Details

If is a square-free positive integer and and are integers, then linear combinations of angles with tangents of the form (called pure geodetic angles) form a vector space over the rationals. A basis for the space is formed by and certain angles for prime . If or if and , then is defined only when is congruent to a square modulo . Express as for the smallest possible positive . Then . Rational linear combinations of pure geodetic angles for all are called mixed geodetic angles. With fixed we get a vector subspace in the space of all mixed geodetic angles.

An application: For the dihedral angle of a dodecahedron we have , so . This means that is linearly independent of . If is an additive function over the reals, such that , rational, we could choose . The Dehn invariant of the dodecahedron is , while for a cube it is . So the dodecahedron is not equidecomposable with the cube.

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.