Spherical polyhedron¶
A finite edge-to-edge covering of a round sphere by great-arc-bounded spherical polygon faces.
Core Idea¶
A spherical polyhedron is a finite edge-to-edge covering of a complete round sphere by spherical polygon faces bounded by great-circle arcs. The closed faces cover every point; interiors remain disjoint while faces share whole edges or vertices. Digons are admitted, but regularity, a fixed face count and projection from a convex solid are not required.[ref-5bbe7084931c][ref-d0a4699ed934]
Scope of Application¶
The cited cases are complete round-sphere cellulations, not manufactured shells or measured volumes.
- Spherical dodecahedron: Twelve pentagonal faces cover the sphere in a dodecahedral incidence pattern.[ref-5bbe7084931c][ref-d0a4699ed934][^ref-94fa5cf99776]
- Three-sided hosohedron: Three great-semicircle meridians bound three digon lunes between two poles.[ref-5bbe7084931c][ref-d0a4699ed934][^ref-94fa5cf99776]
Clarity¶
A great-circle edge curves through ambient space, unlike a straight chord in a related Euclidean solid. “Disjoint” applies to face interiors, since closed neighboring faces share boundaries. The hosohedron's attempted straight-edge, flat-face Euclidean realization collapses; its spherical surface still bounds a ball.[ref-d0a4699ed934][ref-94fa5cf99776]
Manages Complexity¶
Four checks organize membership: a specified round carrier, finite great-arc network, spherical polygon faces meeting in whole cells, and exhaustive closed-face cover with disjoint interiors. A dome patch may satisfy the local edge and face checks while missing the global cover. Digons are permitted, not required.[ref-5bbe7084931c][ref-d0a4699ed934]
Abstract Reasoning¶
First fix the metric before calling an edge geodesic; then test shared-cell incidence and full cover separately. Comparing the dodecahedron with the hosohedron shows why pentagons, regularity and positive volume for a planar-faced realization are not membership conditions. The latter failure concerns that attempted Euclidean realization, not the spherical cellulation.[ref-5bbe7084931c][ref-d0a4699ed934][^ref-94fa5cf99776]
Knowledge Transfer¶
The definition transfers literally to other tilings only when a complete round sphere, great arcs, shared-cell incidence and full cover remain. Topological Surface is an internal carrier of every instance, with Topological Space inherited through it. A generic partition or polyhedral graph preserves only some roles; convex projection is a conditional route, not a universal origin.[ref-5bbe7084931c][ref-d0a4699ed934]
Relationships to Other Abstractions¶
Current abstraction Spherical polyhedron Domain-specific
Parents (1) — more general patterns this builds on
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Spherical polyhedron is part of Topological Surface Domain-specific
Every admitted round-sphere cellulation contains the connected boundaryless topological two-surface S² as its complete identity-bearing carrier.
Hierarchy paths (5) — routes to 3 parentless roots
- Spherical polyhedron → Topological Surface → Topological Space → Closure
- Spherical polyhedron → Topological Surface → Topological Space → Set and Membership
- Spherical polyhedron → Topological Surface → Topological Space → Topology
- Spherical polyhedron → Topological Surface → Topological Space → Intersection → Set and Membership
- Spherical polyhedron → Topological Surface → Topological Space → Union → Set and Membership
Neighborhood in Abstraction Space¶
Spherical polyhedron sits in a sparse region of the domain-specific corpus (84th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Euclidean & Geometric Structures (25 abstractions)
Nearest neighbors
- Lens space — 0.81
- Reuleaux polygon — 0.81
- Kline sphere characterization — 0.81
- Ideal polyhedron — 0.81
- Polygon partition — 0.81
Computed from structural-signature embeddings · 2026-10-17
References¶
[^ref-5bbe7084931c]: Hoi Ping Luk, Roman Nedela and Christopher Purcell. (2025). Tiling the Sphere with Regular Polygons (arXiv 2512.05577v1), arXiv:2512.05577v1, submitted 5 December 2025; manuscript cover dated 8 December 2025, original author preprint PDF. Relevant one-based PDF p. 2 Figure 1, p. 5 §2 spherical-tiling and conditional-projection paragraphs plus Figure 5, and p. 6 digon definition and Propositions 2.1–2.2. This is the inspected 44-page preprint; its regular-tiling theorem is narrower than the class defined here. [^ref-d0a4699ed934]: Jonn Angel L. Aranas and Mark L. Loyola. (2020). “Geometric realizations of abstract regular polyhedra with automorphism group H3,” Acta Crystallographica Section A 76, 358–368, DOI: 10.1107/S2053273320001564, full original article. Relevant §2.3 abstract dodecahedral incidence; §3.2 after Theorem 3.1 hosohedron and collapsed straight-segment realization; §3.3.1 spherical vertices, open great-arc edges and open spherical-polygon facets. Its Figure 3(d) is a great-stellated, not ordinary, dodecahedral realization. [^ref-94fa5cf99776]: Allen Liu. The Stars Above Us: Regular and Uniform Polytopes up to Four Dimensions. Undated in the inspected Harvard Mathematics-hosted thesis PDF. Relevant one-based PDF p. 26 §4 text and spherical-dodecahedron caption, pp. 26–27 hosohedron and dihedron discussion, and p. 27 flat-faced proper-volume convention. The cited thesis text and captions were inspected; its figure pixels and publication year were not verified.