This table is copied directly out of Danzer's paper, with some corrections. For each edge on each tile, the dihedral angle between the faces that share that edge is given in degrees, followed by the length of the edge. a = sqrt(10+2*sqrt(5))/4, b = sqrt(3)/2, T = (1+sqrt(5))/2, and Ti = 1/T. The colors correspond to Danzer's edge colorings.
| Edge | ||||||
|---|---|---|---|---|---|---|
| Tetrahedron | 1-2 | 2-3 | 3-1 | 2-4 | 1-4 | 3-4 |
| A | 36, a | 60, Tb | 72, Ta | 108, a | 90, 1 | 60, b |
| B | 36, a | 36, Ta | 60, Tb | 120, b | 108, Tia | 90, 1 |
| C | 36, Tia | 60, Tb | 90, T | 120, b | 72, a | 36, a |
| K | 36, a | 60, b | 72, Tia | 90, T/2 | 90, 1/2 | 90, Ti/2 |
The solid angle is the area on the surface of the unit sphere which is subtended by a surface. That is, if you draw a ray from the origin to every point on some patch in space, then the area of all the points where those rays intersect the unit sphere centered at the origin is the solid angle. The total solid angle is the area of the unit sphere, 4*pi. Thus the sum of the solid angles of all of the vertices involved in a vertex configuration is always 4*pi = 720 degrees.
Jeff Weeks' book The Shape of Space explains how to compute the area of a triangle on the surface of the unit sphere. If the angles of the triangle are a, b, and c, then the area is a+b+c-pi. We will not give the proof here, though it is quite simple and elegant. This provides us an easy way to calculate the solid angle of the corner of a tetrahedron given its dihedral angles; since the angle between two faces is the same as the angle formed between the intersections of those faces with the surface of the sphere, the solid angle is just the sum of the dihedral angles minus pi.
Below is a table of the solid angle in (square?) degrees for each vertex of each prototile. The colors correspond to the vertex classes defined earlier.
| Vertex | Key | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Tetrahedron | 1 | 2 | 3 | 4 | Color | Vertex Class | ||||||||||
| A | 18 | 24 | 12 | 78 | 1 | |||||||||||
| B | 24 | 12 | 6 | 138 | 2 | |||||||||||
| C | 18 | 36 | 6 | 48 | 3 | |||||||||||
| K | 18 | 6 | 42 | 90 | 4 | |||||||||||
Back to Danzer tiling