Fixed Point
Since -1/i = i, we find that r(i) is a fixed point of the T rotation; it’s also true that r((i+1)/2) = STSr(i), and we can visualize both of these facts by applying the T rotation to the stereographically projected icosahedron and following the red and blue points.
It’s easy to compute this value algebraically; e.g., the transformation property immediately implies the computation via the quadratic formula.
However, we wanted to use the geometry of the icosahedron to compute this special value.
We begin by recalling the construction of the icosahedron by three mutually orthogonal golden rectangles.
In particular, since the long diagonal in the purple rectangle is 2, we can easily compute the lengths of the two green segments below to be
Finally, we use similar triangles and the Pythagorean theorem to compute the length of the red segment
