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Shapes of polyhedra and triangulations of the sphere
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The space of shapes of a polyhedron with given total angles less than 2\pi at each of its n vertices has a Kaehler metric, locally isometric to complex hyperbolic space CH^{n-3}. The metric is not complete: collisions between vertices take place a finite distance from a nonsingular point. The metric completion is a complex hyperbolic cone-manifold. In some interesting special cases, the metric completion is an orbifold. The concrete description of these spaces of shapes gives information about the combinatorial classification of triangulations of the sphere with no more than 6 triangles at a vertex.
Forward citations
Cited by 2 Pith papers
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Neoplatonic solids
Every 6-net with at most 50 vertices has an undented Euclidean realization from unit equilateral triangles, and prime 6-nets have unique convex ideal hyperbolic realizations.
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Triangulations of the Sphere
Baez outlines Thurston's results constructing triangulations of the sphere via Eisenstein integers and shows the moduli space of associated flat metrics is open and dense in an orbifold defined by a quadratic form on C^10.
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