This book draws on elements from everyday life, architecture, and the arts to provide the reader with elementary notions of geometric topology. Pac Man, subway maps, and architectural blueprints are the starting point for exploring how knowledge about geometry and, more specifically, topology has been consolidated over time, offering a learning journey that is both dense and enjoyable. The text begins with a discussion of mathematical models, moving on to Platonic and Keplerian theories that explain the Cosmos. Geometry from Felix Klein's point of view is then presented, paving the way to an introduction to topology. The final chapters present the concepts of closed, orientable, and non-orientable surfaces, as well as hypersurface models. Adopting a style that is both rigorous and accessible, this book will appeal to a broad audience, from curious students and researchers in various areas of knowledge to everyone who feels instigated by the power of mathematics in representing our world - and beyond.
Author(s): Ton Marar
Publisher: Springer
Year: 2022
Language: English
Pages: 123
City: Cham
Foreword
Preface
Contents
1 Mathematical Models
References
2 Ancient Greek Big Bang Theory
2.1 Platonic Solids
2.2 Arithmetic, Geometric and Harmonic Means
2.3 The Golden Proportion
2.3.1 Golden Rectangles and Architecture
2.3.2 The Golden Number in the Regular Pentagon
2.3.3 Golden Rectangles and the Sum of Squares
2.3.4 The Golden Number in Trigonometry
2.3.5 The Golden Number and the Quasicrystals
2.3.6 The Golden Number and the Means
2.3.7 Golden Rectangles and the Regular Dodecahedron
2.3.8 The Golden Number and the Fibonacci Sequence
2.3.9 Johannes Kepler
References
3 Geometry: From Disorder to Order
3.1 Euclidean
3.2 Euclidean and Non-Euclidean
3.3 Felix Klein
3.4 Points at Infinity
References
4 Topology
4.1 A Kind of Geometry
4.1.1 A Brief History
4.1.2 Cutting and Gluing
4.1.3 Basic Surfaces
4.1.4 Connected Sum of Surfaces
4.1.5 The Fundamental Identity
4.1.6 Planar Models
4.1.7 Word Representation of Surfaces
4.2 Topological Classification of Surfaces
4.3 Surface Identification
4.3.1 Tripartite Unity
4.3.2 An Olympic Surface
4.3.3 Surfaces with a Quadrilateral Planar Model
4.3.4 Francis Torus
4.3.5 Surfaces with a Planar Model with Six Edges
4.4 Shape of Objects
References
5 The Fourth Dimension
5.1 Flatland and Spaceland
5.2 Beyond the Third Dimension
5.3 Four-Dimensional Place
5.4 The Fourth Dimension in Arts and Literature
References
6 Non-orientable Surfaces
6.1 Model Making in Three-Dimensional Space
6.2 The Sphere with a Cross-Cap
6.3 The Steiner Roman Surface
6.4 The Boy Surface
6.5 Non-orientable Surfaces in Other Areas
References
7 Hypersurfaces
7.1 From Surfaces to Hypersurfaces
7.2 On the Shape of the Universe
7.3 Three-Dimensional Objects
7.3.1 The Hypersphere
7.3.2 Regular Hyperpolyhedra
7.3.3 The 3-Torus
7.4 A Model of the Universe
References
