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This is a Quantitative Study study.

October 9, 2025Open Access

Regular 3-polytopes of type \n, n\

Key Points

Constructed self-dual regular 3-polytopes with 2^n n flags, addressing fundamental questions in polytope existence.
Explicit realization of the automorphism group as a semidirect product of two well-known groups with a full presentation.
Advancing systematic constructions enhances understanding of polytopes with specific symmetries and structures.
Findings support further exploration into geometry, providing necessary tools to manage symmetry and classification.

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 Schl\"afli types. The automorphism group \ (Aut (P) \) is explicitly realized as the semidirect product \ (F₂^n-1 D₂₍ \), where \ (D₂₍ \) 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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Cite This Study

Li et al. (Wed,) studied this question.

synapsesocial.com/papers/68e8439a9989581a2fd4e042 https://doi.org/https://doi.org/10.48550/arxiv.2505.09505

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