The Elements of Enteger Geometry

Abstract

In [5] we described three kinds of number tree, giving their growth rules and some of their number properties. The third kind we called either the regular knot tree (RKT) or the rational number tree (RNT). The first name arose naturally, since the arrangement of numbers (pairs of integers) on its nodes reflects precisely the manner by which string-runs of regular knots — a class of braids — evolve (see [2,5]) from the unknot. Alternatively, if each pair of integers on a node is taken as a ratio, or rational number, then the same tree generates all the rational numbers, uniquely and in lowest form; hence the second name we gave to the tree. In the RNT guise, its properties have been studied by others: for example, in [4] Schroeder gives many results and mentions applications for its left-subtree, which is known as the Farey tree. In [1, page 63] Knuth, in a problem refers to it as the Stern-Peirce tree, and gives three early references.