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

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

Named alongside it

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

Continued fractions Lowest terms Mediant Stern brocot tree Bijection Counting two ways Diophantine approximation Ford circles Irrationality Rational approximation Tangency Unimodular

All concepts