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PreprintarXiv licencearXiv

Equifacetted polytopes whose facets are products of simplices: The simplicial prism case

  • Federico Castillo
  • Guillermo Piñeda-Villavicencio
  • José Alejandro Samper
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Abstract

We study \emph{prismatical} polytopes, those all of whose facets are combinatorially isomorphic to the simplicial prism Δd−2×Δ1. In dimension three these are exactly the cubical polytopes, so the class sits naturally between simplicial and cubical polytopes. In dimension four, Δ2×Δ2 is the simplest prismatical polytope and we can construct infinitely many by using connected sums. By fibreing a triangle over a triangulated polygon we construct infinitely many connected-sum-prime prismatical $4$-polytopes. Finally, we show that there are no prismatical d-polytopes for d≥5, answering a question of Perles and Shephard.

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Cite

BibTeXRIS
BibTeX
@misc{castillo2026equifacetted,
  title = {{Equifacetted polytopes whose facets are products of simplices: The simplicial prism case}},
  author = {Federico Castillo and Guillermo Piñeda-Villavicencio and José Alejandro Samper},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32035},
  eprint = {2610.32035},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32035},
}
RIS
TY  - UNPB
TI  - Equifacetted polytopes whose facets are products of simplices: The simplicial prism case
AU  - Federico Castillo
AU  - Guillermo Piñeda-Villavicencio
AU  - José Alejandro Samper
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32035
UR  - https://arxiv.org/abs/2610.32035
AB  - We study \emph{prismatical} polytopes, those all of whose facets are combinatorially isomorphic to the simplicial prism Δ^d−2×Δ^1. In dimension three these are exactly the cubical polytopes, so the class sits naturally between simplicial and cubical polytopes. In dimension four, Δ^2×Δ^2 is the simplest prismatical polytope and we can construct infinitely many by using connected sums. By fibreing a triangle over a triangulated polygon we construct infinitely many connected-sum-prime prismatical $4$-polytopes. Finally, we show that there are no prismatical d-polytopes for d≥5, answering a question of Perles and Shephard.
ER  -