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The space of Euclidean cone metrics on centrically symmetric octahedra with fixed cone angles  θi  &lt; 2 π , 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 2 π  −  θi .

Shapes of centrally symmetric octahedra with prescribed cone-deficits