A Regge transformation is one of three involutions on the six edge lengths or six dihedral angles
of a labeled tetrahedron. List the coordinates in
opposite-edge order as . The transformation
fixing the first opposite pair is
where . The other two transformations are obtained
by choosing either of the other opposite-edge pairs to be fixed. Applied to the edge
lengths of a tetrahedron, each transformation gives
the edge lengths of a second tetrahedron related
to the first by the Regge symmetry. On dihedral-angle
data, the transformation preserves rational multiples of , so it maps rational
tetrahedra to rational tetrahedra.
Akopyan, A. and Izmestiev, I. "The Regge Symmetry, Confocal Conics, and the Schläfli Formula." Bull. London Math. Soc.51,
765-775, 2019. https://doi.org/10.1112/blms.12276.Kedlaya,
K. S.; Kolpakov, A.; Poonen, B.; and Rubinstein, M. "Space Vectors Forming
Rational Angles." Nov. 28, 2020. https://arxiv.org/abs/2011.14232.Regge,
T. "Symmetry properties of Racah's coefficients." Nuovo Cim.11,
116-117, 1959. https://doi.org/10.1007/BF02724914.
. The transformation
fixing the first opposite pair is
. The other two transformations are obtained
by choosing either of the other opposite-edge pairs to be fixed. Applied to the edge
lengths of a tetrahedron, each transformation gives
the edge lengths of a second tetrahedron related
to the first by the Regge symmetry. On dihedral-angle
data, the transformation preserves rational multiples of 
, so it maps rational
tetrahedra to rational tetrahedra.
