Max Dehn showed that one type of these figures has a non-zero Dehn invariant in his proof of Hilbert's third problem. The pentachoron contains 5 of these cells in four dimensions, and one subclass of this figure includes disphenoids. The volume of this figure is computed as 
Max Dehn showed that one type of these figures has a non-zero Dehn invariant in his proof of Hilbert's third problem. The pentachoron contains 5 of these cells in four dimensions, and one subclass of this figure includes disphenoids. The volume of this figure is computed as one-sixth of the scalar triple product of its sides, and its (*) regular type has surface area equal to s squared times the square root of three. It is generalized as a 3-simplex, and its regular variety is the simplest platonic solid. For ten points, identify this polyhedron with four triangular faces.
Max Dehn showed that one type of these figures has a non-zero Dehn invariant in his proof of Hilbert's third problem.