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.


This Demonstration calculates Dehn invariants for disjoint unions of Platonic and Archimedean solids having octahedral symmetry (and edge length 1). In this case, the dihedral angles are supplementary to the angles between suitable axes of rotational symmetry.

The disjoint union of a tetrahedron and a truncated tetrahedron has 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.

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