dance company co-directed by the author to incorporate polyhedra into dance works led to these constructions, and to similar designs with loops of rope, fingers and hands, and the bodies of dancers. Just as mathematical concepts often suggest artistic explorations for those involved in the interplay between these fields, performance problems may suggest mathematical questions, in this case involving finding efficient and symmetrical ways to construct the skeletons, or edge sets, of the Platonic solids. -
Treb All De Fide Gra U
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by UPCommons. Portal del coneixement obert de la UPC Grau en Matematiques` T´ıtol:Tilings and the Aztec Diamond Theorem Autor: David Pardo Simon´ Director: Anna de Mier Departament: Mathematics Any academic:` 2015-2016 TREBALL DE FI DE GRAU Facultat de Matemàtiques i Estadística David Pardo 2 Tilings and the Aztec Diamond Theorem A dissertation submitted to the Polytechnic University of Catalonia in accordance with the requirements of the Bachelor's degree in Mathematics in the School of Mathematics and Statistics. David Pardo Sim´on Supervised by Dr. Anna de Mier School of Mathematics and Statistics June 28, 2016 Abstract Tilings over the plane R2 are analysed in this work, making a special focus on the Aztec Diamond Theorem. A review of the most relevant results about monohedral tilings is made to continue later by introducing domino tilings over subsets of R2. Based on previous work made by other mathematicians, a proof of the Aztec Dia- mond Theorem is presented in full detail by completing the description of a bijection that was not made explicit in the original work. MSC2010: 05B45, 52C20, 05A19. iii Contents 1 Tilings and basic notions1 1.1 Monohedral tilings............................3 1.2 The case of the heptiamonds.......................8 1.2.1 Domino Tilings.......................... 13 2 The Aztec Diamond Theorem 15 2.1 Schr¨odernumbers and Hankel matrices................. 16 2.2 Bijection between tilings and paths................... 19 2.3 Hankel matrices and n-tuples of Schr¨oderpaths............ 27 v Chapter 1 Tilings and basic notions The history of tilings and patterns goes back thousands of years in time. -
Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity∗
Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity∗ Erik D. Demaine, Martin L. Demaine MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, MA 02139, USA, {edemaine,mdemaine}@mit.edu Dedicated to Jin Akiyama in honor of his 60th birthday. Abstract. We show that jigsaw puzzles, edge-matching puzzles, and polyomino packing puzzles are all NP-complete. Furthermore, we show direct equivalences between these three types of puzzles: any puzzle of one type can be converted into an equivalent puzzle of any other type. 1. Introduction Jigsaw puzzles [37,38] are perhaps the most popular form of puzzle. The original jigsaw puzzles of the 1760s were cut from wood sheets using a hand woodworking tool called a jig saw, which is where the puzzles get their name. The images on the puzzles were European maps, and the jigsaw puzzles were used as educational toys, an idea still used in some schools today. Handmade wooden jigsaw puzzles for adults took off around 1900. Today, jigsaw puzzles are usually cut from cardboard with a die, a technology that became practical in the 1930s. Nonetheless, true addicts can still find craftsmen who hand-make wooden jigsaw puzzles. Most jigsaw puzzles have a guiding image and each side of a piece has only one “mate”, although a few harder variations have blank pieces and/or pieces with ambiguous mates. Edge-matching puzzles [21] are another popular puzzle with a similar spirit to jigsaw puzzles, first appearing in the 1890s. In an edge-matching puzzle, the goal is to arrange a given collection of several identically shaped but differently patterned tiles (typically squares) so that the patterns match up along the edges of adjacent tiles. -
Embedded System Security: Prevention Against Physical Access Threats Using Pufs
International Journal of Innovative Research in Science, Engineering and Technology (IJIRSET) | e-ISSN: 2319-8753, p-ISSN: 2320-6710| www.ijirset.com | Impact Factor: 7.512| || Volume 9, Issue 8, August 2020 || Embedded System Security: Prevention Against Physical Access Threats Using PUFs 1 2 3 4 AP Deepak , A Krishna Murthy , Ansari Mohammad Abdul Umair , Krathi Todalbagi , Dr.Supriya VG5 B.E. Students, Department of Electronics and Communication Engineering, Sir M. Visvesvaraya Institute of Technology, Bengaluru, India1,2,3,4 Professor, Department of Electronics and Communication Engineering, Sir M. Visvesvaraya Institute of Technology, Bengaluru, India5 ABSTRACT: Embedded systems are the driving force in the development of technology and innovation. It is used in plethora of fields ranging from healthcare to industrial sectors. As more and more devices come into picture, security becomes an important issue for critical application with sensitive data being used. In this paper we discuss about various vulnerabilities and threats identified in embedded system. Major part of this paper pertains to physical attacks carried out where the attacker has physical access to the device. A Physical Unclonable Function (PUF) is an integrated circuit that is elementary to any hardware security due to its complexity and unpredictability. This paper presents a new low-cost CMOS image sensor and nano PUF based on PUF targeting security and privacy. PUFs holds the future to safer and reliable method of deploying embedded systems in the coming years. KEYWORDS:Physically Unclonable Function (PUF), Challenge Response Pair (CRP), Linear Feedback Shift Register (LFSR), CMOS Image Sensor, Polyomino Partition Strategy, Memristor. I. INTRODUCTION The need for security is universal and has become a major concern in about 1976, when the first initial public key primitives and protocols were established. -
