Internal and external duality in abstract polytopes

TL;DR

This paper introduces the concept of internal self-duality in abstract regular polytopes, explores its structural implications, and constructs numerous examples demonstrating the existence of internally and externally self-dual polytopes across various types and ranks.

Contribution

It defines internal self-duality as a symmetry-realizable form of self-duality, and provides the first extensive construction of such polytopes, including new families across different ranks.

Findings

Existence of internally self-dual regular polyhedra of each type {p,p} for p ≥ 3.

Presence of infinitely many internally and externally self-dual polyhedra of type {p,p} for even p.

Construction of new families of internally self-dual polytopes in each rank.

Abstract

We define an abstract regular polytope to be internally self-dual if its self-duality can be realized as one of its symmetries. This property has many interesting implications on the structure of the polytope, which we present here. Then, we construct many examples of internally self-dual polytopes. In particular, we show that there are internally self-dual regular polyhedra of each type $\{p, p\}$ for $p \geq 3$ and that there are both infinitely many internally self-dual and infinitely many externally self-dual polyhedra of type $\{p, p\}$ for $p$ even. We also show that there are internally self-dual polytopes in each rank, including a new family of polytopes that we construct here.