Bounding the density of spherical polygon packings
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866910160357687296 |
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| author | Filho, Fernando Mário de Oliveira Spomer, Andreas Vallentin, Frank |
| author_facet | Filho, Fernando Mário de Oliveira Spomer, Andreas Vallentin, Frank |
| contents | We determine putative optimal packings of regular spherical polygons via optimization on smooth manifolds. For several cases, we establish maximality by extending the Lovász theta number to Cayley graphs on the special orthogonal group ${\rm SO}(3)$. To this end, we introduce an algebraic criterion characterizing when congruent regular spherical polygons have disjoint interiors, leading to a unified formulation of the packing constraints. Using harmonic analysis on ${\rm SO}(3)$, we reduce the theta number to a trigonometric sum-of-squares problem, which can be solved via semidefinite programming. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21451 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bounding the density of spherical polygon packings Filho, Fernando Mário de Oliveira Spomer, Andreas Vallentin, Frank Metric Geometry Optimization and Control 52C17, 90C23, 90C30 We determine putative optimal packings of regular spherical polygons via optimization on smooth manifolds. For several cases, we establish maximality by extending the Lovász theta number to Cayley graphs on the special orthogonal group ${\rm SO}(3)$. To this end, we introduce an algebraic criterion characterizing when congruent regular spherical polygons have disjoint interiors, leading to a unified formulation of the packing constraints. Using harmonic analysis on ${\rm SO}(3)$, we reduce the theta number to a trigonometric sum-of-squares problem, which can be solved via semidefinite programming. |
| title | Bounding the density of spherical polygon packings |
| topic | Metric Geometry Optimization and Control 52C17, 90C23, 90C30 |
| url | https://arxiv.org/abs/2604.21451 |