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Concept

Farey sequence

Every fraction in the unit interval with denominator at most a given bound, listed in increasing order. Adjacent entries always satisfy a determinant condition of one, which is what makes their circles tangent and what keeps every entry in lowest terms.

Named by 5 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 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.

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 The 46 Farey fractions of order 12, and how far each strays from even spacing. Farey fractions of order 12 against 46 evenly spaced points, with the deviation of each drawn as a bar; largest deviation 0.0616.

How evenly the fractions spread

List every fraction between nought and one with denominator at most n, in order. They spread across the interval almost evenly, and how fast the unevenness shrinks as n grows is — exactly, provably — the Riemann hypothesis. The link runs through a second fact: set the fractions round a circle and add them as arrows, and what is left is a whole number.

number · Stern brocot The tree writes 4/17 as four unit fractions, and a search finds three. Left-neighbour chain 4/17, 3/13, 2/9, 1/5, 0/1, gaps 1/221, 1/117, 1/45, 1/5; three-term decomposition 1/5 + 1/30 + 1/510.

Three unit fractions for every four over n

Two neighbouring fractions in the tree always differ by a unit fraction, so stepping down the tree writes any fraction as a sum of them — four over n in at most four steps. Erdős and Straus asked in 1948 whether three always suffice. Three identities settle every n except those leaving remainder 1 on division by 24, a search settles every one anyone has tried, and nobody has a proof.

number · Stern brocot

Named alongside it

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

MediantStern brocot treeContinued fractionsLowest termsUnimodularBijectionConjectureCounting two waysDeterminantDiophantine approximationDiscrepancyEquidistribution

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