Geometry in Our Three-Dimensional World

✍ Scribed by Alfred S Posamentier, Bernd Thaller, Christian Dorner

• Publisher: World Scientific Publishing Company
• Year: 2021
• Tongue: English
• Leaves: 441
• Series: Problem Solving in Mathematics and Beyond 25
• Category: Library

✦ Synopsis

The book presents a comprehensive overview of various aspects of three-dimensional geometry that can be experienced on a daily basis. By covering the wide range of topics — from the psychology of spatial perception to the principles of 3D modelling and printing, from the invention of perspective by Renaissance artists to the art of Origami, from polyhedral shapes to the theory of knots, from patterns in space to the problem of optimal packing, and from the problems of cartography to the geometry of solar and lunar eclipses ― this book provides deep insight into phenomena related to the geometry of space and exposes incredible nuances that can enrich our lives.

The book is aimed at the general readership and provides more than 420 color illustrations that support the explanations and replace formal mathematical arguments with clear graphical representations.

Readership: All people interested in geometry and science. General audience with a basic knowledge of high-school math. Teachers seeking ideas for a modern approach to teaching geometry with applications.

✦ Table of Contents

Contents
Preface
About the Authors
Chapter 1 Our World is Three-Dimensional
1.1 Shortest Distance on the Globe
1.2 The Wrong Tilt of the Moon
1.3 Coordinates, Dimension, and Space
1.4 Philosophy of Space
1.5 Optical Illusions and the Limits of Spatial Imagination
Chapter 2 The Eight Basic Elements of Spatial Thinking
2.1 Spatial Skills — How Can They Help Us?
2.2 Visual System
2.3 Visual Perception
2.4 Spatial Ability
2.5 The Eight Basic Elements of Spatial Thinking
2.6 Element 1: Visualization
2.7 Element 2: Form Constancy
2.8 Element 3: Position in Space
2.9 Element 4: Transformation in Space
2.10 Element 5: Object Combination
2.11 Element 6: Dynamics
2.12 Element 7: Fine Motor Skills
2.13 Element 8: Spatial Orientation
2.14 Crosslinks to Mathematics and Writing/Reading
A. Appendix — Solutions of the Problems
A.1 Solution of the Visualization Problems
A.2 Solution of the Form Constancy Problems
A.3 Solution of the Position-in-Space Problems
A.4 Solution of the Transformation-in-Space Problems
A.5 Solution of the Object-Combination Problems
A.6 Solution of the Dynamics Problems
A.7 Solution of the Fine-Motor-Skills Problems
A.8 Solution of the Spatial-Orientation Problems
Chapter 3 Spatial Representation
3.1 Parallel Projection
3.2 Central Projection and Perspective
3.3 Perspective Drawing Using Vanishing Points
3.4 Two-Point Perspective
3.5 The Pinhole Camera
3.6 Optical Distortions in Photography
Chapter 4 The Geometry on a Sphere
4.1 History of Spherical Geometry
4.2 Basic Properties of the Sphere: Surface and Volume
4.3 Great Circles
4.4 Great Circles as Straight Lines on the Sphere
4.5 Measuring Distances and Angles
4.6 Axis and Poles
4.7 Regular Digons on the Sphere
4.8 Circumference and Area of a Lune
4.9 Spherical Triangles
4.10 The Area of a Spherical Triangle and Girard’s Theorem
4.11 Coordinates on the Sphere
4.12 Applications of Spherical Coordinates
4.13 Coordinates in Geography
4.14 Making Maps
4.15 Navigation and the Loxodrome
4.16 Properties of the Mercator Projection and Local Distortions
4.17 Orthographic and Stereographic Projection
4.18 Other Projections
Chapter 5 The Earth, the Sun and the Moon
5.1 Why the Earth is not Flat
5.2 Day and Night
