Golden ratio
Named by 31 essays across 8 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
Three gaps and no more
Turn a circle by the same irrational angle over and over. The points never repeat and never settle, and yet at every single stage the gaps they leave take at most three different lengths — never four, at any number of steps, for any angle.
Two hundred and forty directions
The quaternions have twenty-four units and they are the vertices of the most symmetric object in four dimensions. Eight dimensions has two hundred and forty of them, and the quaternions turn out to be how they are built — twice over, with a hundred and ninety-two left to explain.
A matrix that counts the returns
Draw a map straight through the cycle 0 → 1/3 → 1 and its two pieces carry each other in a fixed pattern: the left piece only across the right, the right across both. The orbits' words are then walks on a two-node graph, and the number of points that come back after n steps is the trace of that graph's matrix to the nth power — 1, 3, 4, 7, 11, 18 — each one checked by solving for the points exactly.
The folds that measure chaos
Apply the logistic map six times and its graph goes up and down 38 times at r = 3.5 and 64 times at r = 4. How fast that number of folds multiplies with each further step is the map's topological entropy: zero through the whole cascade of period doublings, log of the golden ratio in the window of three, log 2 at the top — and it never decreases as r rises.
![The integers of ℚ(√5), with ℤ[√5] inside them. Points a + bφ plotted against their conjugates for small whole a and b, with the index-two sublattice ℤ[√5] filled and the basic cells of both lattices shaded, of areas √5 and 2√5.](/f/the-integers-of-q-5-with-z-5-inside-them-a4f067e7.svg)
The integers a field contains
Inside the field of numbers a + b√5, the obvious integers are those with whole a and b. They are not all of them: the golden ratio has a one-half in it and satisfies x² = x + 1, a monic equation with whole coefficients, exactly as an integer should. The right integers form a lattice twice as dense as the obvious one — and a whole-number matrix proves they are closed under addition.
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.
The length the derivative never sees
The Cantor function climbs from 0 to 1 with a slope of zero almost everywhere, so the formula ∫√(1 + f′²) dx says its graph has length 1 — the length of a flat line. The inscribed polygons say 2. Mix it half and half with the diagonal and the length becomes exactly the golden ratio, while the formula still reports only the part the slope can see.
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.
The diagonal no unit measures
Draw the five diagonals of a regular pentagon and they make a star with a smaller pentagon at its centre. Subtract the side from the diagonal and what is left is the smaller pentagon's diagonal; subtract that from the side and what is left is its side. The pentagon has handed back a smaller copy of itself, and it will do so for ever — which means no unit, however small, measures both the side and the diagonal an exact whole number of times.
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.
Two counts that agree for no visible reason
Write 10 as a sum of whole numbers that differ from each other by at least two, and there are six ways. Write 10 as a sum of numbers that each leave 1 or 4 on division by 5, and there are six ways. The same happens for 20 (thirty-one each), for 40 (three hundred and seventy-four each), for every number anyone has checked and every number there is. The two lists look nothing alike, and no one has found a simple way to turn one into the other.
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.
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.
Counting in a base that is not a whole number
Multiply by β and keep the fractional part, over and over, and the whole parts you throw away are the digits of the starting number in base β — even when β is the golden ratio. Which digit strings can ever appear is decided by one string alone: the way the number 1 is written in that base. In base φ it is .11, so 11 is the only thing forbidden; in base 1.8 it never ends, and no finite list of rules describes what is allowed.
Where a base-β orbit spends its time
Follow x ↦ βx mod 1 for a million steps and count how often the orbit visits each part of the interval. For the doubling map the answer is evenly; for the golden ratio it is a staircase with one step, spending 1.17 times the average near 0 and 0.72 times it near 1. The step sits exactly where the orbit of the number 1 lands, and in a base whose 1 never has a finite expansion the staircase has infinitely many steps.
A growth rate no step contains
Start with 1, 1 and at every step either add the last two numbers or subtract them, choosing by a coin. The sequence wanders, shrinks, even returns near zero — and grows, almost surely, like 1.13198824 to the power of the step. That number is a Lyapunov exponent of a product of random matrices, and it cannot be found by averaging anything about a single step: the two steps do not commute, and the rate depends on which way the vector is pointing when each one is applied.
Fractions repeat and roots look random
Almost every number between 0 and 1 has base-φ digits with the frequencies Parry's measure predicts. Which particular numbers do? Every fraction, provably, does not: its expansion repeats from the first digit, with a period equal to the period of the Fibonacci numbers modulo its denominator. And √2 − 1 and 1/π, computed exactly to twelve thousand digits, match every predicted frequency to within sampling error — which proves nothing about them at all.
How nearly equal an odd number of triangles can be
A square cannot be cut into an odd number of triangles of equal area. It can be cut into five triangles whose areas differ by about two hundredths, seven that differ by three thousandths, nine by a ten-thousandth and a half — the spread falling by a factor of seven or more with every two triangles added, closing on equality and never reaching it. A search for the closest finds that the best five it can make have the golden ratio in their areas.
