Regular Tetrahedra Formed by Lattice Points Equidistant from the Origin

pick a tetrahedron

1

norm = 1

3

edge length = 2

2

From an integer lattice, pick four points at the same distance from the origin (called the norm) that determine the vertices of a regular tetrahedron. Add the condition that the set of all 12 coordinates (

(x,y,z)

for each of the four vertices) does not have a common factor. Solutions we have found so far always appear to have a norm of the form

(2n-1)

3

.