A Regge symmetry is the correspondence between a tetrahedron and the tetrahedron obtained by applying a Regge transformation to its edge lengths. It is a symmetry of tetrahedral metric data and is distinct from the more general Regge calculus, which approximates manifolds by simplicial complexes. Both concepts are named for Tullio Regge. For every tetrahedron in Euclidean geometry, spherical geometry, or hyperbolic geometry, the transformed numbers are the edge lengths of a tetrahedron in the same geometry. Labeling the vertices 1, 2, 3, and 4, its dihedral angles satisfy the analogous relation in opposite-edge order
 |
where 
.
The two tetrahedra have equal volume.
In the Euclidean case, they also have the same Dehn
invariant and are scissors-congruent (Akopyan and Izmestiev 2019).
The three Regge transformations, together with the 24 relabelings of the tetrahedron, generate
the Regge group, which is isomorphic to the direct
product 
(Kedlaya et al. 2020).
Regge discovered the Regge transformations as additional symmetries of Racah W-coefficients, equivalently the Wigner 6j-symbols (Regge 1959). Ponzano and Regge (1968) gave them their Euclidean tetrahedral interpretation. Akopyan and Izmestiev (2019) proved the Euclidean, spherical, and hyperbolic cases geometrically using confocal conics and the Schläfli differential formula. An application of the symmetry is the organization and generation of rational tetrahedra.
