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
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References Bulatov, V. "Compounds of Uniform Polyhedra." http://bulatov.org/polyhedra/uniform_compounds/ . Coffin,
S. T. The
Puzzling World of Polyhedral Dissections. New York: Oxford University Press,
1990. Coffin, S. T. and Rausch, J. R. The
Puzzling World of Polyhedral Dissections CD-ROM. Puzzle World Productions,
1998. 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 --A Wolfram Web Resource. https://mathworld.wolfram.com/PolyhedronDissection.html
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