# Dehn Invariant of Some Disjoint Unions of Polyhedra with Octahedral Symmetry

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The Dehn invariant of a polyhedron is , where is the length of the edge , is the corresponding dihedral angle, and is an additive functional defined on a certain finite-dimensional vector space of reals over the rationals for which [1]. A polyhedron has Dehn invariant 0 if and only if it is equidecomposable with a cube of same volume.

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Contributed by: Izidor Hafner (October 2014)

Open content licensed under CC BY-NC-SA

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## Details

That some combinations of Platonic and Archimedean solids have Dehn invariant 0 was shown in [1].

Reference

[1] J. H. Conway, C. Radin, and L. Sadun, "On Angles Whose Squared Trigonometric Functions Are Rational," *Discrete & Computational Geometry*, 22(3), 1999 pp. 321–332. doi:10.1007/PL00009463.