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Concept

Rational approximation

A fraction standing in for a number that is not one, and the question of how much accuracy a given denominator buys. How well a number can be approximated is measured against the denominator, and the golden ratio is the worst case.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

φ as a continued fraction. The nested fraction, one quotient per step, descending to the right.

A fraction that never closes

Euclid's algorithm throws away everything except the number of squares it peeled at each step. Those counts are a second name for the number it started from — one that terminates exactly when the ratio is a ratio.

number · Euclidean algorithm 8 multiples of φ in 7 boxes. The fractional parts of the first multiples of a number, dropped into equal boxes along the unit interval.

How close a fraction can get

Drop eight points into seven boxes and two of them share. That one line, applied to the multiples of an irrational number, proves that every irrational has infinitely many astonishingly good rational approximations — and no construction is needed anywhere.

number · Pigeonhole How closely a fraction can come, and the barrier that says no closer. Two panels at very different scales: the approximations to √2, which stay above the barrier a degree-two number obeys, and the truncations of a constructed number, which fall below every barrier drawn.

Approached too fast to be algebraic

An algebraic number of degree d cannot be approached by fractions faster than the denominator's dth power. So a number that is approached faster than that is the root of no polynomial at all — and one can be built by choosing where its decimal digits go.

number · Irrationality 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 tangent built from a continued fraction. The curve tan x on (−1.55, 1.55) with 4 of Lambert's convergents: a straight line, then rational curves that bend ever closer to the tangent and follow it towards its poles.

The fraction Lambert built for the tangent

The first proof that π is not a fraction, from 1761, does not look at π at all. It writes the tangent as an endless continued fraction, shows that the fraction's value at any rational point other than zero cannot be rational — because its tails are trapped between nothing and one — and then notes that tan(π/4) = 1.

number · Irrationality The continued fraction of e. Bars for the first 30 continued-fraction terms of e: mostly ones, with every third bar rising in a straight staircase, 2, 1, 2, 1, 1, 4, 1, 1, 6, ….

The pattern in e's continued fraction

Written as a continued fraction, e is 2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8 — two ones, then the next even number, for ever. Euler found the pattern and proved it with a differential equation. A proof from 2006 needs only three integrals, each of which turns out to be exactly the error of one of e's own convergents.

number · Irrationality Riemann's function on one period, with its one kind of smooth point. R(x) = Σ sin(n²x)/n² on [0, 2π]; R(π) = 0 with derivative −1/2; R(π/2) = 1.2336, a cusp.

The slope a Gauss sum leaves behind

Riemann is said to have offered sin x + sin(4x)/4 + sin(9x)/9 + … as a continuous function with no derivative anywhere. It has one after all, at π and at every π times an odd number over an odd number, and the slope there is always exactly −1/2. Computed with a million terms, the function's behaviour at every fraction is read off a single number — a quadratic Gauss sum — and the slope appears exactly where that number is zero.

analysis · Uniform convergence The tree of solutions of x² + y² + z² = 3xyz. Markov triples from (1, 2, 5) to depth 4; 231 Markov numbers below 10^15; first 1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, 1325.

The numbers that come after the golden ratio

The golden ratio is the number worst approximated by fractions: no constant larger than √5 works in Hurwitz's theorem. Set it aside and √2 is worst, with √8. Set that aside and the next is a number whose continued fraction repeats 2, 2, 1, 1, with √221/5. Markov proved in 1879 that the list continues for ever below 3, one entry for each solution of x² + y² + z² = 3xyz.

geometry · Golden ratio

Named alongside it

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

Continued fractionsDiophantine approximationGolden ratioIrrationalityContinued fraction convergente, the numberExistence proofLiouville numberPeriodicityTranscendenceAlgebraic numberBinary trees

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