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?
Thanks everybody! Here's the matlab code to graph the solution; it can be freely rotated too:
x=[0.67 0.67 -0.33 1.00 -0.33 1.00 -1.33 -1.33];
y=[-0.50 0.50 -0.50 -0.83 0.50 0.83 -0.50 0.50];
z=[0.50 -0.50 -0.50 -0.83 0.50 0.83 0.50 -0.50];
t=[1 2 1 3 1 4 2 3 2 4 3 4 1 2 1 5 1 6 1 2 5 2 6 2 5 6 2 3 5 3 7 3 8 3 5 7 5 8 7];
%Edges are: AB AC AD BC BD CD AB AE AF BE BF EF CE CG CH EG EH GH
plot3(x(t),y(t),z(t));
xlabel('x'); ylabel('y'); zlabel('z');
text(.67,-.5,.5,'A');
text(.67,.5,-.5,'B');
text(-.33,-.5,-.5,'C');
text(1,-.83,-.83,'D');
text(-.33,.5,.5,'E');
text(1,.83,.83,'F');
text(-1.33,-.5,.5,'G');
text(-1.33,.5,-.5,'H');