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Construction of Toroidal Polyhedra corresponding to perfect Chains of wild Tetrahedra

Reymond Akpanya, Vanishree Krishna Kirekod, Alice C. Niemeyer, Daniel Robertz

Abstract

In 1957, Steinhaus proved that a chain of regular tetrahedra, meeting face-to-face and forming a closed loop does not exist. Over the years, various modifications of this statement have been considered and analysed. Weakening the statement by only requiring the tetrahedra of a chain to be wild, i.e. having all faces congruent, results in various examples of such chains. In this paper, we elaborate on the construction of these chains of wild tetrahedra. We therefore introduce the notions of chains and clusters of wild tetrahedra and relate these structures to simplicial surfaces. We establish that clusters and chains of wild tetrahedra can be described by polyhedra in Euclidean 3-space. As a result, we present methods to construct toroidal polyhedra arising from chains and provide a census of such toroidal polyhedra consisting of up to 20 wild tetrahedra. Here, we classify toroidal polyhedra with respect to self-intersections and reflection symmetries. We further prove the existence of an infinite family of toroidal polyhedra emerging from chains of wild tetrahedra and present clusters of wild tetrahedra that yield polyhedra of higher genera.

Construction of Toroidal Polyhedra corresponding to perfect Chains of wild Tetrahedra

Abstract

In 1957, Steinhaus proved that a chain of regular tetrahedra, meeting face-to-face and forming a closed loop does not exist. Over the years, various modifications of this statement have been considered and analysed. Weakening the statement by only requiring the tetrahedra of a chain to be wild, i.e. having all faces congruent, results in various examples of such chains. In this paper, we elaborate on the construction of these chains of wild tetrahedra. We therefore introduce the notions of chains and clusters of wild tetrahedra and relate these structures to simplicial surfaces. We establish that clusters and chains of wild tetrahedra can be described by polyhedra in Euclidean 3-space. As a result, we present methods to construct toroidal polyhedra arising from chains and provide a census of such toroidal polyhedra consisting of up to 20 wild tetrahedra. Here, we classify toroidal polyhedra with respect to self-intersections and reflection symmetries. We further prove the existence of an infinite family of toroidal polyhedra emerging from chains of wild tetrahedra and present clusters of wild tetrahedra that yield polyhedra of higher genera.

Paper Structure

This paper contains 13 sections, 7 theorems, 44 equations, 14 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Theoretical background
  3. Simplicial surfaces
  4. Embedding simplicial surfaces into $\mathbb{R}^3$
  5. Clusters of wild tetrahedra
  6. Relation between clusters of wild tetrahedra and simplicial surfaces
  7. Constructing embeddings of multi-tetrahedral surfaces
  8. Constructing toroidal polyhedra
  9. Toroidal polyhedra with reflection symmetry
  10. Toroidal polyhedra without reflection-symmetry
  11. Infinite family of multi-tetrahedral tori
  12. Multi-tetrahedral surfaces of higher genera
  13. Notes on implementation

Key Result

Theorem 1

There exists a perfect chain of wild tetrahedra without self-intersections consisting of $14$ tetrahedra.

Figures (14)

  • Figure 1: Chains of regular tetrahedra consisting of three (a) and six (b) regular tetrahedra.
  • Figure 2: Various views of a perfect chain of wild tetrahedra with $14$ tetrahedra.
  • Figure 3: The simplicial tetrahedron (a) and a wild-coloured simplicial tetrahedron with red, blue and green edges (b).
  • Figure 4: Polyhedron arising from a $(1,\tfrac{9}{10},\tfrac{3}{4})$-embedding of the wild-coloured simplicial tetrahedron
  • Figure 5: Various views of two chains ((a) and (b)) and a cluster (c) of wild tetrahedra consisting of four regular tetrahedra.
  • ...and 9 more figures

Theorems & Definitions (46)

  • Theorem
  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 36 more