TITLE: Tetrahedral torus
QUESTION [9 upvotes]: 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)?

REPLY [3 votes]: 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.