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).

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.

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Polyhedron Dissection

Cite this as:

Weisstein, Eric W. "Polyhedron Dissection." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PolyhedronDissection.html

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