Rational and Heron tetrahedra

Abstract

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.

Constructing $7$-clusters

“…Due to the formula for A∆(a, b, c) for an integer sided triangle with characteristic 1, the area is rational and may in principle be non-integral. Nevertheless one may consider the cases of the side lengths modulo 8 (see [6]) and conclude that such triangles have to be integral. We summarize these findings in: Corollary 2.6.…”

On the generation of Heronian triangles

“…It is a conjecture of [8] that no non-degenerated solution exists in this case. For the cases 4(iv), 4(vii), 5(ii), and 6(i) sporadic solutions are known.…”

“…Here it is possible that the edge lengths of a tetrahedron are integral and that the volume is genuinely rational. If the edge lengths are integral and the face areas and the volume are rational, then all values must be integral, see [8]. In [3] it was also shown that a perfect pyramid with at most two different edge lengths cannot exist.…”

“…For three different edge lengths a parameter solution of an infinite family is given. More comprising the authors of [3,8] consider all possibilities of coincidences of edge lengths. Up to symmetry we have the following configurations: In the cases 1(i), 2(i-v), 3(i), 3(iii), 3(iv), 3(vii), and 4(i) there is no solution.…”

“…They proved that tetrahedron with integral edge lengths, rational face areas and rational volume do not exist. If only one face area is forced to be rational then there exist the example (a, b, c, d, e, f ) = (10,8,6,7,11,9), which is conjectured to be unique up to scaling. We have verified this conjecture up to diameter 600000.…”