Imaginary elements in geometry : according to Von Staudt and Klein

Format:

Book

Author/Creator:

Boer

Language:

English

Physical Description:

1 online resource (180 pages)

Edition:

1st ed.

Place of Publication:

Verlag am Goetheanum 2024

Summary:

A remarkable fact in mathematics is the accordance between algebra and geometry: since the time of Descartes it is possible to express geometric phenomena in terms of numbers. And after doing calculations with these numbers, we can draw geometric conclusions from them. However, soon it appeared that, for instance, a circle and a line outside that circle have 'imaginary' meeting points: points that have imaginary coordinates but can not be found in the figure. Karl von Staudt found a brilliant way to visualize imaginary geometric points, lines and planes, and Felix Klein simplified and extended his method. In this book imaginary elements appear as Klein-movements in our real space. Apart from its mathematical importance, these movements may be seen as an alternative for the remarkable occurrences of complex numbers in physics, viz. quantum physics.

Contents:

Cover

Impressum

Preface

Contents

I 1-dimensional geometry

1 The theory of Von Staudt

1.1 The common points of conic and line

1.2 Orientation of the line

1.3 Imaginary points

1.4 Coordinates

2 The theory of Klein

2.1 Definition

2.2 Coordinates

2.3 Klein versus Von Staudt

3 Groups of automorphisms

3.1 Elliptic maps

3.2 Drawback

3.3 Application to physics

4 Constructions

4.1 The fundamental construction

4.2 The dual construction

4.3 The fundamental construction in space

4.4 Construction of an imaginary point

4.5 Construction of an imaginary line

4.6 Construction of an involution

4.7 Constructing a Klein-map with a conic

II 2-dimensional geometry

5 The Klein-plane

5.1 Imaginary elements

5.2 Ordering

5.3 The Staudt-plane

6 The numerical plane

7 The bijection

7.1 The coordinate map κ

7.2 Consistency

7.3 The inverse of κ

7.4 The invariance of ≺

7.5 More objections

8 Conic and line

8.1 Orientation on a conic

8.2 Von Staudt's method

8.3 Klein's method

8.4 An example

8.5 Conversion Klein-Staudt

9 Extension to the plane?

9.1 The cubic

9.2 W-curves

III 3-dimensional geometry

10 The numerical space

10.1 Definitions

10.2 Plücker-coordinates

10.3 Ordering, join, meet

10.4 Types of lines

10.5 Projective maps

10.6 The linear congruence

11 The Klein-space

11.1 The synthetic space

11.2 Low imaginary elements

11.3 The high imaginary line

11.4 The matrix of a high imaginary line

11.5 Sets and numbers

11.6 Ordering

11.7 Meet and join

11.8 Is our Klein-space a projective one?

12 The bijection for 3-d

12.1 Point and plane coordinates

12.2 The coordinates of a line

12.3 The inverse of κ

13 Invariance of dim and ≺

13.1 Dimension

13.2 Containment.

14 W-curves in space

IV Appendix

15 Klein's original text

16 Projective spaces

16.1 Definition of 'projective space'

16.2 Subspaces

16.3 Isomorphic spaces

16.4 Eigenspaces

17 Geometry of the line

17.1 Orientation

17.2 Separation

17.3 Cross Ratio

17.4 Maps of the real line

17.5 Maps of the complex line

17.6 Splitting matrices

17.7 The standard elliptic map

18 The Klein-arrow

18.1 Arrow and twist in the plane

18.2 Extension to space

19 Various

19.1 Non-integer powers of a matrix

19.2 An image of the linear congruence

19.3 Postponed proofs

19.4 Three non-concurrent lines in the plane

19.5 The projective image of a line

Bibliography

List of symbols

Index

Rückseite Umschlag.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

3-7235-1747-1

OCLC:

1526481166