Regular 3-polytopes of type $\{n,n\}$
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910943891423232 |
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| author | Li, Mingchao Zhang, Wei-Juan |
| author_facet | Li, Mingchao Zhang, Wei-Juan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regular 3-polytopes of type $\{n,n\}$ Li, Mingchao Zhang, Wei-Juan Combinatorics Group Theory 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. |
| title | Regular 3-polytopes of type $\{n,n\}$ |
| topic | Combinatorics Group Theory |
| url | https://arxiv.org/abs/2505.09505 |