Basis for Pure Geodetic Angles

​
a
4
b
5
d
1
2
3
5
6
-1
tan
5
3
4
= π-
-1
tan
3
2
-
-1
tan
2
3

p
<p
>
3
7
-1
tan
3
2
13
-1
tan
2
3

19
-1
tan
3
4
31
-1
tan
3
3
2
37
-1
tan
2
3
5
43
-1
tan
3
3
4
61
-1
tan
2
3
7
67
-1
tan
3
8
73
-1
tan
4
3
5
79
-1
tan
5
3
2
This Demonstration illustrates the theorem: If
tan(θ)=(a/b)
d
, with square-free positive integer
d
and relatively prime
a
and
b
, and if the prime factorization of
2
a
+d
2
b
is
k
1
p
1
k
2
p
2
...
k
n
p
n
, then we have
θ=tπ±
k
1
<
p
1
>
d
±
k
2
<
p
2
>
d
±…±
k
n
<
p
n
>
d
for some rational
t
.

Details

If
d
is a square-free positive integer and
a
and
b
are integers, then linear combinations of angles with tangents of the form
b/a
d
(called pure geodetic angles) form a vector space over the rationals. A basis for the space is formed by
π
and certain angles
<p
>
d
for prime
p
. If
p>2
​
or if
p=2
and
d≡7(mod8)
, then
<p
>
d
is defined only when
-d
is congruent to a square modulo
p
. Express
4
s
p
as
2
a
+d
2
b
for the smallest possible positive
s
. Then
<p
>
d
=1/s
-1
tan
(
d
ba)
. Rational linear combinations of pure geodetic angles for all
d
are called mixed geodetic angles. With fixed
d
we get a vector subspace in the space of all mixed geodetic angles.
An application: For the dihedral angle
α
of a dodecahedron we have
α=
-1
cos
-
5
5
, so
α=
-1
tan
(-2)=π-
-1
tan
(2)=π-<5
>
2
. This means that
α
is linearly independent of
π
. If
f
is an additive function over the reals, such that
f(qπ)=0
,
q
rational, we could choose
f(α)=1
. The Dehn invariant of the dodecahedron is
30f(α)=30
, while for a cube it is
12f(π/2)=0
. So the dodecahedron is not equidecomposable with the cube.

References

[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.

External Links

Rational Linear Combinations of Pure Geodetic Angles
Rational Linear Combinations of Pure Geodetic Angles, Part 2
Dehn Invariant (Wolfram MathWorld)
Platonic Solid (Wolfram MathWorld)

Permanent Citation

Izidor Hafner
​
​"Basis for Pure Geodetic Angles"​
​http://demonstrations.wolfram.com/BasisForPureGeodeticAngles/​
​Wolfram Demonstrations Project​
​Published: April 19, 2011