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Stern brocot — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · number
  2. 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.

    part 2 · number
  3. Every positive rational, in one sequence. The first 32 terms of Stern's diatomic sequence as bars, with the ratios of consecutive terms beneath. Every ratio is in lowest terms, no two agree, and each term counts the hyperbinary representations of its index.

    Every rational in one sequence

    The tree lists every positive fraction once and needs a tree to do it. One recursion on the whole numbers lists them in a single row — and each term of it counts something nobody was asking about, which is why the enumeration works.

    part 3 · number
  4. 13 record approximations in 26 turns. The distance from π to each fraction the descent passes, against its denominator, on logarithmic axes. 13 of them beat every fraction with a smaller denominator.

    The fractions that beat every smaller one

    Walking down the tree towards a number produces a sequence of fractions closing in on it. Most of them are steps along the way; a few are the best approximations there are — closer than every fraction with a smaller denominator — and which few is decided by where the turns change direction.

    part 4 · number
  5. 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.

    part 5 · number
  6. Minkowski's question-mark function. The graph of Minkowski's function ?(x) on the unit interval: continuous and increasing, sending each Stern–Brocot fraction to the binary fraction in the same position. It sends √2 − 1 to 2/5 and φ − 1 to 2/3.

    The function that sends fractions to binary

    The Stern–Brocot tree and the tree of binary fractions have exactly the same shape, so there is a function that sends each fraction to the binary fraction in the same position. It is continuous and increasing, it turns every quadratic irrational into an ordinary fraction, and it does all of its rising on a set of numbers so thin that at almost every point its slope is nought.

    part 6 · number

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