The space of Euclidean cone metrics on centrically symmetric octahedra with fixed cone angles θi < 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 .

中文翻译：

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具有规定锥缺的中心对称八面体的形状

具有固定锥角θi<2π的中心对称八面体上的欧几里德圆锥度量空间，具有总表面积1，具有自然的双曲度量，并且局部等距于双曲3空间。该空间的度量完成度与双曲面理想四面体等轴测，该四面体的二面角为圆锥赤字2π-θi的一半。