#### Volume 1 (1998)

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Shapes of polyhedra and triangulations of the sphere

### William P Thurston

Geometry & Topology Monographs 1 (1998) 511–549
DOI: 10.2140/gtm.1998.1.511
 arXiv: math.GT/9801088
##### Abstract
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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 Kähler 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.

##### Keywords
polyhedra, triangulations, configuration spaces, braid groups, complex hyperbolic orbifolds
##### Mathematical Subject Classification
Primary: 51M20
Secondary: 51F15, 20H15, 57M50