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Concept

Periodicity

The property of a pattern that repeats itself exactly after a fixed step. It is what a Fourier series assumes, and imposing it on a function that lacks it is what produces the artefacts at the ends.

Named by 11 essays across 5 fields — each of them below, with the objects they name alongside it.

A circle unrolled into a sine wave. On the left a radius turns through an angle; on the right the height of its tip is plotted against the angle, tracing a sine curve.

A sine wave is a circle seen from the side

Sine is introduced as a ratio in a right triangle, which is true and explains nothing about why its graph is a wave. There is a better picture.

analysis · Circular functions Partial sums of the square wave. Approximations using 1, 3, 7, 21 terms; the corners sharpen but a fixed overshoot remains.

A square wave built entirely out of round ones

Add enough sine waves together and flat tops and vertical cliffs appear from nothing. Almost — there is a 9% overshoot that never goes away, and it is not a bug.

analysis · Fourier series Arithmetic on a dial of 12. A dial with 12 positions. Starting at 8 and stepping forward 9 places lands on 5, because the walk passes the top 1 time on the way.

Numbers that wrap

A clock does arithmetic. It has finitely many numbers, addition never leaves it, and multiplication behaves entirely differently depending on one property of the size of the dial.

discrete · Modular arithmetic 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 One number, two dials: 3 and 5. A grid of remainder pairs, each cell holding the smallest number that leaves those two remainders.

Two dials at once

Watch one number on two clocks with different faces. If the faces share no factor, every pair of readings occurs exactly once — so two remainders name a number, and a hard calculation can be split into two easy ones.

number · Modular arithmetic A rule for moving between 3 states. 3 states drawn as circles with an arrow for every move the rule allows, labelled with its chance; a dashed loop is the chance of staying put.

The rule that forgets where it came from

A walk between a few states, with the next step decided by the current one and nothing else. Run it long enough and the starting point stops mattering — but only when two conditions hold, and both of them have a picture in which they fail.

probability · Markov chains The spectrum of a pulse train, as the period grows. The same pulse repeated at three different intervals, with its spectrum below each. The lines move closer together as the period lengthens and the curve they lie on does not move at all.

When the period grows without bound

A repeating signal has a spectrum of separate lines. Stretch the gap between repeats and the lines crowd together while the curve they sit on stays exactly where it is — and at infinite period the lines are gone and the curve is the whole answer.

analysis · Fourier series A 4-bit register that visits all 15 nonzero states. The first 15 states of a 4-bit linear feedback shift register with taps at 4 and 1, with the bit that leaves the register at each step; the output shows every nonzero window of 4 bits exactly once.

A memory of four bits

A register holding four bits, shifting them along and adding two of them back, runs through all fifteen nonzero states before it repeats. Which two are added back is a question about a polynomial, and getting it wrong costs fourteen of the fifteen.

computation · De bruijn The states the continued fraction of √61 can be in. A grid of whole-number pairs with the band of reduced states shaded, the pairs that qualify marked, and the cycle of states visited by the expansion of the square root numbered in order.

Why the expansion has to repeat

The continued fraction of √61 runs 7; 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14 and then starts again. It must: each step's state is a pair of whole numbers trapped in a small band, and only 14 pairs fit. The expansion of √61 visits 11 of them in a cycle, the other 3 form a cycle of their own, and the period reads the same backwards before its last term, which is twice the first.

number · Pell Lissajous figures for every coprime pair of frequencies up to 4. A 4 by 4 grid of Lissajous figures x = sin(pt + 0.3), y = sin(qt), with the crossing count 2pq − p − q under each and the non-coprime pairs left blank.

When two circular motions come home

Drive a point across with one sine wave and up and down with another. If the two frequencies are in a whole-number ratio the point retraces a closed figure whose crossings can be counted in advance — 2pq − p − q of them — and if they are not, it never comes back and fills the square, spending twenty times longer in the corners than in the middle.

analysis · Circular functions 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

Named alongside it

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

Continued fractionsGreatest common divisorSineConvergenceCyclic groupLimitModular arithmeticModulusPiRational approximationRecurrenceRemainder

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