Regular 3-polytopes of type \n,n\

2025/05/14 by Mingchao Li, Li, Mingchao, Wei-Juan Zhang +1
Engineering · Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2505.09505

openalex publication_date 2025/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For each integer \( n ≥ 3 \), we construct a self-dual regular 3-polytope \( P \) of type \( \n, n\ \) with \( 2ⁿ n \) flags, resolving two foundamental open questions on the existence of regular polytopes with certain Schläfli types. The automorphism group \( Aut(P) \) is explicitly realized as the semidirect product \( \mathbbF₂^n-1 \rtimes D_2n \), where \( D_2n \) is the dihedral group of order \( 2n \), with a complete presentation for \( Aut(P) \) is provided. This advances the systematic construction of regular polytopes with prescribed symmetries.

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