We give a rigorous geometric proof of the Murakami-Yano formula for thevolume of a hyperbolic tetrahedron. In doing so, we are led to considergeneralized hyperbolic tetrahedra, which are allowed to be non-convex, and havevertices `beyond infinity'; and we uncover a group, which we call 22.5K, of23040 scissors-class-preserving symmetries of the space of (suitably decorated)generalized hyperbolic tetrahedra. The group 22.5K contains the Reggesymmetries as a subgroup of order 144. From a generic tetrahedron, 22.5Kproduces 30 distinct generalized tetrahedra in the same scissors class,including the 12 honest-to-goodness tetrahedra produced by the Regge subgroup.The action of 22.5K leads us to the Murakami-Yano formula, and to 9 others,which are similar but less symmetrical. From here, we can derive yet othervolume formulas with pleasant algebraic and analytical properties. The key tounderstanding all this is a natural relationship between a hyperbolictetrahedron and a pair of ideal hyperbolic octahedra.
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