An even number of (at least 8) tetrahedra can be connected along their edges to form a ring in a way that allows them to be continuously rotated “inside-out” without disconnecting. Such configurations are commonly referred to as kaleidocycles. Shown above are kaleidocycles with 8, 10, and 12 tetrahedra exhibiting 4, 5, and 6-fold rotational symmetry, respectively. There has to be at least 8 tetrahedra, because any less would result in the tetrahedra colliding into each other at certain instances of the rotation. You can even make your own paper model using this guide.