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Concept

Fourier series

A periodic function written as a sum of sines and cosines at whole-number multiples of one frequency. It turns a condition on a shape into a condition on which frequencies may appear, and differentiating becomes multiplication by the frequency.

Named by 11 essays across 6 fields — each of them below, with the objects they name alongside it.

6 cosines, and a curve with no tangent anywhere. Partial sums of a sum of cosines whose amplitudes shrink geometrically and whose frequencies grow faster. Each term adds finer detail; the curve converges and its slopes do not.

A curve with a corner at every point

Continuity means a curve can be drawn without lifting the pen. Differentiability means it has a tangent. The first was assumed to nearly imply the second until 1872, when Weierstrass exhibited a curve that is continuous everywhere and has a tangent nowhere — and it is a sum of cosines.

analysis · The derivative The two numbers that make up a width. A Reuleaux polygon with 3 sides, its centre marked as the origin, and the two supporting lines with normals 33° and 213°. The perpendicular distances from the origin to the two lines are marked; they add to the width.

The shape described from outside

A convex shape can be given by its boundary or by the family of lines that touch it, and the second description turns the constant-width condition into one line of arithmetic — after which the perimeter falls out, and curves with no corners at all can simply be written down.

geometry · Constant width Four shapes of perimeter 300 between their inner and outer circles. A square, an ellipse, a Reuleaux triangle and a stadium, each drawn with the largest circle inside it and the smallest circle around it, the ring between the two shaded, with the ring's width and the widest ring allowed.

Nearly the most means nearly round

A shape that holds almost as much as a circle of the same perimeter must almost be a circle. Bonnesen made that exact: the ring between a convex shape's largest inscribed circle and smallest enclosing circle is never wider than √(L² − 4πA)/π. Three quite different shapes holding 99% of the circle's area all have rings under 9.55 wide, and not one of 200 random convex shapes breaks the bound.

geometry · Isoperimetric How long four orbits of the doubling map spend in each quarter. Grouped bars for four starting points of the doubling map, each group showing the fraction of time the orbit spends in each quarter of the unit interval, against a dashed line at one quarter.

Almost every orbit is fair

Double a number and keep the fractional part, and do it again, and again. Where the orbit goes is written in the number's binary digits, and for almost every starting point it spends exactly a quarter of its time in each quarter of the interval. The proof is a short argument about Fourier coefficients being pushed to infinity — and it leaves room for a set of exceptions with no length at all, which includes every fraction and, for all anyone can prove, every number anyone has ever named.

dynamics · Iteration Integration error against the number of points: lattice, Halton and random, on a smooth periodic function. Log-log plot of integration error against N for random points (slope -0.47), Halton points (-0.96) and Fibonacci lattices (-1.83).

Points on a lattice that see almost nothing

Average a function over the points of a carefully tilted lattice and the error can fall like one over the square of the number of points — far faster than random sampling, and faster than the most evenly spread sequences. The reason is that a lattice rule is blind to only a thin set of frequencies, its dual lattice, and a smooth periodic function has almost nothing there.

probability · Monte Carlo A disc of radius 0.2 and an ellipse of size 1.22 around the same interval. For 1/(1 + 25x²): the interval from −1 to 1 on the real axis, poles at 0 + 0.2i and 0 − 0.2i, the Taylor disc at 0 of radius 0.20, and the Bernstein ellipse with foci ±1 through the poles, with ρ = 1.2198.

An ellipse, not a disc

A Taylor series converges on a disc, and the disc's radius is the distance to the nearest singularity. Ask instead how well polynomials can follow a function on an interval, and the answer is an ellipse with the interval's ends as its foci — the largest one the function is smooth inside.

analysis · Taylor series A countable closed set stripped of its isolated points, 3 times. Cantor–Bendixson derivatives of the set of sums of up to 2 reciprocals: sizes 750, 40, 1, 0 in the truncation, ending empty.

Closed sets obey the continuum hypothesis

Whether some set of reals is bigger than the whole numbers and smaller than the line is a question the axioms cannot answer. For closed sets it has an answer, found in 1883: strip away the isolated points, again and again, and what is left is either nothing or a perfect set, which is as large as the line — so every closed set is countable or of the line's size. The same answer holds for Borel and analytic sets, and fails to be provable exactly one step further up.

logic · Cardinality The primitive 15th roots of unity, raised to a power and added. c₁₅(1) = 1; c₁₅(3) = −2; c₁₅(5) = −4; c₁₅(6) = −2; c₁₅(10) = −4; c₁₅(15) = 8.

The sums of primitive roots are always whole

Take the roots of unity of order exactly q, raise each to the power n and add them. The answer is always a whole number, it depends on n only through what n shares with q, and as n varies the sums behave like the sines and cosines of a Fourier series — so well that Ramanujan could rebuild the sum of the divisors of any number from them, and prove that their weighted total is nought, a fact that at n = 1 is the prime number theorem.

algebra · Roots of unity 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 Leading digits of products of random numbers, against Benford's law. 1 factor: ones 11.1%, nines 11.1%; 2 factors: ones 24.1%, nines 3.4%; 3 factors: ones 30.1%, nines 4.2%; 6 factors: ones 30.1%, nines 4.6%; the law gives 30.1% and 4.6%.

Multiplying makes the digit one common

Multiply a few random numbers together and the product starts with 1 about 30 per cent of the time and with 9 under 5 per cent — Benford's law, which no factor contains. The logarithm of a product is a sum, the central limit theorem spreads that sum across many powers of ten, and once it is spread its fractional part is uniform. The approach is geometric, at a rate fixed by a single number for each kind of factor; sums never get there, and the powers of two get there with no randomness at all.

probability · Central limit Heights that shrink to nothing, slopes that grow without bound. Two stacked panels: sin(n²x)/n for n = 2, 3, 5 shrinking towards zero, and their derivatives n·cos(n²x) growing in amplitude.

Close in height and nowhere close in slope

Uniform convergence carries continuity to the limit and carries integrals to the limit. It carries slopes nowhere. The functions sin(n²x)/n flatten to nothing while their slopes grow without bound; smooth curves converge uniformly to a corner; a Fourier series converges and its differentiated series diverges. What does carry a slope is uniform convergence of the slopes themselves — and the reason is that integration averages wiggles away while differentiation multiplies them.

analysis · Uniform convergence

Named alongside it

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

ContinuityConvergenceConvex hullConvexityCounterexampleEllipseUniform convergenceAnalytic functionApproximationBinary expansionCantor setCardinality

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