Regular 3-polytopes of type $\{n,n\}$

TL;DR

This paper constructs a family of self-dual regular 3-polytopes of type n, n for all integers n 3, with explicit automorphism groups, solving key open questions in the field.

Contribution

It introduces a systematic method to construct regular 3-polytopes of type n, n with explicit automorphism groups, addressing previously unresolved existence questions.

Findings

Constructed self-dual regular 3-polytopes of type n, n for all n 3.

Explicitly realized automorphism groups as semidirect products involving dihedral groups.

Provided complete presentations for the automorphism groups.

Abstract

For each integer \( n \geq 3 \), we construct a self-dual regular 3-polytope \( \mathcal{P} \) of type \( \{n, n\} \) with \( 2^n n \) flags, resolving two foundamental open questions on the existence of regular polytopes with certain Schl\"afli types. The automorphism group \( \operatorname{Aut}(\mathcal{P}) \) is explicitly realized as the semidirect product \( \mathbb{F}_2^{n-1} \rtimes D_{2n} \), where \( D_{2n} \) is the dihedral group of order \( 2n \), with a complete presentation for \( \operatorname{Aut}(\mathcal{P}) \) is provided. This advances the systematic construction of regular polytopes with prescribed symmetries.