Math and Art

An Introduction to Visual Mathematics

Sasho Kalajdzievski

• Publisher: Chapman and Hall/CRC
• Year: 2021
• Print ISBN:
  9780367076115
  9780367076139
• Edition: 2
• eBook ISBN:
  9780429664427
• Topic: Mathematics
• Subtopic: Art Techniques
• Index: Mathematics

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. It includes numerous illustrations, computer-generated graphics, photographs, and art reproductions to demonstrate how mathematics can inspire or generate art.

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.

Features

• Provides an accessible introduction to mathematics in art
• Supports the narrative with a self-contained mathematical theory, with complete proofs of the main results (including the classification theorem for similarities)
• Presents hundreds of figures, illustrations, computer-generated graphics, designs, photographs, and art reproductions, mainly presented in full color
• Includes 21 projects and approximately 280 exercises, about half of which are fully solved
• Covers Euclidean geometry, golden section, Fibonacci numbers, symmetries, tilings, similarities, fractals, cellular automata, inversion, hyperbolic geometry, perspective drawing, Platonic and Archimedean solids, and topology

New to the Second Edition

• New exercises, projects and artworks
• Revised, reorganized and expanded chapters

• More use of color throughout

CHAPTER 1 Euclidean Geometry

DOI: 10.1201/9780429021602-1

1.0INTRODUCTION

The history of ages teaches us that virtually nothing in this world changes linearly, and that nothing expands uniformly in time. Human civilizations are no exception; there were times when they grew in their scope and depth, and there were dark epochs when calamities, catastrophes, and cataclysms left magnificent ancient cities in ruins. These ruins, sometimes buried under desert sand, sometimes covered by jungle vegetation, and sometimes inundated by rising seas and oceans, are silent witnesses of the existence of ancient civilizations with superior practical knowledge of rules of geometry (Figures 1.0.1 and 1.0.2).

Figure 1.0.1 The sphinx and the great pyramid, both of undetermined antiquity, photo by Hajor, 2002.

Figure 1.0.2 The temple of Quetzalcoatl, also known as the step pyramid of Chichen Itza.

According to orthodox archeology, the onset of modern civilization happened at about 3500 BC, in Sumer, situated in the fertile lands between the rivers Tigris and Euphrates. There are relatively few written records left of the Sumerian civilization, but we do know that those people possessed at least a sophisticated practical knowledge of geometry. For example, the statement of (what we now call) the Pythagorean theorem was discovered on a Babylonian tablet dated a long time ago, sometime around 1900 BC.

It was the ancient Greeks who made the next giant step in the direction of better understanding and interpreting the abstract and material universe. Greek philosophers were virtually obsessed with the beauty of geometrical objects and their properties, and they found in geometry a perfect medium for abstract analysis of various phenomena.¹ The above-mentioned Pythagorean theorem is a case in point: Pythagoras (fifth century BC) was a Greek geometer, mathematician, and philosopher, who realized that it is inadequate to accept the validity of properties just by empirical observations or measurements. For example, checking that the square of the hypotenuse of a right-angled triangle is the sum of the squares of the other two sides by simple measurements in many triangles is unsatisfactory, because it does not preclude the possibility of existence of some exotic triangle where that property fails. Pythagoras wanted to understand why that property is satisfied. Thus, the notion of proof was born.

A visual proof of the Pythagorean theorem is given in illustrated Steps 1 to 6 below.

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1 For example, according to Plato (as stated in his book Timaeus), the earth was made of tiny cubes, while the air was an aggregate of small tetrahedrons. ...

Table of contents

• Cover Page
• Half-Title Page
• Title Page
• Copyright Page
• Dedication Page
• Contents
• Introduction and Acknowledgements
• CHAPTER 1 ◾ EUCLIDEAN GEOMETRY
• CHAPTER 2 ◾ PLANE TRANSFORMATIONS
• CHAPTER 3 ◾ SIMILARITIES, FRACTALS, AND CELLULAR AUTOMATA
• CHAPTER 4 ◾ HYPERBOLIC GEOMETRY
• CHAPTER 5 ◾ PERSPECTIVE
• CHAPTER 6 ◾ SOME THREE-DIMENSIONAL OBJECTS
• CHAPTER 7 ◾ TOPOLOGY
• APPENDIX: CLASSIFICATION THEOREM FOR SIMILARITIES
• SOLUTIONS OF ODD-NUMBERED EXERCISES
• INDEX