Basis for Pure Geodetic Angles

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This Demonstration illustrates the theorem: If , with square-free positive integer
and relatively prime
and
, and if the prime factorization of
is
, then we have
for some rational
.
Contributed by: Izidor Hafner (April 2011)
Open content licensed under CC BY-NC-SA
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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.
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