Can Regular Tetrahedra be Glued Together Face to Face to form a Ring?

Abstract

Suppose someone gave you a large collection of identical regular tetrahedra. Would it be possible to glue them together face to face in such a way that you formed a nontrivial ring, or doughnut, in the sense that, during the glueing process, you could glue a tetrahedron both to your last tetrahedron and your first? Tetrahedra would be allowed to cut into each other if necessary.