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

Mingchao Li; Wei-Juan Zhang

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arXiv · 2505.09505

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

Mingchao Li·Wei-Juan Zhang

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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.

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Mingchao Li, Wei-Juan Zhang. 2025-05-14. Regular 3-polytopes of type $\{n,n\}$. https://arxiv.org/abs/2505.09505

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