Closing a Platonic Gap

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 we close a gap in this area by presenting such rings for tetrahedra. The problem is that, in 1958, S. Świer-czkowski [SS, SS1] proved that no such ring exists! 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 each two tetrahedra in the chain have disjoint interiors. We 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 (assume centimeter-long edges through-out) having a gap of 20 mm. It seemed that the overall error could be reduced by using perturbations t