Concept

Modular group

The group of transformations z ↦ (az + b)/(cz + d) with whole-number entries and ad − bc = 1, acting on the upper half of the complex plane. It carries any pair of neighbouring fractions to any other, and its motions tile the half-plane with congruent curved triangles.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

$The tree as words in two matrices. 6 nodes of the Stern–Brocot tree, each as the word of turns reaching it, the matrix that word multiplies out to, its two columns as fractions, and the mediant of those columns.$

Two matrices that generate the tree

A node of the Stern–Brocot tree is not really a fraction — it is the pair of fractions it lies between. Written as the columns of a matrix, the two turns of the tree become two multiplications, and the determinant that kept everything in lowest terms becomes a property of a product.

number · Stern brocot

The Farey tessellation, and a line down to √2 − 1. Semicircles over the unit interval joining every pair of Farey neighbours with denominators up to 13, and a vertical line at √2 − 1. The 6 arcs it crosses are the intervals of the Stern–Brocot descent to √2 − 1, and their turns spell LRRLL.

The arcs a line crosses on its way to a number

Draw a semicircle over every pair of neighbouring fractions and the half-plane above the number line is cut into curved triangles that never overlap. A straight line dropped towards any number crosses those arcs one after another, and the arcs it crosses, and the side it leaves each triangle by, are exactly the steps of the Stern–Brocot descent towards that number.

number · Stern brocot

Named alongside it

The objects these essays reach for when they reach for this one.

Determinant Mediant Stern brocot tree Unimodular Bijection Continued fractions Farey sequence Lowest terms Matrix

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