The following has come up during an undergraduate research project. It's rather elementary, but we are having trouble finding anywhere that mentions it. It might have something to do with rigidity.
Consider two tetrahedrons that share a base. What is the relationship between the lengths of its sides and the diagonal? We suspect there should be some polynomial relationship between the lengths (that's what happens in the plane case). By introducing Cartesian coordinates and assuming the triangle is ABC is in the xy-plane, A is the origin, and AB is on the x-axis, I was able to compute the coordinates of C using sines and cosines. I'm assuming I know the lengths of all sides except for the orange diagonal in the picture. I can then look at the intersection of the three spheres centered at A,B and C with radii |AD|, |BD|, and |CD| to compute |D|. I can do the same thing for E, and then I can use the distance formula to solve for |DE|. I can replace my sine in compute C with $\sqrt{1-\cos^2(\theta)}$, and then use the law of cosines to write cosines in terms of the lengths of the triangle. In this way I get a relationship just between the lengths of the tetrahedron. I'm not very interested in computing |DE| from the lengths of the other sides, we're really just after a relationship between the sides that is an invariant for all "double tetrahedrons" (if anyone knows what this shape is called, please let me know).
Computing the relationship in the manner I did above worked, with the help of Maple, the the expression was had lots of square roots in it. The problem is that the y coordinate of C contained a square root ($\sqrt{1-cos^2(\theta)}$) where $\theta$ is the ABC angle and $\cos(\theta)$ turns out to be $\frac{AB^2+BC^2-AC^2}{2AB\cdot BC}$. My professor suggested to take my original relation, solve to the y coordinate of C, and square it and hope this leaves up with a polynomial. Maple, however, was unable to solve for this.
Does anybody have any idea on how to approach this problem, or any reference to similar problems? In the plane case - with polygons and diagonals - we were able to work out the relationships between the lengths for pentagons and were able to give results for $n$-gons. The relationships turn out to be cubic polynomials with the square of the lengths as variables). This stuff is very elementary, but we cannot find anyplace where it has been done. I've ran out of time right now, but I will return later tonight to try and make the problem more clear or go through more details. Thanks in advance for any insights or comments.