# Rational $d$-simplices

Define a rational $d$-simplex as a simplex in $\mathbb{R}^d$ such that the measure of all its $k$-dimensional faces, $k \ge 1$, is rational. So a rational triangle has rational edge lengths and rational area. A rational tetrahedron has six rational edge lengths, four rational face areas, and rational volume. And so on. (By scaling these could all be natural numbers.)

Q. For which $d$ is it known that there are an infinite number of non-similar rational simplices in $\mathbb{R}^d$ ?

(The non-similar restriction is to exclude scalings.) This is in some sense a request to update Richard Guy's Unsolved Problems In Number Theory, Problem D22. But I am specifically interested in the boundary between an infinite number of examples, and a finite class of distinct examples. Is that boundary: Yes ($\infty$) for $d \le 3$, and No (or unknown?) for $d > 3$? Or is the boundary: $d \le 2$ vs. $d > 2$? Or some other divison?

(Image from MathWorld. Face areas $1170, 1800, 1890, 2016$; volume $18144$.)

Update. Answered in the comments by j.c. and Moritz Firsching: Infinite families are known up to $d=3$, but not for $d \ge 4$ (and it seems unlikely even in $d=4$).

• It may be worth linking your previous highly related question mathoverflow.net/a/71968 . Also, the paper of Buchholz "Perfect Pyramids" dx.doi.org/10.1017/S0004972700030252 cited by Guy in Problem D22 gives an infinite family of rational 3-simplices, according to the abstract of a more recent paper by Chisholm and MacDougall "Rational and Heron tetrahedra" dx.doi.org/10.1016/j.jnt.2006.02.009 . – j.c. Apr 23 '16 at 1:15
• Buchholz examines 4-dimensional pentahedra and writes: "Since any simplex has 10 edges, 10 faces, 5 volumes and 1 hypervolume, the occurrence of a perfect 4-pyramid, which requires all these quantities to be integral, is probably a rarity at best." It gets worse for higher dimensions. (He goes on to find the n-simplices that have rational edges lenght and rational (full-dimensional) volume. – Moritz Firsching Apr 23 '16 at 6:59