Maximal Volume Ideal Polyhedra and the Arithmetic Angle Phenomenon
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909956337303552 |
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| author | Rivin, Igor |
| author_facet | Rivin, Igor |
| contents | We present a software suite for the analysis and optimization of ideal convex polyhedra in hyperbolic 3-space $\mathbb{H}^3$. Using Rivin's variational characterization of ideal polyhedra, we develop efficient algorithms for checking combinatorial realizability and finding volume-maximizing configurations. Our systematic computational study reveals two striking phenomena: (1) maximal volume ideal polyhedra consistently exhibit dihedral angles that are rational multiples of $π$ -- a property with no obvious explanation from the optimization formulation; and (2) the distribution of volumes for random configurations is well-approximated by a Beta distribution, with mean normalized volume converging to approximately $\ln 2 \approx 0.69$ as the vertex count increases. We provide complete data for small vertex counts, including vertex positions, triangulations, and verified rational angle structures. An interactive implementation is publicly available. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_10087 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maximal Volume Ideal Polyhedra and the Arithmetic Angle Phenomenon Rivin, Igor Geometric Topology 52A55, 11F06, 52B10 We present a software suite for the analysis and optimization of ideal convex polyhedra in hyperbolic 3-space $\mathbb{H}^3$. Using Rivin's variational characterization of ideal polyhedra, we develop efficient algorithms for checking combinatorial realizability and finding volume-maximizing configurations. Our systematic computational study reveals two striking phenomena: (1) maximal volume ideal polyhedra consistently exhibit dihedral angles that are rational multiples of $π$ -- a property with no obvious explanation from the optimization formulation; and (2) the distribution of volumes for random configurations is well-approximated by a Beta distribution, with mean normalized volume converging to approximately $\ln 2 \approx 0.69$ as the vertex count increases. We provide complete data for small vertex counts, including vertex positions, triangulations, and verified rational angle structures. An interactive implementation is publicly available. |
| title | Maximal Volume Ideal Polyhedra and the Arithmetic Angle Phenomenon |
| topic | Geometric Topology 52A55, 11F06, 52B10 |
| url | https://arxiv.org/abs/2512.10087 |