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Desargues's and Pappus's hexagon theorems on translation-type surfaces in Thurston geometries

Jenő Szirmai

Abstract

In \cite{Sz25} we generalized the famous Menelaus' and Ceva's theorems for translation triangles in each non-constant curvature Thurston geometry. In this paper based on the described method and results, we prove that the classical Desargues's and Pappus's hexagon theorems are true not only in classical geometries with constant curvature, but also in Thurston geometries with non-constant curvature on the translation surfaces. In our work we use the unified projective models of Thurston geometries.

Desargues's and Pappus's hexagon theorems on translation-type surfaces in Thurston geometries

Abstract

In \cite{Sz25} we generalized the famous Menelaus' and Ceva's theorems for translation triangles in each non-constant curvature Thurston geometry. In this paper based on the described method and results, we prove that the classical Desargues's and Pappus's hexagon theorems are true not only in classical geometries with constant curvature, but also in Thurston geometries with non-constant curvature on the translation surfaces. In our work we use the unified projective models of Thurston geometries.
Paper Structure (9 sections, 9 theorems, 12 equations, 3 figures)

This paper contains 9 sections, 9 theorems, 12 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Pappus's and Desargues's theorems for translation triangles in $\mathbf{Nil}$ geometry
  3. On the projective model of $\mathbf{Nil}$ geometry
  4. Translation curves
  5. The surfaces of translation triangles and generalizations of Menelaus' theorem
  6. Desargues's theorem in $\mathbf{Nil}$ geometry
  7. Pappus's hexagon theorem in $\mathbf{Nil}$ geometry
  8. The extensions of the Desargues's and Pappus's hexagon theorems
  9. Extension of the theorems of Pappus and Desargues to additional Thurston geometries

Key Result

Lemma 2.1

$\square$

Figures (3)

  • Figure 1: Desargues's theorem on translation-type surface in $\mathbf{Nil}$ geometry.
  • Figure 2: Pappus's hexagon theorem on translation-type surface in $\mathbf{Nil}$ geometry.
  • Figure 3: The proof of the Pappus's hexagon theorem using Menelaus' theorem on $\mathbf{Nil}$ translation-type surface.

Theorems & Definitions (18)

  • Remark 1.1
  • Lemma 2.1
  • Remark 2.2
  • Definition 2.3: Sz25
  • Lemma 2.4: Sz25
  • Definition 2.5: Sz25
  • Theorem 2.6: Menelaus's theorem for translation triangles in $\mathbf{Nil}$ space Sz25
  • Remark 2.7
  • Definition 2.1
  • Theorem 2.8: Desargues's theorem in $\mathbf{Nil}$ space
  • ...and 8 more