Harmonic Ratio of the Tetrahedron

Using again the notations above, we find in Euclid Book XIII, Prop. 13, D2 = (3/2)*S2, or
[1] R2 = (3/2)*s2
To relate s and r, we again resort to a central triangle.

The central triangle argument for the tetrahedron

Consider a central triangle: center of tetrahedron to vertex, vertex to center of a face, and center of face to center of tetrahedron. As in the case of the cube, the circum-radius, R, and the in-radius, r, are related by
[2] R2 = r2 + c2,
where c is the distance from the center of a face to an adjacent vertex.

In the case of the cube, the face is a square, and we know where the center of the square is, without resorting to Euclid or Pythagoras. But for the tetrahedron (and all other regular polyhedra excepting the dodecahedron) the face is an equilateral triangle. And where is the center of an equilateral triangle? Fortunately, we have already dealt with this above, so we know that,

[3] c = 2*s/sqrt(*3)
Combining these equations, we have,
[4] R2 = r2 + 4*s2/3 = r2 + 4*(2/3)*(R2)/3
or (1/9)*R2 = r2, or R = 3*r. So the harmonic ratio of the tetrahedron is 3.
Revised 18 jan 2002 by ralph abraham