Concept

Unimodular

Having determinant one, for a matrix of whole numbers. Such a matrix has a whole-number inverse and takes the integer lattice exactly onto itself, and the condition is what keeps a mediant in lowest terms and what makes two Ford circles touch.

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

The Stern–Brocot tree to depth 4. Every positive rational, each appearing exactly once, generated by taking mediants.

Every fraction, exactly once

Take two fractions, add the tops and add the bottoms. That is not how fractions are added, it is not an average, and repeating it produces every positive rational exactly once, already in lowest terms.

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

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

Named alongside it

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

Bijection Lowest terms Mediant Stern brocot tree Continued fractions Counting two ways Determinant Farey sequence Ford circles Matrix Modular group Tangency

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