Mediant
Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
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.
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.
The fraction written on every bulb
Round the edge of the Mandelbrot set's cardioid hangs one bulb for every fraction between nought and one. The fraction is the bulb's address, its period, and the turn its cycle makes round a fixed point, all at once, and it sets the bulb's size as well: sin(πp/q)/q², which 245 computed bulbs obey to within fourteen per cent.
A small quotient for every denominator
Pick a denominator, any whole number. Among the fractions with that denominator, is there always one whose continued fraction has no large entries — whose path down the Stern–Brocot tree never turns the same way more than five times running? Stanisław Zaremba guessed so in 1972, for a reason to do with sampling a square evenly. Every denominator up to a hundred thousand obeys, with only 6, 54 and 150 needing a five. The guess is proved for almost every denominator and not for all, and the reason it ought to be true is a dimension: the reals whose quotients stay at most five fill a set of dimension 0.837, enough to give each denominator thousands of good fractions on average, and the open question is whether averages are kept everywhere.
Named alongside it
The objects these essays reach for when they reach for this one.
Stern brocot treeFarey sequenceContinued fractionsLowest termsBijectionUnimodularCounting two waysDeterminantModular groupBinaryConjectureCountability