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.