I put in some time [too much, probably] on rational simplices over the holiday: here's a summary. Edge-lengths in order [AB,AC,BC,AD,BD,CD,AE,BE,CE,DE,...] where vertices are A,B,C,D,E,...; cases differing only by a permutation of the vertices considered equivalent. A simplex spanning n dimensions is "proper" when every subset with n vertices spans n-1 dimensions; "noncyclic" when none with n+1 lies on a (n-1)-sphere. [The latter implies the former if an extra vertex at inversive infinity be attached --- this will assumed]. 4 vertices in 2 dimensions --- For edges not exceeding 32, there are 513 distinct proper integer cases. Approx. 80 percent of these are trapezia or parallelograms, each constituting a rational 4-parameter class: quadratic for trapezia [a*b, (b*d+a*c)/2, (b*d-a*c)/2, (b*d-a*c)/2, (b*d+a*c)/2, c*d]; cubic for parallelograms [(p^2+q^2)*t+4*p*q*s, (p^2-q^2)*t+(p^2+q^2)*s, (2*p*q)*t+(p^2+q^2)*s, (2*p*q)*t+(p^2+q^2)*s, (p^2-q^2)*t+(p^2+q^2)*s, (p^2+q^2)*t+2*(p^2-q^2)*s]. The latter is a linear combination of rectangle and rhombus, both constructed from Pythagorean triples. [Care is required with both to ensure that the results are proper and integer.] 5 vertices in 2 dimensions --- All but 129 of the 513 are concyclic, and it has been earlier established that there are concyclic proper integer m-vertex configurations for any m; however, the construction does not generalise to more than 2 dimensions. This suggests exploring noncyclic cases: it transpires that there is just one such plane 5-vertex with edges up to 32: [26, 25, 9, 18, 20, 13, 13, 15, 18, 19]. 5 vertices in 3 dimensions --- for edges up to 10, there are 63 distinct proper integer cases. Just 2 of these are conspheric [with the same radius] [9, 9, 6, 8, 7, 5, 7, 8, 4, 7], [9, 8, 7, 8, 5, 4, 7, 8, 7, 9]. In contrast, there is also the single ambiguous pair [7, 6, 2, 6, 2, 3, 5, 4, 4, 4], [7, 6, 2, 6, 2, 3, 4, 5, 4, 4]. 6 vertices in 3 dimensions --- none for edges up to 10. Although there are numerous pairs which may be glued together at a common tetrahedron, either the single new edge created is irrational, or else one of the component tetrahedra turns out to be planar. Almost certainly a considerably more extensive search is required to find such an object, if it exists. No proper integer case has emerged with any edge of length unity; just one with edge 2 has been found in 2 dimensions, and only 12 out of 63 in 3 dimensions. Fred Lunnon