Volume 6: The Berlin Years: Writings, 1914-1917 (English translation supplement) page 153

The Quadrahelix: A Nearly Perfect Loop of Tetrahedra

In 1958, S. Świerczkowski 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 ε, there is a tetrahedral loop such that its difference from a closed loop is less than ε. 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.