The rectangle that eats itself

Most named ratios are the answer to a measurement. The golden ratio is the answer to a demand: find the rectangle whose shape is unchanged by having a square cut off it.

That is an odd thing to ask for, and it is not obvious that anything satisfies it. A square with a square removed is nothing. A long thin rectangle with a square removed is a longer, thinner rectangle. Somewhere between those two behaviours is a proportion that comes back to itself.

The demand, written down #

Let the rectangle be $\varphi$ long and $1$ high. Cut off the $1 \times 1$ square. What is left is $1$ high and $\varphi - 1$ long — so, lying on its side, it is $1$ long and $\varphi - 1$ high.

For the shape to be unchanged, the ratios must match:

$$\frac{\varphi}{1} = \frac{1}{\varphi - 1}.$$

Multiply out and it is a quadratic:

$$\varphi^2 - \varphi - 1 = 0, \qquad \varphi = \frac{1 + \sqrt 5}{2} = 1.6180339887\ldots$$

Nothing about that derivation is deep, and it is worth saying so plainly, because a great deal of writing about this number treats its existence as remarkable. It is the positive root of a quadratic with small whole-number coefficients. There are a great many such numbers and this is one of them.

What makes it worth a picture is not the value but the property — and the property is genuinely unusual, in a way the next several sections are about.

The squares are the Fibonacci numbers #

Run the peeling backwards. Start with a square, glue another square of the same size onto its side, then a square on the long side of the resulting rectangle, then again, turning a quarter turn each time.

Each new square has to span the long side of everything built so far, which is the previous square’s side plus the one before that. So the sides obey

$$F_{n+1} = F_n + F_{n-1},$$

and the Fibonacci numbers arrive not as a sequence anyone chose but as the only sizes that fit.

The enclosing rectangle after $n$ squares is $F_n$ by $F_{n+1}$, so its ratio is a ratio of consecutive Fibonacci numbers. Those ratios do not equal $\varphi$ — no ratio of whole numbers ever will — and they close on it very fast.

The same sequence turns up in Pascal’s triangle, along its shallow diagonals, for a reason with no rectangles in it at all: the number of ways to write $n$ as an ordered sum of ones and twos obeys the same recursion, because the last term is either a one or a two. Two constructions, one recursion — the situation the conic sections are the cleanest instance of, and it is worth expecting whenever a sequence shows up twice.

The alternation is worth noticing rather than skipping. $1/1 = 1$ is below, $2/1 = 2$ is above, $3/2 = 1.5$ is below, $5/3 = 1.667$ is above. Every convergent overshoots in the opposite direction from the one before, which means each consecutive pair brackets the answer — the error is never larger than the gap between two successive ratios, and the gap can be read off. That is a property of continued fractions generally, and it is the reason approximations of this kind come with their own error bar.

Drawn at whole numbers, on purpose #

There is a decision buried in these figures worth stating, because it is the kind of thing that goes wrong quietly.

The squares here have integer sides — $1, 1, 2, 3, 5, 8, 13$ — rather than sides in the exact ratio $\varphi$. Drawn at the true golden ratio, every square would be an irrational multiple of the one before, and the tiling would close only to within rounding. That is a problem, because the tiling closing exactly is precisely what the figure is claiming. A drawing whose squares nearly fit would be assuming what it was asked to demonstrate.

At Fibonacci sizes the squares tile their rectangle exactly, at every depth, and the generator checks it — no gap, no overlap, nothing outside. What the picture then shows is a true statement about integers that approaches a true statement about $\varphi$, rather than an approximate statement about $\varphi$ dressed up as an exact one.

The same care applies to reading it. Each frame is a fact about a rectangle that is not golden. The golden rectangle is what they converge to, and it is not any of them.

The spiral is not the spiral #

The quarter-circle arcs threaded through the squares are the famous image, and they are not a logarithmic spiral.

A true logarithmic spiral has constant curvature-per-turn: its curvature changes smoothly and by the same factor everywhere. The quarter-circle construction has constant curvature within each square and then jumps — the radius changes discontinuously at every quarter turn, from one Fibonacci number to the next. The two curves are close, they touch at the joins, and they are different objects. The arc version is not even smooth in the sense that matters: its curvature is a step function.

The size of the jump is easy to state. At each join the radius changes by a factor of $\varphi$, so the curvature — which is one over the radius — drops instantly to $1/\varphi \approx 0.618$ of what it was. A physical curve doing that would be a road with a kink in its steering: the driver’s hands are in the same place either side of the join, and the rate at which they are turning is not. The two curves agree in position and in direction at every join, and disagree in curvature at every one of them.

That would be a footnote if the difference did not carry most of the popular claims. The nautilus shell is a logarithmic spiral, which is well established and unsurprising — a shell grown by adding material at a constant proportional rate has no other shape available. Whether it is a golden logarithmic spiral, growing by $\varphi$ every quarter turn, is a separate and much stronger claim, and measurement does not support it. Surveys of actual shells put the growth ratio nearer $1.3$ per quarter turn, and it varies between individuals and along a single shell. The golden spiral is a spiral; the nautilus is a spiral; they are not the same spiral.

