The Quadrahelix: A Nearly Perfect Loop of Tetrahedra

TL;DR

This paper proves the conjecture that nearly perfect closed loops of congruent tetrahedra can be constructed with arbitrarily small gaps, using simple patterns and Diophantine relations, advancing understanding of polyhedral loops.

Contribution

The paper introduces a simple pattern to generate nearly closed tetrahedral loops with arbitrarily small gaps, and provides explicit examples with extremely small errors.

Findings

Existence of tetrahedral loops with gaps less than 10^{-100}

Explicit construction methods using Diophantine relations

Demonstration of arbitrarily small gaps in tetrahedral loops

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.