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The 10 antipodal pairings of strongly involutive polyhedra

Javier Bracho, Eric Paulí Pérez, Luis Montejano, Jorge Luis Ramírez-Alfonsín

Abstract

It is known that strongly involutive polyhedra are closely related to self-dual maps where the antipodal function acts as duality isomorphism. Such a family of polyhedra appears in different combinatorial, topological and geometric contexts, and is thus attractive to be studied. In this note, we determine the 10 antipodal pairings among the classification of the 24 self-dual pairings $Dual(G)\rhd Aut(G)$ of self-dual maps G. We also present the orbifold associated to each antipodal pairing and describe explicitly the corresponding fundamental regions. We finally explain how to construct two infinite families of strongly involutive polyhedra (one of them new) by using their doodles and the action of the corresponding orbifolds.

The 10 antipodal pairings of strongly involutive polyhedra

Abstract

It is known that strongly involutive polyhedra are closely related to self-dual maps where the antipodal function acts as duality isomorphism. Such a family of polyhedra appears in different combinatorial, topological and geometric contexts, and is thus attractive to be studied. In this note, we determine the 10 antipodal pairings among the classification of the 24 self-dual pairings of self-dual maps G. We also present the orbifold associated to each antipodal pairing and describe explicitly the corresponding fundamental regions. We finally explain how to construct two infinite families of strongly involutive polyhedra (one of them new) by using their doodles and the action of the corresponding orbifolds.
Paper Structure (9 sections, 2 theorems, 4 equations, 11 figures, 1 table)

This paper contains 9 sections, 2 theorems, 4 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Antipodal pairings
  3. Spherical isometries
  4. Detecting antipodal pairings
  5. Orbifolds
  6. Constructing strongly involutive maps
  7. The $q$-multi hyperwheel
  8. The $q$-multi wheel
  9. Concluding Remarks

Key Result

Theorem 1

Every strongly involutive map corresponds to one of the following self-dual pairings: $[q]\triangleleft [2,q],[q]^{+}\triangleleft [2,q^{+}]$ for $q$ even; ${[q]\triangleleft [2^{+},2q]}, {[q]^{+}\triangleleft [2^{+},2q^{+}]}$ for $q$ odd, $[2,2]^{+}\triangleleft [2,2], [2^{+},4]\triangleleft [2,4],

Figures (11)

  • Figure 1: (Left) 5-wheel. (Right) 4-hyperwheel.
  • Figure 2: Reflection groups.
  • Figure 3: Zig-zag described by $(\rho\gamma)^i$ in $[2^{+},6]$. The exterior dotted circle represents one of the poles, the central black circle the other pole and the middle dotted circle the equator.
  • Figure 4: The 4-multi hyperwheel with 3 levels $\mathcal{O}_4^3$.
  • Figure 5: (Left) A fundamental region (shaded) of $[4]\triangleleft [2,4]$. (Right) The doodle of the 4-hyperwheel.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 1
  • proof
  • Remark 1