The Parthenon, the Great Pyramid, and various paintings receive the same treatment, and the same objection applies with more force. A rectangle drawn around a building can be placed many ways, the ratio depends on which features are chosen as its corners, and $1.618$ is close enough to $1.6$, to $5/3$, and to $8/5$ that almost any deliberate proportion lands near it. When a claim can be confirmed by choosing where to put the ruler, confirming it establishes nothing — which is the same trap as reading a pattern off a spiral of primes and concluding the primes are explained.

What the number genuinely does #

Set against all that, there is one property of $\varphi$ that is real, sharp, and does explain something observed in the world. It is not the one usually advertised.

Every irrational number has a continued fraction expansion, and $\varphi$’s is as simple as one can be:

$$\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cdots}}}$$

which follows immediately from the defining equation $\varphi = 1 + 1/\varphi$, substituted into itself forever.

Now recall what Euclid’s algorithm drawn as squares says about continued fractions: the terms are the run-lengths of equal squares peeled off, and a large term means a long run, which means one rational approximation is doing an unusually good job. $\pi$’s expansion begins $[3; 7, 15, 1, 292, \ldots]$, and the $292$ is why $355/113$ is so extraordinarily accurate.

$\varphi$’s terms are all $1$. There are no large terms, ever, because there is nothing larger than a $1$ available. So no rational approximation of $\varphi$ is ever unusually good — every convergent is about as bad as a convergent has to be. The golden ratio is the hardest number to approximate by fractions. That is a theorem, it is exact, and it makes $\varphi$ genuinely special in a way no proportion in a painting could.

And it has a consequence outdoors. A plant putting out leaves around a stem wants each new leaf to miss the ones below it, which means the turn between successive leaves should be a fraction of a full turn that is badly approximated by simple fractions — because a turn near $p/q$ means leaves stack up in $q$ columns and shade each other. The angle hardest to approximate is $\varphi$’s, and the angle observed in a great many plants is $360°/\varphi^2 \approx 137.5°$. Here the golden ratio is not decoration. It is the answer to an optimisation, and the optimisation is the one the continued fraction describes.

Where it fails to be special #

It is worth naming the boundary of that argument too, since the previous section is the strongest case for the number and it is easy to over-extend.

The “most irrational” property is a statement about how well any rational $p/q$ approximates $\varphi$ relative to $q^2$, and it identifies $\varphi$ up to a family of numbers that behave the same way. Every number whose continued fraction ends in all $1$s — that is, every number of the form $(a\varphi + b)/(c\varphi + d)$ with integer entries and determinant $\pm 1$ — is exactly as hard to approximate. The extremal property belongs to a whole equivalence class, not to $\varphi$ alone. $\varphi$ is the simplest member of it, which is why it is the one that gets named.

Nor is the property about beauty, or growth, or proportion. It is about the tail of a continued fraction. Anything claimed for $\varphi$ that does not route through that tail — or through the quadratic it satisfies — is a claim that has to be made on its own evidence, and usually is not.

What the picture cannot show #

The figures are finite and the property is a limit, in the way the tiling of an irrational rectangle is. Nine squares are drawn and the tenth would be a few pixels across; nothing visible distinguishes a process that never terminates from one that stops two steps past the edge of the page.

More particularly, the picture cannot show the difference it most needs to. A Fibonacci rectangle at $34:21$ and a true golden rectangle differ by one part in three thousand, which at any drawn size is a fraction of a line width. Everything interesting here — that the shape is exactly reproduced rather than nearly, that the approximations never get unusually good — lives in that gap, and the gap is smaller than the ink.

The self-similarity is also invisible in the honest sense. What the eye reads off the figure is these particular squares fit, which is a fact about the drawn integers. That the construction would continue forever on the limiting shape is an argument about the defining equation, and the drawing is silent about it.

The ladder from here #

Rungs above this one: the continued fraction proved rather than quoted, with the convergents’ error bounded. Hurwitz’s theorem, which makes “hardest to approximate” a precise inequality and identifies the equivalence class that shares it. The logarithmic spiral in its own right, with the growth-per-turn made a parameter and the nautilus measured against it. Phyllotaxis, with the packing argument drawn and the $137.5°$ derived rather than reported. Penrose tilings, where $\varphi$ appears as the ratio of the two rhombs and the tiling is aperiodic because of it. The Fibonacci numbers’ closed form, and why a formula full of $\sqrt5$ returns integers. Lucas sequences, where the same recursion runs from different seeds. And the pentagon and pentagram, where $\varphi$ is the ratio of diagonal to side and the Greeks first ran into it — the same encounter that produced the incommensurability crisis, and the place where this ladder meets the fifth roots of unity, since $\varphi$ is what those five equally spaced points measure.

The habit #

There is a general lesson in the gap between what this number does and what it is said to do.

$\varphi$ has one sharp, provable, consequential property, and it is technical: its continued fraction is all ones, so it resists rational approximation better than anything else. Everything defensible about it follows from that or from the quadratic. Everything else — the shells, the buildings, the faces, the paintings — is a claim about the world that would need the ordinary evidence any claim about the world needs, and mostly does not have it.

The picture is complicit in this. A rectangle peeling into squares with a spiral through it is a genuinely lovely image, and a lovely image is persuasive about propositions it does not contain. The defence is the same one that applies to diagonals in a spiral of primes: ask what exactly the drawing establishes, and then ask separately whether the sentence next to it is that.