I've got 8 points - A, B, C, D, E, F, G, H - and I need three specific sets of four (ABCD, ABEF, and CEGH) to describe tetrahedra of equal edge length in some multidimensional space.
I can embed ABCD and ABEF in $\mathbb R^3$, but I can't also put CEGH in $\mathbb R^3$, because then the edge connecting C and E is longer than the others, and there's nowhere to put G and H.
Of course I can go crazy and use the 7-simplex in $\mathbb R^7$ - i.e., so that all possible pairs of four are tetrahedra - but I'd like to avoid using extra dimensions.
The question: What is the minimal dimensionality I need for these 3 tetrahedra? And, any pointers as to how to find the cartesian coordinates of the vertices?