5.3 A Formula for the Length of a Day
5.4 Seasons
5.5 Phases of the Moon
5.6 Solar Eclipse
5.7 Size of the Moon’s Shadow
5.8 Annular Solar Eclipse
5.9 Frequency of Eclipses
5.10 Lunar Eclipses
5.11 Summary
Chapter 6 Simple Solid Shapes
6.1 Pyramids
6.2 Platonic Solids
6.3 What Can Be Learned from Folding Vertices
6.4 Angular Defect and Euler Characteristic
6.5 Existence of Platonic Solids and the Golden Ratio
6.6 Duality
6.7 Kepler-Poinsot Solids
6.8 Archimedean Solids
6.9 Duals of Archimedean Solids
Chapter 7 How to Sew a Ball — A Polyhedral Approach
7.1 Ball Games
7.2 Platonic Solids and Archimedean Solids
7.3 The Four Common Balls and Their Associated Solids
7.4 The Soccer Ball
7.5 The Beach Ball
7.6 The Basketball
7.7 The Volleyball
Chapter 8 Patterns: Voronoi, Kindergarten, Economy, and Epidemics
8.1 Honeycomb Pattern
8.2 Voronoi Diagrams
8.3 Voronoi Diagrams in Three Dimensions
8.4 Applications of Voronoi Diagrams to Public Health and Economy
8.5 Algorithms for Creating Voronoi Diagrams
8.6 Delaunay Triangulation
8.7 Dynamic Construction of Voronoi Diagrams
8.8 Voronoi Diagrams in Nature and Art
8.9 Architecture and the Weaire–Phelan Structure
Chapter 9 Packing Problems
9.1 History of Containerization
9.2 What is a Packing Problem?
9.3 Packings in Infinite Space
9.4 The Best Way to Pack Spheres
9.5 The Kissing Problem
9.6 The Kepler Conjecture Solved
9.7 How Many Billiard Balls Fit into a Shipping Container?
9.8 Dense Packings Can Help Save the Climate
9.9 Three-Dimensional Packing Puzzles
9.10 Bin-Packing Algorithms
Chapter 10 Fascinating Shapes
10.1 Magic with Paper Strips
10.2 Experiments with Möbius Strips
10.3 The Topology of Paper Strips
10.4 Surfaces Without Borders
10.5 The Klein Bottle
10.6 Further Examples of One-Sidedness
10.7 The Real Projective Plane
Chapter 11 Manipulating Two Dimensions in a Three-Dimensional World — Origamiin Space
11.1 Flat-Folding and Non-Flat-Folding Origamis
11.2 Modular Origami and Polyhedra: Nets and Folding Units
11.3 Folding Curves and the World of Developable Surfaces
11.4 Some Origami Applications in Modern Technology
Chapter 12 Knot Theory
12.1 Knots in Everyday Life
12.2 The Gordian Knot
12.3 Practical Knots
12.4 Knotting Matters
12.5 Mathematical Knots
12.6 The Trefoil Knot
12.7 The Origins of Knot Theory
12.8 Are Atoms Composed of Knots?
12.9 Knot Diagrams and Reidemeister Moves
12.10 The Recognition Problem and Knot Invariants
12.11 Tricolorability of a Knot
12.12 Applications of Knot Theory
Chapter 13 Kinematics — The Geometry of Motion
13.1 Degrees of Freedom
13.2 Gruebler’s Equation, Robot-Arms and the Stewart-Gough Platform
13.3 A Fascinating Application: The Turbula® Mixer
13.4 Involute Gears
Chapter 14 The Art of Designing Three-Dimensional Models
14.1 Introduction
14.2 Creating Virtual Three-Dimensional Models
14.3 Congruence Transformations
14.4 Application: Modeling a “Shark Wheel”
14.5 Scaling Transformations
14.6 Boolean Operations
14.7 Application: Creating an Ambiguous Cylinder
14.8 Loft, Sweep, and Revolve
14.9 Application: Modeling a Klein Bottle
14.10 Giving Birth to the Virtual Model
Chapter 15 A View of a World with More Than Three Dimensions
15.1 Higher-Dimensional Spaces in Mathematics
15.2 The Physics of Space–Time
15.3 Geometry in Four Dimensions
15.4 Thinking in Analogies
15.5 Möbius strip in Flatland
15.6 A Four-Dimensional Shape: The Hypercube
15.7 Perspective View of a Hypercube
15.8 The Hypersphere
15.9 Cross-Sections of a Hypersphere
15.10 Stereographic Projection of a Hypersphere
15.11 The Hypersphere in Physics
15.12 The Hypersphere in Art
Index