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

TL;DR

The paper resolves the existence of self-dual regular 3-polytopes of Schläfli type $\{n,n\}$ for every $n\ge 3$ by constructing a semidirect-product automorphism group $\operatorname{Aut}(\mathcal{P})\cong \mathbb{F}_2^{\,n-1} \rtimes D_{2n}$ and a rank-3 string C-group generated by $\rho_0,\rho_1,\rho_2$ with type $[n,n]$, yielding a polytope with $2^n n$ flags. The construction uses a faithful $D_{2n}$-action on $\mathbb{F}_2^{n-1}$ realized through matrices $U,V$, and an explicit presentation for the automorphism group is derived. Self-duality follows from an automorphism $\delta$ exchanging $\rho_0$ and $\rho_2$, ensuring the polytope is isomorphic to its dual. As consequences, the results provide, for all odd $m\ge 3$, self-dual polytopes of type $\{m,m\}$ with $2^m m$ flags, and, for primes $p\ge 2$ and any $s\ge 0$, polytopes of type $\{2^s p,2^s p\}$ with $2^{(2^s p+s)}p$ flags, unifying odd and even Schläfli entries within a unified framework.

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 Schläfli 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.

Theorems & Definitions (18)

• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Theorem 2.3: MS
• Proposition 2.4
• Proposition 2.5
• proof
• Lemma 3.1
• proof
• Lemma 3.2