# 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)?

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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.

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If you have a JSTOR subscription (or access to a library which does), the link is jstor.org/stable/3616971 –  Willie Wong Jan 28 '11 at 20:38
Interesting to note that the author also allows immersions: the tetrahedra are allowed to cut into one another. It may seem strange that this "unphysical case" is allowed at first, but in fact by doing so the question becomes simpler (one can re-word the problem purely group theoretically). –  Willie Wong Jan 28 '11 at 20:44
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.