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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?

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    @CHCH The construction of the 600-cell in $\mathbb{R}^4$ works very similarly, except that instead of closing the gap between two triangles around a point you're closing the gap between two tetrahedra around an edge. There's less folding to be done - five tetrahedra come closer to completing a circle around your edge than five triangles do around your point - so you need more tetrahedra to eventually 'close up' the shape, which is why it has 600 tetrahedral faces (as opposed to the icosahedron's 20 triangles).2012-07-26

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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');