Tetrahedra with congruent face pairs

If the four triangular facets of a tetrahedron can be partitioned into pairs having the same area, then the triangles in each pair must be congruent to one another. A Heron-style formula is then derived for the volume of a tetrahedron having this kind of symmetry. Mathematics Subject Classification: 52B10, 52B12, 52B15, 52A38. From elementary geometry we learn that two triangles are congruent if their edges have the same three lengths. In particular, there is only one congruence class of equilateral triangles having a given edge length. Said differently, any pair of equilateral triangles in the Euclidean plane are similar, differing at most by an isometry and a dilation. Meanwhile, triangles that are symmetric under a single reflection have two congruent sides and are said to be isosceles. The situation is more complicated in higher dimensions. Indeed an analogous characterization of 3-dimensional tetrahedra already leads to 25 different symmetry classes [19]. These tetrahedral symmetry classes are of special interest in organic chemistry [6, 7, 18], and conditions for tetrahedral symmetry based on the measures of dihedral angles have also been explored [20]. A tetrahedron in R is equilateral or regular if all of its edges have the same length. More generally, a tetrahedron is said to be isosceles if all four triangular facets are congruent to one another, or, equivalently, if opposing (non-incident) edges have the same length. Isosceles tetrahedra are also known as disphenoids [3, p. 15]. It has been shown that if all four facets of a tetrahedron T have the same area, then T must be isoceles [8, p. 94][9, 13]. Consider the following more general symmetry class of tetrahedra: A tetrahedron T will be called reversible if its facets are congruent in pairs; that is, if the facets of T can be labelled f1, f2, f3, f4, where f1 f2 and f3 f4.