Harmonic Ratio of the Dodecahedron, ~1.2584

From Euclid XIII.17 (Heath, v.3, p. 501) we have (from the golden triangle),
[1] S = (R/sqrt(3))*(sqrt(5) - 1)
Again, to relate S and r, we will resort to the central triangle.

The central triangle argument for the dodecahedron

Again, this triangle connects the center of the dodecahedron to a vertex, the side R, to the center of a regular pentagonal face, the side r, and across the pentagonal face, side c. Applying the Pythagorean theorem to this triangle, we have,
[2] R2 = r2 + c2,
In this case, c is the distance from the center of a regular pentagon to one of its vertices. This distance is related to S, the edge of the pentagon, by Euclid XIII.11 (Heath, v.3, p. 466). Euclid gives this distance as (his BA is our S, his r is our c),
[3] S = (c/2)*sqrt(10 - 2*sqrt(5))
which is the minor we have seen before. [Note: The side R in equation [1] of this previous discussion is called c here.] Writing [1] in the form S = a*R (a ~= 0.713644) and [3] in the form S = b*c (b ~= 1.175571), [2] becomes,

R2 = r2 + (S/b)2 = r2 + (a*R/b)2, or [1 - (a/b)2]*R2 = r2, or,

[4] R/r = 1/sqrt(1 - a2/b2)
where a2 = [sqrt(5) - 1]2 / 3, and b2 = [10 - 2*sqrt(5)]/4.

Evaluating approximately,

sqrt(5) ~= 2.236068, sqrt(5) - 1 ~= 1.236068, a2 ~= 0.509288
10 - 2*sqrt(5) ~= 5.527864, b2 ~= 1.381966
1-a2/b2 ~= 1-0.368524 = 0.631476

R/r ~= 1/sqrt(0.631476) ~= 1.258409