Rational Linear Combinations of Pure Geodetic Angles

​
a
1
1
a
2
1
b
1
1
b
2
1
d
1
1
2
3
5
7
11
d
2
2
3
5
7
11
13
-1
tan
(
2
+
3
)
1
2
π
2
+
-1
tan
(
2
)
A "pure geodetic" angle is an angle with any of its six squared trigonometric functions rational (or infinite). This Demonstration shows how an angle whose tangent is of the form
b
1
/
a
1
d
1
+
b
2
/
a
2
d
2
can be expressed as a rational linear combination of pure geodetic angles and an integral multiple of
π/2
, that is, it finds rational
q
1
and
q
2
such that
-1
tan

b
1

a
1
d
1
+
b
2

a
2
d
2

is a sum of
kπ/2
, where
k∈{-1,0,1}
and a rational linear combination of
-1
tan

q
1
d
1

and
-1
tan

q
2
d
2

.

Details

​

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

Linear Combination (Wolfram MathWorld)

Permanent Citation

Izidor Hafner
​
​"Rational Linear Combinations of Pure Geodetic Angles"​
​http://demonstrations.wolfram.com/RationalLinearCombinationsOfPureGeodeticAngles/​
​Wolfram Demonstrations Project​
​Published: December 23, 2010