The number of tetrahedra sharing the same metric invariants via symbolic and numerical computations

Abstract

The problem consists in bounding the number of tetrahedra that share the same volume V, circumradius R, and face areas A1,A2,A3,A4. A symbolic computation approach is used to prove that the upper bound of the number of tetrahedra is eight. First, the problem is reduced to a counting root problem using the Cayley-Menger determinant formula for the volume of a simplex and some metric equations of the tetrahedron to construct a system of algebraic equations. Then, an investigation of its real roots is done by analyzing the discriminant sequence and extensive numerical computations.