Eschenburg J.-H. Geometry - Intuition and Concepts: Imagining, understanding, thinking beyond. An introduction for students

Wiesbaden: Springer, 2022. — 168 p.This book deals with the geometry of visual space in all its aspects. As in any branch of mathematics, the aim is to trace the hidden to the obvious; the peculiarity of geometry is that the obvious is sometimes literally before one's eyes.Starting from intuition, spatial concepts are embedded in the pre-existing mathematical framework of linear algebra and calculus. The path from visualization to mathematically exact language is itself the learning content of this book. This is intended to close an often lamented gap in understanding between descriptive preschool and school geometry and the abstract concepts of linear algebra and calculus. At the same time, descriptive geometric modes of argumentation are justified because their embedding in the strict mathematical language has been clarified.
The concepts of geometry are of a very different nature; they denote, so to speak, different layers of geometric thinking: some arguments use only concepts such as point, straight line, and incidence, others require angles and distances, still others symmetry considerations. Each of these conceptual fields determines a separate subfield of geometry and a separate chapter of this book, with the exception of the last-mentioned conceptual field "symmetry", which runs through all the others:
- Incidence: Projective geometry - Parallelism: Affine geometry - Angle: Conformal Geometry - Distance: Metric Geometry - Curvature: Differential Geometry - Angle as distance measure: Spherical and Hyperbolic Geometry - Symmetry: Mapping Geometry.
The mathematical experience acquired in the visual space can be easily transferred to much more abstract situations with the help of the vector space notion. The generalizations beyond the visual dimension point in two directions: Extension of the number concept and transcending the three illustrative dimensions.
This book is a translation of the original German 1st edition Geometrie – Anschauung und Begriffe by Jost-Hinrich Eschenburg, published by Springer Fachmedien Wiesbaden GmbH, part of Springer Nature in 2020. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors.What Is Geometry?
Parallelism: Affine Geometry
From Affine Geometry to Linear Algebra
Definition of the Affine Space
Parallel and Semi-Affine Mappings
Parallel Projections
Affine Representations, Ratio, Center of Gravity
Incidence: Projective Geometry
Central Perspective
Points at Infinity and Projection Lines
Projective and Affine Space
Semiprojective Mappings and Collineations
Theorem of Desargues
Conic Sections and Quadrics; Homogenization
Theorem of Brianchon
Duality and Polarity; Pascal's Theorem
Projective Determination of Quadrics
The Cross-Ratio
Distance: Euclidean Geometry
The Pythagorean Theorem
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Isometries of Euclidean Space
Classification of Isometries
Platonic Solids
Symmetry Groups of Platonic Solids
Finite Subgroups of the Orthogonal Group, Patterns, and Crystals
Metric Properties of Conic Sections
Curvature: Differential Geometry
Smoothness
Fundamental Forms and Curvatures
Characterization of Spheres and Hyperplanes
Orthogonal Hypersurface Systems
Angle: Conformal Geometry
Conformal Mappings
Inversions
Conformal and Spherical Mappings
The Stereographic Projection
The Space of Spheres
Möbius and Lie Geometry of Spheres
Angular Distance: Spherical and Hyperbolic Geometry
Hyperbolic Space
Distance on the Sphere and in Hyperbolic Space
Models of Hyperbolic Geometry
Exercises
Affine Geometry (Chap.2)
Projective Geometry (Chap.3)
Euclidean Geometry (Chap.4)
Differential Geometry (Chap.5)
Conformal Geometry (Chap.6)
Spherical and Hyperbolic Geometry (Chap.7)
Solutions
Literature (Small Selection)
Mainly Used Literature
Classics
Historical Works
Geometry and Art
Some Textbooks