# Determining if two tetrahedra in $\mathbb R^3$ are identical or have reflection symmetry

I have two tetrahedra in $\mathbb R^3$, $T_1$ and $T_2$, and access to the coordinates of their vertices. $T_1$ and $T_2$ are tetrahedra in the sense that they each have four vertices, each vertex is connected to every other vertex, and no three edges lie along the same line. There are no further restrictions on their geometry.

Is there a simple method of determining whether the two tetrahedra are identical up to some rotation and translation operation? What if I wish to test whether $T_1$ and $T_2$ have some reflection (mirror) symmetry? One can, of course, exhaustively test for these things by attempting every possible mapping of $T_2$ onto $T_1$, which is my current strategy. However, I'm sure a better method must exist.

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Tetrahedra are determined up to congruence by the edge lengths, so you can just solve for the isometry sending one to the other and check if it preserves orientation. – Louis Dec 31 '11 at 13:29
Just to simplify things, how about translating and rotating both tetrahedra such that one vertex is the origin, the longest edge lies along one axis, and the face with largest area lies in a coordinate plane? – J. M. Dec 31 '11 at 13:31
See link.springer.com/article/10.1134%2FS0001434612030248#page-1 to note that edge length congruence is not enough. – bossylobster Jul 1 '14 at 16:47

$\frac{1}{6}(\vec{AB} \times \vec{AC}) \bullet \vec{AD}$.
One note: make sure to label the tetrahedrons' vertices isomorphically. In other words, for the purposes of the formula, if the side $AB$ has length $7$ on one tetrahedron, then the corresponding side should have length $7$ on the other tetrahedron too.