The points <,0 0, 0>, <0, 1, 1>, <1, 0, 1>, and <1, 1, 0>
are the vertices of a regular tetrahedron.
Cross products of edge vectors yield normals to the faces:
<0, 1, 1> × <1, 0, 1> = <1, 1, –1>.
Another pair yields <1, –1, 1> and the dot product of these two
is –1 and the angle between them, α is such that
cos α
|<1, –1, 1>| |<1, 1, –1>|
= <1, –1, 1> • <1, 1, –1>.
We thus learn that cos α = –1/3
or α = cos–1(–1/3).
This is the exterior dihedral angle;
the interior angle is cos–1(1/3).
From classic spherical trigonometry we take the excess angle formula for
the area of a spherical triangle, which is the same as the solid angle at
the vertex when we consider a sphere with its center at the vertex.
The above external dihedral angle is the angle of the spherical triangle.
The solid angle is thus 3cos–1(1/3) – π.