Math and Art: An Introduction to Visual Mathematics

Description

Math and Art: An Introduction to Visual Mathematics explores the potential of mathematics to generate visually appealing objects and reveals some of the beauty of mathematics. With downloadable resources and a 16-page full-color insert, it includes numerous illustrations, computer-generated graphics, photographs, and art reproductions to demonstrate how mathematics can inspire art. Basic Math Topics and Their Visual Aspects Focusing on accessible, visually interesting, and mathematically relevant topics, the text unifies mathematics subjects through their visual and conceptual beauty. Sequentially organized according to mathematical maturity level, each chapter covers a cross section of mathematics, from fundamental Euclidean geometry, tilings, and fractals to hyperbolic geometry, platonic solids, and topology. For art students, the book stresses an understanding of the mathematical background of relatively complicated yet intriguing visual objects. For science students, it presents various elegant mathematical theories and notions. Comprehensive Material for a Math in Art Course Providing all of the material for a complete one-semester course on mathematics in art, this self-contained text shows how artistic practice with mathematics and a comprehension of mathematical concepts are needed to logically and creatively appreciate the field of mathematics. Cover Page Half-Title Page Title Page Copyright Page Dedication Page Contents Introduction and Acknowledgements CHAPTER 1 ◾ EUCLIDEAN GEOMETRY 1.0 INTRODUCTION 1.1 THE FIVE AXIOMS OF EUCLIDEAN GEOMETRY 1.2 RULER AND COMPASS CONSTRUCTIONS 1.3 THE GOLDEN RATIO 1.4 FIBONACCI NUMBERS CHAPTER 2 ◾ PLANE TRANSFORMATIONS 2.1 PLANE SYMMETRIES 2.2 PLANE SYMMETRIES, VECTORS, AND MATRICES (OPTIONAL) 2.3 GROUPS OF SYMMETRIES OF PLANAR OBJECTS 2.4 FRIEZE PATTERNS 2.5 WALLPAPER DESIGNS AND TILINGS OF THE PLANE 2.6 TILINGS AND ART CHAPTER 3 ◾ SIMILARITIES, FRACTALS, AND CELLULAR AUTOMATA 3.1 SIMILARITIES AND SOME OTHER PLANAR TRANSFORMATIONS 3.2 COMPLEX NUMBERS AND TRANSFORMATIONS (OPTIONAL) 3.3 FRACTALS: DEFINITION AND SOME EXAMPLES 3.4 JULIA SETS 3.5 CELLULAR AUTOMATA CHAPTER 4 ◾ HYPERBOLIC GEOMETRY 4.1 NON-EUCLIDEAN GEOMETRIES: BACKGROUND AND SOME HISTORY 4.2 INVERSION 4.3 HYPERBOLIC GEOMETRY 4.4 SOME BASIC CONSTRUCTIONS IN THE POINCARÉ MODEL 4.5 TILINGS OF THE HYPERBOLIC PLANE CHAPTER 5 ◾ PERSPECTIVE 5.1 PERSPECTIVE: A BRIEF OVERVIEW OF THE EVOLUTION OF THE RULES OF PERSPECTIVE 5.2 PERSPECTIVE DRAWING AND CONSTRUCTIONS OF SOME TWO-DIMENSIONAL (PLANAR) OBJECTS 5.3 PERSPECTIVE IMAGES OF THREE-DIMENSIONAL OBJECTS 5.4 MATHEMATICS OF PERSPECTIVE DRAWING: A BRIEF OVERVIEW (OPTIONAL) CHAPTER 6 ◾ SOME THREE-DIMENSIONAL OBJECTS 6.1 REGULAR AND OTHER POLYHEDRA 6.2 SPHERE, CYLINDER, CONE, AND CONIC SECTIONS 6.3 GEOMETRY, TILINGS, FRACTALS, AND CELLULAR AUTOMATA IN THREE AND HIGHER DIMENSIONS CHAPTER 7 ◾ TOPOLOGY 7.1 HOMOTOPY OF SPACES: AN INFORMAL INTRODUCTION 7.2 TWO-MANIFOLDS AND THE EULER CHARACTERISTIC 7.3 NON-ORIENTABLE TWO-MANIFOLDS AND THREE-MANIFOLDS APPENDIX: CLASSIFICATION THEOREM FOR SIMILARITIES SOLUTIONS OF ODD-NUMBERED EXERCISES INDEX