Polycubes Triangulations And Polyhexes Over Zn

Author : Raymond R. Fletcher III
Publisher : Xlibris Corporation
Page : 191 pages
File Size : 55,8 Mb
Release : 2024-07-02
Category : Mathematics
ISBN : 9781796030761

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Polycubes, Triangulations and Polyhexes over Zn by Raymond R. Fletcher III Pdf

A mathematical puzzle book that involves labeling points or edges on three types of graphs: (1) polycubes, which consist of cubes attached face-to-face, (2) polyhexes, which consist of regular hexagons joined edge to edge, and (3) triangulations, which consist of triangles attached edge to edge. For a polycube with n points, the puzzle consists of labeling each vertex with the full set of integers (mod n), (Zn) so that the four labels belonging to each face of every component cube have the same sum. For a polyhex with n points, the vertices are to be labeled with elements from Zn so that the six labels assigned to each component hexagon have the same sum. For puzzles involving triangulation, the edges, instead of the vertices, are to be labeled with elements of Zn. To solve the puzzle, a labeling must be found so that the edges of each component triangle have the same sum. Unlike sudoku, these puzzles have many solutions; the solutions provided in the back of the book are included just to show that a solution exists. Readers are encouraged to create and solve their own puzzles in any of the three genres. Questions of mathematical interest are provided throughout.

Author : Raymond R. Fletcher
Publisher : Unknown
Page : 0 pages
File Size : 44,7 Mb
Release : 2023-04-26
Category : Mathematics
ISBN : 1959379542

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Puzzles Involving Polycubes, Triangulations and Polyhexes by Raymond R. Fletcher Pdf

Three related types of puzzles are featured in this book. All involve labeling the vertices or edges of graphs. The first type of graph is a polycube. The vertices are labeled with a specific set of numbers and the puzzle involves obtaining a labeling in which the four vertices belonging to each face of a constituent cube have the same sum. The second type of graph is a plane triangulation. The puzzle involves labeling the edges of each constituent triangle so that the sum is the same for each triangle. The third type of graph is a polyhex. For this type of graph the vertices are labeled with integers so that the sum of the six points bounding each hexagon is the same. Problems and research questions are included at the end of each Section, and all puzzle solutions are provided.

Author : Raymond R. Fletcher III
Publisher : Xlibris Corporation
Page : 245 pages
File Size : 51,8 Mb
Release : 2024-07-02
Category : Mathematics
ISBN : 9781796097894

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Polyparallelogram Puzzles and Tiling Problems by Raymond R. Fletcher III Pdf

A mathematical puzzle book which involves labeling vertices of a plane graph with elements from an abelian group G such that the labels applied to the bounding vertices of each region have the same sum in G. The plane graphs are those formed by the edges and vertices of various types of parallelograms fitted together edge to edge. We call these polyparallelograms. The polyparallelogram puzzles are organized into the following categories according to the types of parallelograms involved: (i) polysquares (or polyominoes; (ii) 60/120 rhombs; (iii) 36/144 and 72/108 rhombs (Penrose rhombs); (iv) various lattice parallelograms. The reader is introduced to the theory of polyparallelogram tilings of polygons, and challenged with problems and research questions at the end of each Chapter. All that is required of the elementary theory of graphs and abelian groups is included in the text. Also included are a variety of pure tiling problems involving nonparallelogram lattice quadrilaterals. “Polyparallelogram Puzzles and Tiling Problems” is a sequel to “Polycubes, Triangulations and Polyhexes over Zn” and features a similar type of puzzle but is more engaging mathematically.

Author : Raymond R. Fletcher III
Publisher : Unknown
Page : 128 pages
File Size : 40,7 Mb
Release : 2021-09-25
Category : Electronic
ISBN : 1685360939

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Puzzles Involving Polycubes, Triangulations and Polyhexes by Raymond R. Fletcher III Pdf

Author : Raymond R. Fletcher III
Publisher : Xlibris Us
Page : 202 pages
File Size : 44,6 Mb
Release : 2020-04-24
Category : Mathematics
ISBN : 179609790X

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Polyparallelogram Puzzles and Tiling Problems by Raymond R. Fletcher III Pdf

A mathematical puzzle book which involves labeling vertices of a plane graph with elements from an abelian group G such that the labels applied to the bounding vertices of each region have the same sum in G. The plane graphs are those formed by the edges and vertices of various types of parallelograms fitted together edge to edge. We call these polyparallelograms. The polyparallelogram puzzles are organized into the following categories according to the types of parallelograms involved: (i) polysquares (or polyominoes; (ii) 60/120 rhombs; (iii) 36/144 and 72/108 rhombs (Penrose rhombs); (iv) various lattice parallelograms. The reader is introduced to the theory of polyparallelogram tilings of polygons, and challenged with problems and research questions at the end of each Chapter. All that is required of the elementary theory of graphs and abelian groups is included in the text. Also included are a variety of pure tiling problems involving nonparallelogram lattice quadrilaterals. "Polyparallelogram Puzzles and Tiling Problems" is a sequel to "Polycubes, Triangulations and Polyhexes over Zn" and features a similar type of puzzle but is more engaging mathematically.

