An asymptotically closed loop of tetrahedra (Q1684352)

A tetrahedral chain is a finite sequence of congruent regular tetrahedra \(T_1,\dots , T_n\) in Euclidean 3-space such that \(T_k\) and \(T_{k+1}\) have strictly one facet in common and \(T_k\neq T_{k+2}\) (i.\,e., tetrahedra meet face to face, but never double back through a just-used face). In [Colloq. Math. 7, 9--10 (1959; Zbl 0092.38701)], \textit{S. Świerczkowski} proved that a tetrahedral chain cannot be a closed loop (his proof is free from additional assumption that the chain is embedded, i.\,e., has no self-intersection). The discrepancy of a chain from a closed loop is the smallest \(d\) such that there are two tetrahedra, \(T_i\) and \(T_j\), in the chain, their facets, \(F_i\subset T_i\) and \(F_j\subset T_j\), and an isometry \(A\) of Euclidean 3-space such that \(A(F_i)=F_j\), \(A(T_i)\neq T_j\), and \(\| x- Ax\|\leq d\) for all \(x\in F_i\). The main result of the paper under review reads as follows: for any \(\varepsilon>0\) there is an embedded tetrahedral chain, so that the discrepancy from a closed loop is less than \(\varepsilon\). In order to prove this statement, the authors explicitly construct a suitable embedded tetrahedral chain. Continued fractions also play a role in the proof as well as graphic and algebraic capabilities of \texttt{Mathematica}.