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
See also Cube Dissection ,
Dehn Invariant ,
Diabolical Cube ,
Polycube ,
Soma Cube ,
Wallace-Bolyai-Gerwien Theorem Explore with Wolfram|Alpha References Bulatov, V. "Compounds of Uniform Polyhedra." http://bulatov.org/polyhedra/uniform_compounds/ . Coffin, S. T. The Puzzling World of Polyhedral Dissections. Coffin, S. T. and Rausch, J. R. The Puzzling World of Polyhedral Dissections CD-ROM. Conway, J. H.; Radin, C.; and Sadun, L. "On Angles Whose Squared Trigonometric Functions Are Rational." Discr. Computat. Geom. 22 , 321-332, 1999. Referenced on Wolfram|Alpha Polyhedron Dissection Cite this as: Weisstein, Eric W. "Polyhedron Dissection." From MathWorld https://mathworld.wolfram.com/PolyhedronDissection.html
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