A dihedral angle is called rational if its measure is a rational multiple of . A tetrahedron is called rational if all six of its dihedral angles are rational. In , the authors proved that rational tetrahedra fall into two infinite families and a set of 59 others called sporadic, which are shown in this Demonstration.
In the graphics, each edge is labeled with the fraction , where the dihedral angle at that edge is . In addition, the solid angle at each vertex is labeled in the same way. Twice the sum of the dihedral angles minus the sum of the solid angles equals 4, or in symbols .
Drag the figure to better see the fractions.
The first 15 tetrahedra are in the algebraic space of the golden ratio. After those 15, the following pairs have similar triangles: 19-20, 21-22, 23-24 and 36-37, 38-39, 40-41, 42-43.