Concept

Liouville number

A number approachable by fractions faster than any fixed power of the denominator allows. Every one of them is transcendental, since an algebraic number of degree d cannot be approached better than the dth power permits, and one is built by spacing decimal digits factorially.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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

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

The coordinate change that makes the map a rotation. Graphs of the departure from the identity of the conjugacy between the circle map and the golden rotation, at three nonlinearities, each smooth and growing steeper with K.

How smooth the disguise is

An unlocked circle map is a rigid rotation in different coordinates, and whether those coordinates are smooth is decided by arithmetic. Solving for the change of coordinates means dividing, harmonic by harmonic, by numbers that come arbitrarily close to nought — and the golden rotation number keeps them far enough away that the result is not merely smooth but analytic, while a number lying close to a fraction makes it buckle.

dynamics · Mode locking

Named alongside it

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

Continued fractions Diophantine approximation Existence proof Rational approximation Transcendence Algebraic number Analytic function Circle map Conjugacy Construction Continued fraction convergent Counting argument

All concepts