Author : James F. Peters
Publisher : Springer
Page : 0 pages
File Size : 47,6 Mb
Release : 2016-04-29
Category : Technology & Engineering
ISBN : 3319302604

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Computational Proximity by James F. Peters Pdf

This book introduces computational proximity (CP) as an algorithmic approach to finding nonempty sets of points that are either close to each other or far apart. Typically in computational proximity, the book starts with some form of proximity space (topological space equipped with a proximity relation) that has an inherent geometry. In CP, two types of near sets are considered, namely, spatially near sets and descriptivelynear sets. It is shown that connectedness, boundedness, mesh nerves, convexity, shapes and shape theory are principal topics in the study of nearness and separation of physical aswell as abstract sets. CP has a hefty visual content. Applications of CP in computer vision, multimedia, brain activity, biology, social networks, and cosmology are included. The book has been derived from the lectures of the author in a graduate course on the topology of digital images taught over the past several years. Many of the students have provided important insights and valuable suggestions. The topics in this monograph introduce many forms of proximities with a computational flavour (especially, what has become known as the strong contact relation), many nuances of topological spaces, and point-free geometry.

Author : Sherman Stein,Sandor Szabó
Publisher : Cambridge University Press
Page : 236 pages
File Size : 53,9 Mb
Release : 1994
Category : Mathematics
ISBN : 0883850281

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Algebra and Tiling by Sherman Stein,Sandor Szabó Pdf

A concise investigation into the connections between tiling space problems and algebraic ideas, suitable for undergraduates.

Author : Somashekhar A. Naimpally,James F. Peters
Publisher : World Scientific
Page : 294 pages
File Size : 43,8 Mb
Release : 2013
Category : Mathematics
ISBN : 9789814407663

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Topology with Applications by Somashekhar A. Naimpally,James F. Peters Pdf

The principal aim of this book is to introduce topology and its many applications viewed within a framework that includes a consideration of compactness, completeness, continuity, filters, function spaces, grills, clusters and bunches, hyperspace topologies, initial and final structures, metric spaces, metrization, nets, proximal continuity, proximity spaces, separation axioms, and uniform spaces.This book provides a complete framework for the study of topology with a variety of applications in science and engineering that include camouflage filters, classification, digital image processing, forgery detection, Hausdorff raster spaces, image analysis, microscopy, paleontology, pattern recognition, population dynamics, stem cell biology, topological psychology, and visual merchandising.It is the first complete presentation on topology with applications considered in the context of proximity spaces, and the nearness and remoteness of sets of objects. A novel feature throughout this book is the use of near and far, discovered by F Riesz over 100 years ago. In addition, it is the first time that this form of topology is presented in the context of a number of new applications.

Author : Lorenzo Adlai Sadun
Publisher : American Mathematical Soc.
Page : 131 pages
File Size : 47,9 Mb
Release : 2008
Category : Mathematics
ISBN : 9780821847275

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Topology of Tiling Spaces by Lorenzo Adlai Sadun Pdf

"This book is an introduction to the topology of tiling spaces, with a target audience of graduate students who wish to learn about the interface of topology with aperiodic order. It isn't a comprehensive and cross-referenced tome about everything having to do with tilings, which would be too big, too hard to read, and far too hard to write! Rather, it is a review of the explosion of recent work on tiling spaces as inverse limits, on the cohomology of tiling spaces, on substitution tilings and the role of rotations, and on tilings that do not have finite local complexity. Powerful computational techniques have been developed, as have new ways of thinking about tiling spaces." "The text contains a generous supply of examples and exercises."--BOOK JACKET.

