Buchholz [R.H. Buchholz, Perfect pyramids, Bull. Austral. Math. Soc. 45 (1991) 353-368] began a systematic search for tetrahedra having integer edges and volume by restricting his attention to those with two or three different edge lengths. Of the fifteen configurations identified for such tetrahedra, Buchholz leaves six unsolved. In this paper we examine these remaining cases for integer volume, completely solving all but one of them. Buchholz also considered Heron tetrahedra, which are tetrahedra with integral edges, faces and volume. Buchholz described an infinite family of Heron tetrahedra for one of the configurations. Another of the cases yields a new infinite family of Heron tetrahedra which correspond to the rational points on a two-parameter elliptic curve.

This paper discusses tetrahedra with rational edges forming an arithmetic progression, focussing specifically on whether they can have rational volume or rational face areas. Several infinite families are found which have rational volume, a face can have rational area only if its edges are themselves in arithmetic progression, and a tetrahedron can have at most one such rational face area.

This paper discusses tetrahedra with rational edges forming a geometric progression, focussing on whether they can have rational volume or rational face areas. We examine the 30 possible configurations of such tetrahedra and show that no face of any of these has rational area. We show that 28 of these configurations cannot have rational volume, and in the remaining two cases there are at most six possible examples, and none have been found.