Polyhedron Dissection

A polyhedron dissection (or decomposition) is a dissection of one or more polyhedra into other shapes.

Two polyhedra can be dissected into each other iff they have equal Dehn invariant and volume. More generally, a set of polyhedra can be dissected into another set of polyhedra (where the two sets need not be of equal size) iff the sums of their Dehn invariants and sums of their volumes are equal.

The following table give sets of unit equilateral polyhedra which are interdissectable (E. Weisstein, Aug. 17, 2023), where Dehn invariants are specified using the basis and notation of Conway et al. (1999).

Dehn invariant	volume	interdissectable polyhedra
		regular icosidodecahedron, pentagonal orthobirotunda
		elongated pentagonal gyrobirotunda, elongated pentagonal orthobirotunda
		gyrate rhombicosidodecahedron, metabigyrate rhombicosidodecahedron, parabigyrate rhombicosidodecahedron, small rhombicosidodecahedron, trigyrate rhombicosidodecahedron
		bigyrate diminished rhombicosidodecahedron, diminished rhombicosidodecahedron, metagyrate diminished rhombicosidodecahedron, paragyrate diminished rhombicosidodecahedron
		metabiaugmented dodecahedron, parabiaugmented dodecahedron
		gyrate bidiminished rhombicosidodecahedron, metabidiminished rhombicosidodecahedron, parabidiminished rhombicosidodecahedron
		pentagonal gyrobicupola, pentagonal orthobicupola
		elongated pentagonal gyrobicupola, elongated pentagonal orthobicupola
		metabiaugmented truncated dodecahedron, parabiaugmented truncated dodecahedron
		metabidiminished icosahedron, pentagonal antiprism
		pentagonal gyrocupolarotunda, pentagonal orthocupolarotunda
		elongated pentagonal gyrocupolarotunda, elongated pentagonal orthocupolarotunda
		cuboctahedron, triangular orthobicupola
		elongated triangular gyrobicupola, elongated triangular orthobicupola
		square gyrobicupola, square orthobicupola
		elongated square gyrobicupola, small rhombicuboctahedron
		metabiaugmented hexagonal prism, parabiaugmented hexagonal prism