<P> A cube can be inscribed in a dodecahedron so that each vertex of the cube is a vertex of the dodecahedron and each edge is a diagonal of one of the dodecahedron's faces; taking all such cubes gives rise to the regular compound of five cubes . </P> <P> If two opposite corners of a cube are truncated at the depth of the three vertices directly connected to them, an irregular octahedron is obtained . Eight of these irregular octahedra can be attached to the triangular faces of a regular octahedron to obtain the cuboctahedron . </P> <P> The cube is topologically related to a series of spherical polyhedra and tilings with order - 3 vertex figures . </P> <Table> <Tr> <Th_colspan="12"> show * n32 symmetry mutation of regular tilings: (n, 3) </Th> </Tr> <Tr> <Th_colspan="4"> Spherical </Th> <Th> Euclidean </Th> <Th_colspan="2"> Compact hyperb . </Th> <Th> Paraco . </Th> <Th_colspan="4"> Noncompact hyperbolic </Th> </Tr> <Tr> <Td> </Td> <Td> </Td> <Td> </Td> <Td> </Td> <Td> </Td> <Td> </Td> <Td> </Td> <Td> </Td> <Td> </Td> <Td> </Td> <Td> </Td> <Td> </Td> </Tr> <Tr> <Td> (2, 3) </Td> <Td> (3, 3) </Td> <Td> (4, 3) </Td> <Td> (5, 3) </Td> <Td> (6, 3) </Td> <Td> (7, 3) </Td> <Td> (8, 3) </Td> <Td> (∞, 3) </Td> <Td> (12i, 3) </Td> <Td> (9i, 3) </Td> <Td> (6i, 3) </Td> <Td> (3i, 3) </Td> </Tr> </Table>

What is the volume of the three dimensional object formed by