# Is $\mathbb R^7$ minimally sufficient for embedding 3 tetrahedra - ABCD, ABEF, and CEGH - of equal edge length?

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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Why doesn't R^3 work? Allow AB to be the Z axis - say A at (0,0,-1) and B at (0,0,1); then start with C=(-2, -1, 0) and D=(-2, 1, 0) - so the tetrahedron ABCD goes in the negative X direction - and E=(2, -1, 0), F=(2, 1, 0), so ABEF goes in the positive X direction. Now you should be able to rotate one of these about the Z axis to give CE whatever length you need. – Steven Stadnicki Jul 25 '12 at 19:20
Yeah, I'm not seeing why you can't force $CE$ to be the same distance as $AB$ in $\mathbb R^3$. – Thomas Andrews Jul 25 '12 at 19:21
Wow, you guys are right. Thank you. I'm terrible at thinking in 3d - rotation was the trick. – CHCH Jul 25 '12 at 19:39
@CHCH In fact, you can pack five tetrahedra around a common edge this way, but not quite perfectly - there's a little bit of space left over. Folding into the fourth dimension to close up that space gives you the 600 cell (en.wikipedia.org/wiki/600-cell ). – Steven Stadnicki Jul 25 '12 at 20:54
Mind=blown. But can you point me in the right direction for getting an intuition for what you mean by "folding into", or thinking more fluidly about this? I guess I should work on understanding R^3 first... – CHCH Jul 26 '12 at 2:33

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');

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