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arXiv · 2611.00118

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 $4 π^2/3$ 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.

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Alexander Kolpakov. 2026-10-28. On the second smallest ideal right-angled hyperbolic four-polytope. https://arxiv.org/abs/2611.00118

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