The space of Euclidean cone metrics on centrically symmetric octahedra with fixed cone angles θi < 2π, with total surface area 1, has a natural hyperbolic metric, and is locally isometric to hyperbolic 3-space. The metric completion of the space is isometric to a hyperbolic ideal tetrahedron whose dihedral angles are half the cone-deficits 2π − θi.
I would like to thank my adviser Richard Schwartz, from whom I learned this topic and got many helpful feedbacks on the ideas in and the structure of this article.
Communicated by: J. Ratcliffe
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