On Heron Simplices and Integer Embedding

In [3] problem D22 Richard Guy asked for the existence of simplices with integer lengths, areas, volumes . . . . In dimension two this is well known, these triangles are called Heron triangles. Here I will present my results on Heron tetrahedra, their connection to the existence of an integer box (problem D18), the tools for the search for higher dimensional Heron simplices and my nice embedding conjecture about Heron simplices, which I can only proof in dimension two, but I verified it for a large range in dimension three. 1 Heron triangles 1.1 Basics Proposition 1.1 (Heron formula) Let a, b and c be the lengths of the sides of a triangle, s the half of the perimeter and A the area. Then holds A = s(s − a)(s − b)(s − c). (1)