Rational Tetrahedron

A rational tetrahedron is a Euclidean tetrahedron whose six dihedral angles each have the form for a rational number . Thus the term rational refers here to the angles, not to the edge lengths. Every rational tetrahedron has zero Dehn invariant, since each of its dihedral angles is zero modulo rational multiples of in the definition of that invariant.

Label the vertices 1, 2, 3, and 4, and list the dihedral angles in opposite-edge order (, , , , , ). Up to similarity, the complete classification consists of the following two one-parameter families

(1)

where is rational, together with 59 sporadic similarity classes (Kedlaya et al. 2020). This classification settles a problem posed by Conway and Jones (1976). Pegg (2023) gives interactive views of the sporadic examples.

Chentouf and Sun (2023) proved that all members of the first family tile three-dimensional Euclidean space, while the only member of the second family that tiles is its endpoint , which also belongs to the first family. They also ruled out 19 of the 59 sporadic classes, leaving at most 40 as possible space-filling tetrahedra.

Regge symmetry is useful in this classification because it preserves rationality of the dihedral angles while grouping apparently different tetrahedra into equivalence classes. The Regge group acts on labeled tetrahedra, and a Regge orbit is the group orbit of a labeled tetrahedron under this action. Of the 59 sporadic classes, 56 lie in orbits containing tetrahedra obtained from four-line subconfigurations of either the 15 unoriented lines through opposite vertices of an icosidodecahedron or the nine lines determined by the root system. Here is the rank-3 root system consisting of the 18 vectors for and for , which determine nine unoriented lines. The remaining three form the Regge orbit represented by the dihedral-angle sextuple (, , , , , ) (Kedlaya et al. 2020).

Starting with only the 15 icosidodecahedral lines, Pegg (2026) obtained 15 essentially different four-line face-normal configurations. Closing these under Regge transformations and all 24 relabelings gives 58 tetrahedral similarity classes. Of these, 52 are sporadic and six are particular members of the two infinite families, so this construction misses seven of the sporadic classes.