is a square-free positive integer and
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
is defined only when
is congruent to a square modulo
for the smallest possible positive
. 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
. This means that
is linearly independent of
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.
 J. H. Conway, C. Radin, and L. Sadun, "On Angles Whose Squared Trigonometric Functions Are Rational," Discrete & Computational Geometry
(3), 1999 pp. 321–332.