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 5 essays across 3 fields — 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

A twist along the horizontal loop, done once. Squares with opposite edges glued and a shaded horizontal band, showing one loop before and after the torus is twisted along the band.

A twist that carries one loop to another

Cut a torus along a loop, turn one side of the cut once round, and glue it back. Nothing is torn, so every loop that did not cross itself still does not — but a loop of class (0, 1) is now a loop of class (1, 1). Two such twists reach every loop that never crosses itself, by Euclid's algorithm, and the symmetries they generate are exactly the whole-number matrices of determinant one.

topology · Homotopy

The 4 reduced forms of discriminant −84, each in its place. The reduced binary quadratic forms of discriminant −84 (x² + 21y²; 2x² + 2xy + 11y²; 3x² + 7y²; 5x² + 4xy + 5y²) plotted at their roots in the upper half-plane, all inside the modular fundamental region.

Counting the classes that break factorisation

The class number measures how badly unique factorisation fails in a field, and defined through ideals it looks impossible to compute. Gauss computed it by hand, for every field he wanted, by counting quadratic forms — and every form can be squeezed, by changes of variable that keep its values, into exactly one small standard shape.

number · Unique factorisation

The class number is a degree

The integers of ℚ(√−23) fall into three classes of ideals, and that 3 can be found by counting forms. It is also the degree of a single number: the modular function j, evaluated at the three forms' points, gives three values that are not whole but are the roots of one cubic with whole-number coefficients. When there is only one class, the value is a whole number outright — and that is why e^(π√163) misses being whole by less than a trillionth.

algebra · Field extensions

Named alongside it

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

Class number Determinant Matrix Mediant Quadratic form Stern brocot tree Unimodular Unique factorisation Algebraic integer Bijection Class group Continued fractions

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