Old and New Topics in Geometry: Volume I: Projective, Neutral and Basic Euclidean Geometry

By Franz Rothe

The present first volume begins with Hilbert's axioms from the \emph{Foundations of Geometry}.

After some discussion of logic and axioms in general, incidence geometries, especially the finite ones, and affine and projective geometry in two and three dimensions are treated. As in Hilbert's system, there follow sections about the

• Language: English
• Publisher: LitPrime Solutions
• Release date: May 8, 2023
• ISBN: 9798887032528

Franz Rothe

Franz Rothe graduated from high school in Karlsruhe and studied mathematics, physics, and music there. Graduated with a diploma in mathematics from E T H Zürich, a doctorate in Tübingen. After some changes in life, a professorship at the University of North Carolina at Charlotte, USA.In addition, Rothe and pianist Thomas Turner have developed a repertoire of classical music for flute and piano, and have recorded and released three CDs. This collection also contains several of their own transcriptions. Now Rothe is retired and keeps writing books about mathematics, and too, just for entertainment.

Book preview

FC.jpg

Contents

I Incidence, Affine and Projective Geometry

I.2 Hilbert’s Axioms of Geometry

I.2.1 Logic

I.2.2 David Hilbert’s axiomatization of Euclidean geometry

I.2.3 Importance and Impact of Hilbert’s Foundations of Geometry

I.2.4 Frege’s Critique and Hilbert’s answer

I.2.5 About the consistency proof for geometry

I.2.6 General remark about models in mathematics

I.2.7 What is completeness?

I.2.8 More metamathematical considerations

I.3 Incidence Geometry

I.3.1 Elementary propositions about incidence planes

I.3.2 Finite incidence geometries

I.3.3 Affine incidence planes

I.3.4 Introduction of coordinates

I.3.5 Finite coordinate planes

I.3.6 Projective incidence planes

I.3.7 The Fano Plane

I.3.8 Projective plane with coordinates

I.3.9 Finite affine and projective incidence planes

I.3.10 Elementary propositions for three-dimensional incidence spaces

I.3.11 Three-dimensional Euclidean incidence geometry

I.4 The Theorems of Desargues and Pappus

I.4.1 Desargues’ Theorem

I.4.2 Theorem of Desargues and related theorems in projective setting

I.4.3 Tiling in perspective view

I.4.4 The Prime Power Conjecture about Non-Desarguesian Planes

I.4.5 Hilbert’s investigation about the Theorem of Desargues

I.4.6 TheMouton plane

I.4.7 Theorem of Pappus and related theorems

I.4.8 Theorem of Hessenberg

I.4.9 Relations to the Little Theorems

I.5 Finite Affine and Projective Incidence Planes and Latin Squares

I.5.1 Latin squares

I.5.2 Latin squares from finite fields

I.5.3 Finite Non-Desarguesian planes

I.5.4 A note on projective spaces

II Neutral Geometry

II.6 The Axioms of Order and Their Consequences

II.6.1 Order of points on a line

II.6.2 Bernays’ Lemma

II.6.3 Plane separation

II.6.4 Four-point and n-point Theorems

II.6.5 Angles

II.6.6 Space separation

II.6.7 Interior and exterior of a triangle

II.6.8 Convexity

II.6.9 Topology of the ordered incidence plane

II.6.10 Left and right, orientation

II.6.11 The restricted Jordan Curve Theorem

II.7 Congruence of Segments, Angles and Triangles

II.7.1 Congruence of segments

II.7.2 Some elementary triangle congruences

II.7.3 Congruence of angles

II.7.4 SSS congruence

II.7.5 Right, acute and obtuse angles

II.7.6 Constructions with Hilbert tools

II.7.7 Remarks about angles

II.7.8 Orientated angles

II.7.9 The exterior angle theorem

II.7.10 Congruence of z-angles

II.7.11 Consequences of the exterior angle theorem

II.7.12 SSA congruence

II.7.13 Reflection

II.7.14 Applied problems

II.7.15 Independence of the SAS-axiom

II.7.16 TheMoulton plane

II.7.17 Restriction of SAS congruence

II.8 Measurement and Continuity

II.8.1 The Archimedean axiom

II.8.2 Axioms related to completeness

II.9 Legendre’s Theorems

II.9.1 The First Legendre Theorem

II.9.2 The Second Legendre Theorem

II.9.3 The alternative of two geometries

II.9.4 What is the natural geometry?

II.10 Neutral Geometry of Circles and Continuity

II.10.1 Immediate consequences of neutral geometry

II.10.2 The tangent is the limiting position of a secant

II.10.3 Mutual placement of two circles

II.10.4 Continuity principles for circles

II.10.5 Continuity principles for circles are independent of Hilbert’s axioms

II.10.6 Derivation of continuity principles from Dedekind’s axiom

II.11 Towards a Natural Axiomatization of Geometry

II.11.1 The Uniformity Theorem

II.11.2 Some strange polygons

II.11.3 Defect and AAA congruence

II.11.4 A hierarchy of planes