An Investigation Into Three Dimensional Probabilistic Polyforms Danielle Marie Vander Schaaf Olivet Nazarene University, [email protected]
Olivet Nazarene University Digital Commons @ Olivet Honors Program Projects Honors Program 5-2012 An Investigation into Three Dimensional Probabilistic Polyforms Danielle Marie Vander Schaaf Olivet Nazarene University, [email protected] Follow this and additional works at: https://digitalcommons.olivet.edu/honr_proj Part of the Algebraic Geometry Commons, and the Geometry and Topology Commons Recommended Citation Vander Schaaf, Danielle Marie, "An Investigation into Three Dimensional Probabilistic Polyforms" (2012). Honors Program Projects. 27. https://digitalcommons.olivet.edu/honr_proj/27 This Article is brought to you for free and open access by the Honors Program at Digital Commons @ Olivet. It has been accepted for inclusion in Honors Program Projects by an authorized administrator of Digital Commons @ Olivet. For more information, please contact [email protected]. An Investigation into Three Dimensional Probabilistic Polyforms Danielle Vander Schaaf Olivet Nazarene University 1 Abstract Polyforms are created by taking squares, equilateral triangles, and regular hexagons and placing them side by side to generate larger shapes. This project addressed three-dimensional polyforms and focused on cubes. I investigated the probabilities of certain shape outcomes to discover what these probabilities could tell us about the polyforms’ characteristics and vice versa. From my findings, I was able to derive a formula for the probability of two different polyform patterns which add to a third formula found prior to my research. In addition, I found the probability that 8 cubes randomly attached together one by one would form a 2x2x2 cube. Finally, I discovered a strong correlation between the probability of a polyform and its number of exposed edges, and I noticed a possible relationship between a polyform’s probability and its graph representation. -
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). -
Biased Weak Polyform Achievement Games Ian Norris, Nándor Sieben
Biased weak polyform achievement games Ian Norris, Nándor Sieben To cite this version: Ian Norris, Nándor Sieben. Biased weak polyform achievement games. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2014, Vol. 16 no. 3 (in progress) (3), pp.147–172. hal- 01188912 HAL Id: hal-01188912 https://hal.inria.fr/hal-01188912 Submitted on 31 Aug 2015 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. Discrete Mathematics and Theoretical Computer Science DMTCS vol. 16:3, 2014, 147–172 Biased Weak Polyform Achievement Games Ian Norris∗ Nándor Siebeny Northern Arizona University, Department of Mathematics and Statistics, Flagstaff, AZ, USA received 21st July 2011, revised 20th May 2014, accepted 6th July 2014. In a biased weak (a; b) polyform achievement game, the maker and the breaker alternately mark a; b previously unmarked cells on an infinite board, respectively. The maker’s goal is to mark a set of cells congruent to a polyform. The breaker tries to prevent the maker from achieving this goal. A winning maker strategy for the (a; b) game can be built from winning strategies for games involving fewer marks for the maker and the breaker. -
Teacher's Guide to Shape by Shape
Using Puzzles to Teach Problem Solving TEACHER’S GUIDE TO SHAPE BY SHAPE Also includes Brick by Brick, Block by Block, and That-A-Way BENEFITS Shape by Shape vs. Tangrams Shape by Shape is an assembly puzzle that is an exciting way to teach elements of Shape by Shape is closely related to Tangrams, a classic — problem solving puzzle invented in China that became a world wide — visual and thinking craze in the early 18th century. The seven simple — geometry and measurement Tangram shapes can be arranged to create thousands — combinatorial reasoning of different silhouettes, such as people, animals, letters, The game is self-directed, with 60 challenge cards that and everyday objects. Tangrams are used in schools to include hints. The activities in this guide will help your stu- teach problem solving, geometry and fractions. dents get more out of Shape by Shape. ABOUT THE GAME Equipment. Shape by Shape includes 14 puzzle pieces (6 orange and 8 yellow), 60 challenge cards with answers on the backs, and a tray that doubles as storage for the cards. Shape by Shape and Tangrams are both based on the same triangular grid. They are similar enough that most Tangram activities work with Shape by Shape. Shape by Shape, however, is bigger and more complex. We recommend that you introduce math concepts with Tangrams first, then use Shape by Shape as a fol- low-up activity to explore concepts in more depth. Shape by Shape is different enough fromTangrams that both are worth owning. The main differences are: In a frame . -
