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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

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The space of shapes of a polyhedron with given total angles less than 2π at each of its n vertices has a Kähler metric, locally isometric to complex hyperbolic space CHn-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.


polyhedra, triangulations, configuration spaces, braid groups, complex hyperbolic orbifolds

Mathematical Subject Classification

Primary: 51M20

Secondary: 20H15, 51F15, 57M50


Received: 15 November 1997
Revised: 27 November 1998
Published: 30 November 1998

William P Thurston
Mathematics Department
University of California at Davis
Davis CA 95616