Closing a

For four of the five Platonic solids, there is a toroidal ring using copies of the solid that meet face to face. This is trivial for cubes and not too hard for octahedra, icosahedra, and dodecahedra (Fig. 1). Here a gap in this area is closed, and such rings are presented for tetrahedra. The problem is that, in 1958, S. Świer-czkowski [SS, SS1] proved that it cannot be done! Figure 1. Octahedral, dodecahedral, and icosahedral tori. 1. Background In 1957 Hugo Steinhaus [HS] asked whether there was a perfectly closed loop of congruent regular tetrahe-dra. Precisely: Is there a finite sequence T1,…, Tn of regular tetrahedra that meet face-to-face and do not double back (such is called a Steinhaus chain), and are such that a face of Tn coincides with a face of T1? Świerczkowski showed that such a closed chain cannot exist. Other approaches were found by Dekker [D] and Mason [M]. General chains are allowed to intersect, so let us call a Steinhaus chain embedded if there are no intersections among the tetrahedra. I wondered how close one could get to a true tetratorus (an embedded tetrahedral torus made from regular tetrahedra). Initial explorations led to a loop of 48 tetrahedra (with unit length edges) having a gap of about 0.20. It seemed that the overall error could be reduced by using perturbations to spread the gap around th