MITOCW | Watch?V=2X9tv1bf2um
MITOCW | watch?v=2X9Tv1bF2UM PROFESSOR: All right. lecture 14 was about two main topics, I guess. We had slender adorned chains, the sort of fatter linkages, and then hinged dissection. Most of our time was actually spent with the slender adornments and proving that that works. But most of our questions today are about hinged dissections because that's kind of the most fun, and there's a lot more to say about them. So first question is, is there any software for hinged dissections? And the short answer is no, surprisingly. So this would definitely be a cool project possibility. There are a bunch of examples-- let me switch to these on the web-- just sort of random examples, cool dissections people thought were so neat that they wanted to animate them. And so they basically constructed where the coordinates were over time in Mathematica, and then put it on the web as a illustration of that. So this is a equilateral triangle to a hexagon-- a regular hexagon. It's a hinged dissection by Greg Frederickson, and then is drawn by have Rick Mabry. Here's another one for an equilateral triangle to a pentagon. Pretty cool. And they're hinged in a tree-like fashion even, which is kind of unusual. Greg Frederickson is one of the masters of hinged dissections, and dissections in general. He's probably the master of dissections in general. And he has three books of different kinds of dissections. This is actually hinged the section on the cover here. The purple and pink pieces hinge like this into a smaller star from the outline of a big star to the interior of a smaller star. -
Heesch Numbers of Unmarked Polyforms
Heesch Numbers of Unmarked Polyforms Craig S. Kaplan School of Computer Science, University of Waterloo, Ontario, Canada; [email protected] Abstract A shape’s Heesch number is the number of layers of copies of the shape that can be placed around it without gaps or overlaps. Experimentation and exhaustive searching have turned up examples of shapes with finite Heesch numbers up to six, but nothing higher. The computational problem of classifying simple families of shapes by Heesch number can provide more experimental data to fuel our understanding of this topic. I present a technique for computing Heesch numbers of non-tiling polyforms using a SAT solver, and the results of exhaustive computation of Heesch numbers up to 19-ominoes, 17-hexes, and 24-iamonds. 1 Introduction Tiling theory is the branch of mathematics concerned with the properties of shapes that can cover the plane with no gaps or overlaps. It is a topic rich with deep results and open problems. Of course, tiling theory must occasionally venture into the study of shapes that do not tile the plane, so that we might understand those that do more completely. If a shape tiles the plane, then it must be possible to surround the shape by congruent copies of itself, leaving no part of its boundary exposed. A circle clearly cannot tile the plane, because neighbouring circles can cover at most a finite number of points on its boundary. A regular pentagon also cannot be surrounded by copies of itself: its vertices will always remain exposed. However, the converse is not true: there exist shapes that can be fully surrounded by copies of themselves, but for which no such surround can be extended to a tiling. -
By Kate Jones, RFSPE
ACES: Articles, Columns, & Essays A Periodic Table of Polyform Puzzles by Kate Jones, RFSPE What are polyform puzzles? A branch of mathematics known as Combinatorics deals with systems that encompass permutations and combinations. A subcategory covers tilings, or tessellations, that are formed of mosaic-like pieces of all different shapes made of the same distinct building block. A simple example is starting with a single square as the building block. Two squares joined are a domino. Three squares joined are a tromino. Compare that to one proton/electron pair forming hydrogen, and two of them becoming helium, etc. The puzzles consist of combining all the members of a set into coherent shapes, in as many different variations as possible. Here is a table of geometric families of polyform sets that provide combinatorial puzzle challenges (“recreational mathematics”) developed and produced by my company, Kadon Enterprises, Inc., over the last 40 years (1979-2019). See www.gamepuzzles.com. The POLYOMINO Series Figure 1: The polyominoes orders 1 through 5. 36 TELICOM 31, No. 3 — Third Quarter 2019 Figure 2: Polyominoes orders 1-5 (Poly-5); order 6 (hexominoes, Sextillions); order 7 (heptominoes); and order 8 (octominoes)—21, 35, 108 and 369 pieces, respectively. Polyominoes named by Solomon Golomb in 1953. Product names by Kate Jones. TELICOM 31, No. 3 — Third Quarter 2019 37 Figure 3: Left—Polycubes orders 1 through 4. Right—Polycubes order 5 (the planar pentacubes, Kadon’s Quintillions set). Figure 4: Above—the 18 non-planar pentacubes (Super Quintillions). Below—the 166 hexacubes. 38 TELICOM 31, No. 3 — Third Quarter 2019 The POLYIAMOND Series Figure 5: The polyiamonds orders 1 through 7.