Author : Jiri Patera
Publisher : American Mathematical Soc.
Page : 303 pages
File Size : 47,7 Mb
Release : 1998
Category : Combinatorial geometry
ISBN : 9780821806821

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Quasicrystals and Discrete Geometry by Jiri Patera Pdf

Comprising the proceedings of the fall 1995 semester program arranged by The Fields Institute at the U. of Toronto, Ontario, Canada, this volume contains eleven contributions which address ordered aperiodic systems realized either as point sets with the Delone property or as tilings of a Euclidean space. This collection of articles aims to bring into the mainstream of mathematics and mathematical physics this developing field of study integrating algebra, geometry, Fourier analysis, number theory, crystallography, and theoretical physics. Annotation copyrighted by Book News, Inc., Portland, OR

Author : N. Pytheas Fogg
Publisher : Springer
Page : 404 pages
File Size : 44,5 Mb
Release : 2003-10-24
Category : Mathematics
ISBN : 9783540457145

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Substitutions in Dynamics, Arithmetics and Combinatorics by N. Pytheas Fogg Pdf

A certain category of infinite strings of letters on a finite alphabet is presented here, chosen among the 'simplest' possible one may build, both because they are very deterministic and because they are built by simple rules (a letter is replaced by a word, a sequence is produced by iteration). These substitutive sequences have a surprisingly rich structure. The authors describe the concepts of quantity of natural interactions, with combinatorics on words, ergodic theory, linear algebra, spectral theory, geometry of tilings, theoretical computer science, diophantine approximation, trancendence, graph theory. This volume fulfils the need for a reference on the basic definitions and theorems, as well as for a state-of-the-art survey of the more difficult and unsolved problems.

Author : Johannes Kellendonk,Daniel Lenz,Jean Savinien
Publisher : Birkhäuser
Page : 428 pages
File Size : 53,6 Mb
Release : 2015-06-05
Category : Mathematics
ISBN : 9783034809030

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Mathematics of Aperiodic Order by Johannes Kellendonk,Daniel Lenz,Jean Savinien Pdf

What is order that is not based on simple repetition, that is, periodicity? How must atoms be arranged in a material so that it diffracts like a quasicrystal? How can we describe aperiodically ordered systems mathematically? Originally triggered by the – later Nobel prize-winning – discovery of quasicrystals, the investigation of aperiodic order has since become a well-established and rapidly evolving field of mathematical research with close ties to a surprising variety of branches of mathematics and physics. This book offers an overview of the state of the art in the field of aperiodic order, presented in carefully selected authoritative surveys. It is intended for non-experts with a general background in mathematics, theoretical physics or computer science, and offers a highly accessible source of first-hand information for all those interested in this rich and exciting field. Topics covered include the mathematical theory of diffraction, the dynamical systems of tilings or Delone sets, their cohomology and non-commutative geometry, the Pisot substitution conjecture, aperiodic Schrödinger operators, and connections to arithmetic number theory.

Author : James F. Peters
Publisher : Springer Science & Business Media
Page : 414 pages
File Size : 49,9 Mb
Release : 2014-01-28
Category : Technology & Engineering
ISBN : 9783642538452

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Topology of Digital Images by James F. Peters Pdf

This book carries forward recent work on visual patterns and structures in digital images and introduces a near set-based a topology of digital images. Visual patterns arise naturally in digital images viewed as sets of non-abstract points endowed with some form of proximity (nearness) relation. Proximity relations make it possible to construct uniform topologies on the sets of points that constitute a digital image. In keeping with an interest in gaining an understanding of digital images themselves as a rich source of patterns, this book introduces the basics of digital images from a computer vision perspective. In parallel with a computer vision perspective on digital images, this book also introduces the basics of proximity spaces. Not only the traditional view of spatial proximity relations but also the more recent descriptive proximity relations are considered. The beauty of the descriptive proximity approach is that it is possible to discover visual set patterns among sets that are non-overlapping and non-adjacent spatially. By combining the spatial proximity and descriptive proximity approaches, the search for salient visual patterns in digital images is enriched, deepened and broadened. A generous provision of Matlab and Mathematica scripts are used in this book to lay bare the fabric and essential features of digital images for those who are interested in finding visual patterns in images. The combination of computer vision techniques and topological methods lead to a deep understanding of images.

Author : Martin W. Liebeck
Publisher : Cambridge University Press
Page : 505 pages
File Size : 42,7 Mb
Release : 1992-09-10
Category : Mathematics
ISBN : 9780521406857

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Groups, Combinatorics and Geometry by Martin W. Liebeck Pdf

This volume contains a collection of papers on the subject of the classification of finite simple groups.

Author : H. S. M. Coxeter,Harold Scott Macdonald Coxeter
Publisher : Courier Corporation
Page : 301 pages
File Size : 49,7 Mb
Release : 1999-01-01
Category : Mathematics
ISBN : 9780486409191

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The Beauty of Geometry by H. S. M. Coxeter,Harold Scott Macdonald Coxeter Pdf

Absorbing essays demonstrate the charms of mathematics. Stimulating and thought-provoking treatment of geometry's crucial role in a wide range of mathematical applications, for students and mathematicians.