The Quadrahelix: A Nearly Perfect Loop of Tetrahedra

Abstract

In 1958, S. \'Swierczkowski proved that there cannot be a closed loop of congruent interior-disjoint regular tetrahedra that meet face-to-face. Such closed loops do exist for the other four regular polyhedra. It has been conjectured that, for any positive \epsilon, there is a tetrahedral loop such that its difference from a closed loop is less than \epsilon. We prove this conjecture by presenting a very simple pattern that can generate loops of tetrahedra in a rhomboid shape having arbitrarily small gap. Moreover, computations provide explicit examples where the error is under $10^{-100}$ or $10^{-10^{6}}$. The explicit examples arise from a certain Diophantine relation whose solutions can be found through continued fractions; for more complicated patterns a lattice reduction technique is needed.