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.
A chaotic map that always comes back
Arnold's cat map stretches a square by the golden ratio squared and wraps it round, and any two nearby points fly apart: a difference of a million-millionth fills the square in twenty-seven steps. Apply the same map to a picture made of pixels and it dissolves into noise within a few steps — and then, after exactly thirty steps on a 61 × 61 grid, every pixel is back where it started. Chaos and exact return are both true, and the Fibonacci numbers decide how long the return takes.
The second letter breaks the pattern
In one letter the constructive logic has infinitely many formulas, but they climb in a column two wide, four new ones for every step a state of knowledge can take. Add a second letter and the pattern is gone: sixteen formulas at the last step, 265,454 one step back, and more than two to the power 265,428 one step further. Every one of those is a genuine formula, and the reason the two-letter lattice has never been drawn is that each layer of the model is about as large as the number of formulas the layer above can tell apart.
The rectangles a square can be cut into
Cut a square into rectangles that all have the same shape — the same ratio of long side to short, in any sizes, lying either way. Two to one is easy. The golden ratio is impossible, however many pieces are used and however small. The ratio φ squared works with six. What decides it is not the ratio but its partner: the number obtained by changing the sign of the square root, which must be positive. Changing that sign is a map that keeps every sum and product and scrambles every size, and areas that must add up to one cannot all turn negative.
A search for room that never ends
The textbook way to find the most that a network can carry is to find any route that still has room, push as much along it as it will take, and repeat until no route has room left. With whole-number capacities that must stop. Give one road a capacity of 0.618…, the golden ratio's reciprocal, and choose the routes badly, and it never stops: every push is positive, the flow converges, and it converges to 4.236 when the network can carry 5. The same choice made carelessly in floating point does stop — because of rounding, at the wrong answer, convinced it has finished.
Two constants at the flat point
Turn up the circle map until its graph goes flat at one point, then hunt for the setting at which an orbit turns by the golden ratio. The settings that lock to 1/2, 2/3, 3/5, 5/8 close in on it, each gap 2.8336 times smaller than the last, and the orbit's returns to its start close in by 1.2886 at every step. Below the flat line the same two numbers are the golden ratio's own. Change the map to a different one with the same kind of flat point and both numbers come back unchanged: they belong to flatness, not to the sine.
Rounding to the nearest remainder
Euclid's algorithm divides and keeps the remainder, and the remainder it keeps is the one left after rounding the quotient down. Rounding to the nearest instead — allowing a remainder to be taken from above — never takes more steps, and Kronecker proved that no mixture of rounding up and down does better. On average it saves three steps in ten, the ratio of the logarithms of the golden ratio and of two. And its slowest inputs are not Fibonacci numbers but Pell numbers, because the worst case of each algorithm is the number its own kind of continued fraction expands most slowly: the golden ratio for one, the square root of two for the other.
![Where the secant method goes, from every starting pair. x^3−x, 180×180 on [−2,2]²: to −1 10164, to 0 11954, to 1 10164, none 118.](/f/where-the-secant-method-goes-from-every-starting-pair-f444ff40.svg)
Newton's method without a derivative
Replace Newton's tangent by the chord through the last two guesses and the method no longer needs a derivative — but it needs two starting values, so its basins live in a plane of pairs instead of on a line. The errors multiply instead of squaring, their logarithms add like Fibonacci numbers, and the digits gained grow by the golden ratio: 1, 3, 5, 9, 15, 25, 41, 67, 109.
![A bracket that shrinks by the golden ratio. 2: [0.00000, 1.00000]; 3: [0.00000, 0.61803]; 4: [0.23607, 0.61803]; 5: [0.23607, 0.47214]; 6: [0.32624, 0.47214]; 7: [0.32624, 0.41641]; 8: [0.36068, 0.41641]; 9: [0.36068, 0.39512]; peak 0.374750.](/f/a-bracket-that-shrinks-by-the-golden-ratio-222753f8.svg)
The search that reuses a point
To find the peak of a function that rises and then falls, evaluate it at two points and throw away the part beyond the lower one. Place the points so that the survivor can be used again, and there is exactly one way to do it: at the golden ratio. Each new evaluation then shrinks the bracket to 0.618 of its width, and with a fixed number of evaluations nothing does better than Fibonacci's numbers.
The largest amount two coins cannot pay
With coins of 5 and 7, the amounts 1, 2, 3, 4, 6, 8, 9, 11, 13, 16, 18 and 23 cannot be paid, and everything else can. Two lines of argument give the formula ab − a − b for the last gap, and half the rest of the board is gaps. For three coins no such formula exists, in a sense that can be proved. And the sets that coins can make have a count that climbs at the golden ratio, a conjecture settled from a table of 67, 118, 204.
Named alongside it
The objects these essays reach for when they reach for this one.
Continued fractionsFibonacci numbersFibonacciEuclidean algorithmExhaustive searchLatticeSelf-similaritySymbolic dynamicsTopological entropyLyapunov exponentPeriodic orbitRational approximation