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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2506.04406 |
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| _version_ | 1866914202245922816 |
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| author | Mochán, Elías |
| author_facet | Mochán, Elías |
| contents | Abstract polytopes generalize the face lattice of convex polytopes. A polytope is semiregular if its facets are regular and its automorphism group acts transitively on its vertices. In this paper we construct semiregular, facet-transitive polyhedra with trivial facet stabilizer, showing that semiregular abstract polyhedra can have an unbounded number of flag orbits, while having as little as one facet orbit. We interpret this construction in terms of operations applied to high rank regular and chiral polytopes, and we see how this same operations help us construct alternating semiregular polyhedra (that is, with two facet orbits and adjacent facets in different orbits). Finally, we give an idea to generalize this construction giving examples in higher ranks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04406 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Semiregular abstract polyhedra with trivial facet stabilizer Mochán, Elías Combinatorics 52B15 52B10 05E18 Abstract polytopes generalize the face lattice of convex polytopes. A polytope is semiregular if its facets are regular and its automorphism group acts transitively on its vertices. In this paper we construct semiregular, facet-transitive polyhedra with trivial facet stabilizer, showing that semiregular abstract polyhedra can have an unbounded number of flag orbits, while having as little as one facet orbit. We interpret this construction in terms of operations applied to high rank regular and chiral polytopes, and we see how this same operations help us construct alternating semiregular polyhedra (that is, with two facet orbits and adjacent facets in different orbits). Finally, we give an idea to generalize this construction giving examples in higher ranks. |
| title | Semiregular abstract polyhedra with trivial facet stabilizer |
| topic | Combinatorics 52B15 52B10 05E18 |
| url | https://arxiv.org/abs/2506.04406 |