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On the second smallest ideal right-angled hyperbolic four-polytope

  • Alexander Kolpakov
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Abstract

The regular ideal right-angled 24-cell has volume 4 π2/3 and is the unique minimum-volume entirely ideal right-angled hyperbolic four-polytope. We prove that the next distinct volume is 8 π2/3 and that its unique equality case is the reflection double of the 24-cell across one octahedral facet. We also show that an entirely ideal right-angled four-polytope whose facets are regular ideal octahedra or their doubles across triangular faces decomposes into a chain of regular 24-cells, with opposite cutting facets in each intermediate cell.

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Cite

BibTeXRIS
BibTeX
@misc{kolpakov2026second,
  title = {{On the second smallest ideal right-angled hyperbolic four-polytope}},
  author = {Alexander Kolpakov},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00118},
  eprint = {2611.00118},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00118},
}
RIS
TY  - UNPB
TI  - On the second smallest ideal right-angled hyperbolic four-polytope
AU  - Alexander Kolpakov
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00118
UR  - https://arxiv.org/abs/2611.00118
AB  - The regular ideal right-angled 24-cell has volume 4 π^2/3 and is the unique minimum-volume entirely ideal right-angled hyperbolic four-polytope. We prove that the next distinct volume is 8 π^2/3 and that its unique equality case is the reflection double of the 24-cell across one octahedral facet. We also show that an entirely ideal right-angled four-polytope whose facets are regular ideal octahedra or their doubles across triangular faces decomposes into a chain of regular 24-cells, with opposite cutting facets in each intermediate cell.
ER  -