TITLE: Which platonic solids can form a topological torus?
QUESTION [12 upvotes]: 8 cubes can be joined face-to-face to form a closed ring with a hole in it, with each cube sharing a face with only two others. The same can be done with 8 dodecahedrons.
Is the same possible with the other platonic solids?
What is the minimum number of each solid needed to form such a loop?  
Given a convex regular d-polytope, is there a general way to determine if it can form a loop, by gluing their d-1 dimensional faces together? (Assuming the loop has a hole and no two objects intersecting, ie no two objects share a d-volume).
And is there a way to compute the minimum number of equal polytopes needed for this?

Edit: added image
I am also looking for software that can be used to check for atleast small N

REPLY [4 votes]: A long time ago a similar question arose in the sci.math newsgroup. I worked it out that in order to make a loop out of dodecahedra, one has to use sets of three dodecahedra in a row, as shown (four times) in the picture above -- otherwise one introduces a set of rotations in R^3 which has no relations which we can use to get the first and last faces to line up. See http://www.math-atlas.org/98/dodec_prf
dave