#### Geometry & Topology Monographs 1 (1998),
The Epstein Birthday Schrift,
paper no. 25, pages 511-549.

## Shapes of polyhedra and triangulations of the sphere

### William P Thurston

**Abstract**.
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.
**Keywords**.
Polyhedra, triangulations, configuration spaces, braid groups,
complex hyperbolic orbifolds

**AMS subject classification**.
Primary: 51M20.
Secondary: 51F15, 20H15, 57M50.

**E-print:** `arXiv:math.GT/9801088`

Submitted: 15 November 1997.
(Revised: 27 November 1998.)
Published: 4 December 1998.

Notes on file formats
William P Thurston

Mathematics Department

University of California at Davis

Davis, CA 95616, USA

Email: wpt@math.ucdavis.edu

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