Dehn Invariant of Some Disjoint Unions of Polyhedra with Icosahedral Symmetry

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Two sets and are equidecomposable (can be dissected into each other) if there are two families of sets and , , such that , the interiors of the are disjoint, , the interiors of the are disjoint, and is congruent to ). More intuitively, can be cut up into finitely many pieces that can be rearranged to form ; here the pieces should be polyhedra.


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.

This Demonstration calculates Dehn invariants for disjoint unions of Platonic and Archimedean solids with icosahedral symmetry and edge length 1. In this case, dihedral angles are supplementary to angles between suitable pairs of adjacent axes of rotational symmetry. The disjoint union of an icosahedron, a dodecahedron, and an icosidodecahedron is an example of a combination of polyhedra with Dehn invariant 0. Find some other combinations with Dehn invariant 0.


Contributed by: Izidor Hafner (October 2014)
Open content licensed under CC BY-NC-SA



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


[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.

[2] N. Do, "Mathellaneous," Gazette of the Australian Mathematical Society, 33(2), 2006 pp. 81–87.

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