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Concept

Fibonacci

The sequence in which each term is the sum of the two before it — 1, 1, 2, 3, 5, 8 and onward. Consecutive terms have ratio approaching the golden ratio, and they are the denominators of that number's continued-fraction convergents.

Named by 9 essays across 5 fields — each of them below, with the objects they name alongside it.

Euclid's algorithm on a 34 by 13 rectangle. The rectangle is tiled by peeling off the largest square that fits, again and again, until nothing is left.

The oldest algorithm, drawn as a tiling

Euclid's method for finding a greatest common divisor is usually presented as a loop. It is also a way of tiling a rectangle with squares, and the tiling explains why it works.

geometry · Euclidean algorithm The whirling squares. Squares with Fibonacci sides 1, 1, 2, 3, 5, 8, 13, each attached to the long side of what came before. They fill a 13 by 21 rectangle exactly.

The rectangle that eats itself

Cut a square off a golden rectangle and what is left is a golden rectangle. That single property is the whole of the golden ratio, and it explains both what the number really does and most of what is wrongly claimed for it.

geometry · Golden ratio 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 A run down a diagonal, and the entry it adds to. 9 rows of Pascal's triangle with 5 entries shaded and the entry they add to marked. The claim is checked by adding the shaded entries: 1 + 3 + 6 + 10 + 15 = 35.

The run that lands one place along

Add up a run of entries down one of Pascal's diagonals and the total is another entry of the triangle — one row further down and one place along. The same triangle holds four more sums of that kind, and each is a different question answered by the same additive rule.

discrete · Pascals triangle Euclid's game on 34 and 21. A 34 by 21 rectangle tiled by the squares of Euclid's algorithm, each run of equal squares shaded by the player who faces it, with the deciding run outlined.

The player who meets the first long run

Turn Euclid's algorithm into a game: two players take turns cutting squares off the rectangle, any number from the current run, and whoever cuts the last one wins. The whole game is decided before it starts — by whether the ratio of the sides is more or less than the golden ratio, which is the same thing as how many runs of length one come first.

geometry · Euclidean algorithm A straight line across a grid, and the word of walls it crosses. A line of slope 0.6180 crossing a unit grid, with each crossing marked V or H, beside the same path folded into a square as a billiard trajectory; the word begins VHVHVVHVHVVHVVHVHVVH.

The word a straight line spells

A ball rolling across a square table, forever, hits walls in some order: V for a side wall, H for the end wall. Unfold the table and the ball becomes a straight line across a grid, and the order of walls becomes a word — VHVHVVHVHVVHV… for the golden slope. That word has exactly n + 1 different blocks of every length n, the fewest any word that never repeats can have; every stretch of it holds its fair share of H's to within one; and its blocks occur with at most three different frequencies.

dynamics · Billiards A patch of Penrose's rhombs. A disc of Penrose rhombus tiling after 5 subdivisions, thick and thin rhombs shaded differently: 550 thick and 340 thin half-rhombs.

Tiles that never repeat

Two rhombs with angles taken from the pentagon, cut each into halves, and cut each half into smaller copies of the two halves by a fixed rule. Repeat, and the pieces fill the plane with no gaps and no overlaps, in a pattern with five-fold stars everywhere and no period anywhere. The reason it cannot repeat is a single number: thick tiles outnumber thin ones by φ, and a repeating pattern would make that ratio a fraction.

geometry · Golden ratio Sign patterns, and the far fewer points they land on. A logarithmic plot of the number of sign patterns, two to the n, against the number of distinct values the golden geometric sum takes, which is a Fibonacci number less one and falls further behind at every step.

Two sign patterns that land together

At λ = 1/φ the sign patterns + − − and − + + land in exactly the same place, because λ² + λ = 1. That one coincidence, repeated wherever it fits, puts 2ⁿ patterns onto a Fibonacci number of points, leaves the random sum's transform ringing at the same height forever, and makes a distribution that fills a whole interval live on a set of no length.

analysis · Harmonic series The standard map as its kick grows. Three square phase portraits of the standard map at increasing strengths, dotted with orbits: curves spanning the square at the smallest, fewer at the critical value, and a scattered sea with islands at the largest.

The last circle to break

Kick a spinning rotor once a turn and most of its motions stay on curves that wind round forever, walls no orbit can cross. As the kick grows the walls break one by one, and the last to go is the one whose winding is the golden ratio — at a kick of 0.9716, found by watching a sequence of periodic orbits approximate it and asking whether they are stable.

dynamics · Mode locking

Named alongside it

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

Golden ratioContinued fractionsTilingEuclidean algorithmGreatest common divisorIncommensurabilityRecursionSelf-similaritySubstitutionTerminationAlgebraic identityAlgebraic integer

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