Tetrahedral torus

Is it possible to form a closed loop by joining regular (platonic) tetrahedrons together side-to-side, with each tetrahedron having two neighbours? It should be a loop with a hole in, as can be done with 8 cubes, or 8 dodecahedrons as shown below. What is the minimum number of tetrahedrons needed?

Edit: Could it be that it is possible to create such a ring by allowing the tetrahedrons to extend in one more spatial dimension (R^4)?

Solution 1:

No, it isn't possible. Here is the reference, which I haven't found online (maybe someone can find a link):

Mason, J. (1972). "Can Regular Tetrahedra Be Glued Together Face to Face to Form a Ring?" Mathematical Gazette 56 (397) p.194-197.

Solution 2:

One can get interesting geometrical "rings" of tetrahedra when they are attached edge to edge: http://www.mathematische-basteleien.de/kaleidocycles.htm On this page some of the examples use regular tetrahedra and others less regular ones.

Solution 3:

To the follow up question about allowing embeddings in $\mathbb{R}^4$: the answer is yes. One simple example is to start with the hexadecachoron or 600-cell and remove some of the constituent tetrahedra.