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The orthogonal connectedness of polyhedral surfaces

Julia Q. Du, Xuemei He, Xiaotian Song, Daniela Stiller, Liping Yuan, Tudor Zamfirescu

Abstract

Using the orthogonal connectedness, we introduce the notion of orthogonal decomposability of convex polytopes and study it in the case of Platonic and Archimedean solids. While doing so, we also encounter polytopes which are not orthogonally decomposable.

The orthogonal connectedness of polyhedral surfaces

Abstract

Using the orthogonal connectedness, we introduce the notion of orthogonal decomposability of convex polytopes and study it in the case of Platonic and Archimedean solids. While doing so, we also encounter polytopes which are not orthogonally decomposable.
Paper Structure (7 sections, 17 theorems, 11 figures)

This paper contains 7 sections, 17 theorems, 11 figures.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. The orthogonal connectedness of polyhedral surfaces
  4. Orthogonal decomposability of Platonic solids
  5. Orthogonal decomposability of Archimedean solids
  6. Orthogonal non-decomposability
  7. Epilogue

Key Result

Theorem 3.1

If the boundary of the simple polytope $P\subset \mathds{R}^{3}$ is orthogonally connected, then $P$ is rich and $\mathcal{G}_{P}$ is connected.

Figures (11)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 6:
  • ...and 6 more figures

Theorems & Definitions (31)

  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • proof
  • Corollary 4.4
  • Lemma 5.1
  • ...and 21 more