Why Does the Golden Ratio Keep Showing Up?

Pick up a pinecone and count the spirals of scales. On many cones you get 8 spirals curling one way and 13 curling the other.

Now divide: 13 ÷ 8 = 1.625.

That's close to 1.618..., a number called the golden ratio. You've heard the list. Pinecones, sunflowers, pineapples, art class. It's everywhere, so it must mean something.

Is 1.618 a mystical signature, or a coincidence?

Neither. It's a number with a reason, and you can build the reason yourself with a ladder and some addition.

The short answer

1.618... is the one ratio left when a ladder of levels has one rule and its first new rung joins two different earlier rungs. The rule is that each rung is the one below it times the same number, so that number is the ratio between neighboring rungs. A join adds two rungs together. And any list of positive numbers that adds its last two drifts to it from any start, which is why it keeps turning up.

Reality keeps a permanent record of every event, because the basic laws of physics never destroy information, and that record is built in levels: each level is a stretch of the record, and each new level is built by joining levels that already exist. Those levels follow the ladder's rules, so 1.618... is the ratio between them. The section "Where this goes" says what a level is, gives the reasons the levels follow the rules, and names the one step still open.

Here's the whole answer in six steps. The rest of this post takes them one at a time and shows how we know each one.

1. A ladder with one rule. Reality's levels sit on a ladder, and every rung is the one below it times the same number. Call that number r.
2. Joins have a price. A new rung is two earlier rungs added together. A join compares its two parts, and its price measures how unequal they are: zero when they match. Joining a rung to itself costs nothing and records nothing. A real join pairs two different rungs, and it costs something.
3. One ratio is left. Rung 2 is r × r, and a real join of rungs 0 and 1 makes it 1 + r, so r × r = 1 + r. Exactly one positive number solves that: 1.618..., the golden ratio. Every rung is then the two below it, added.
4. Each rule earns its place. Take away any of the ladder's rules and other ladders get in: one that doubles, one that runs downhill, one with no single ratio.
5. Cheapest isn't the point. A cheaper ladder exists, and it isn't the one. The price rules out the free join. It doesn't pick the number.
6. Start anywhere, land there. Any list of positive numbers that adds its last two settles to 1.618..., whatever it starts with. That's why the number keeps showing up.

Why this matters: nobody chose this number. Plenty of the numbers in science were measured and then written in a table. List what building a ladder of levels requires, and exactly one ratio survives. That's what we mean by forced: not picked, not fitted, just what's left when the alternatives are crossed out.

And 1.618... is a fact of the same type as the eight in Part 1, a number that falls out of what a rule requires. Lasting structure needs a tie that no smooth motion can undo, the way two linked rings of a chain can't be pulled apart without cutting one. Closed loops link like that only in three dimensions: in two they can't wind around each other, and in four or more any two loops slip apart. So space has three directions, each with two sides, which gives 2 × 2 × 2 = 8, the eight corners of a cube, and a cycle that visits every corner once takes eight steps.

Step 1: A ladder with one rule

Reality's levels sit on a ladder, one level to a rung. The ladder has one rule: every rung is the one below it times the same number.

Here's what that means. Number the rungs from 0. Say rung 0 has size 1. Call the multiplier r. It's also the ratio between neighboring rungs. Then rung 1 is r, rung 2 is r × r, rung 3 is r × r × r, and so on up.

Pick r = 2 and the rungs are 1, 2, 4, 8, 16. Pick r = 1.5 and they're 1, 1.5, 2.25, 3.375. Pick r = 3 and they're 1, 3, 9, 27. Every positive number makes a ladder, so the rule by itself picks nothing.

Why does reality's ladder have just one rule? Because the step from one level to the next does two things. It does to a stack of things what it does to each thing in the stack: if a thing of size 2 becomes size 3, a stack of 5 of them, size 10, becomes size 15. And it keeps every comparison: if one thing in reality's record was bigger than another before the step, it's bigger after.

A step like that can't multiply two things by different numbers. If it did, stack up copies of each until one stack is just barely ahead of the other. The unequal multipliers flip the lead. So the step multiplies everything by the same number. At the end of this post we'll watch that happen with real numbers.

Next, and why: what ties one rung to another. In reality, a new rung is built out of rungs that already exist.

Step 2: Joins have a price

Reality builds each new rung by joining rungs that already exist. A join compares its two parts, and the price of a comparison measures how unequal they are: zero when they match. A join of two equal things costs nothing and records nothing. A join of two different things costs something.

Rungs 0 and 1 are the starting rungs, so rung 2, the first new one, can only be built from them. Join two of them by adding their sizes. A rung can be used twice, so there are exactly three ways to pick the pair. And rung 2 is r × r by the one rule, so each pick gives an equation for r:

• Rung 0 + rung 0. Then r × r = 1 + 1 = 2, so r = 1.414...
• Rung 1 + rung 1. Then r × r = r + r, so r = 2.
• Rung 0 + rung 1. Then r × r = 1 + r. Hold that thought.

Two of the three join a rung to itself. Here's why that matters. A join compares its two parts. For two things whose sizes are in ratio x, the price of that comparison is

J(x) = (x + 1/x) / 2 - 1

Look at what it does. Equal sizes, ratio 1: cost 0. Ratio 2: cost 0.25. Ratio 1/2 also costs 0.25, so it doesn't matter which one you call first. Ratio 1.618...: cost about 0.118.

That price isn't a choice.¹ It has to be zero only when the two sizes match, grow as they differ, and come out the same whichever one you put first. Those requirements, a consistency rule for how the prices of combined comparisons add up, and one calibration (how sharply the price rises near a match) leave exactly one price: J.²

A rung joined to itself has ratio 1, so its join costs exactly 0. That's not a rounding error. The smallest thing that can happen is one thing being told apart from another, a single yes or no. You're doing it right now: every perception and every measurement tells one thing from another. Comparing a thing with a copy of itself tells nothing apart, so there's no yes or no. Reality's record writes down differences. There's nothing to write. A free join isn't an event.

So the first two joins aren't really joins. Only the third pairs two different rungs, and it's the only one that's an event.

Next, and why: put that pair into the ladder and see which ratio it forces.

Step 3: One ratio is left

The only real join builds rung 2 from rung 0 and rung 1, and that leaves exactly one ratio: 1.618..., the golden ratio.

Here's the algebra, with numbers. Rung 2 is r × r, from the one rule. Rung 2 is also rung 0 + rung 1, which is 1 + r, from the join. So r × r = 1 + r.

Try some values. At r = 1.5, r × r is 2.25 but 1 + r is 2.5, so r × r is too small. At r = 1.7, r × r is 2.89 but 1 + r is 2.7, so it's too big. At 1.6 it's 2.56 against 2.6, still too small. At 1.62 it's 2.6244 against 2.62, just too big. You're closing in. The exact answer is

r = (1 + √5) / 2 = 1.6180339887...

That's the golden ratio, written with the Greek letter φ ("phi"). Squaring it and adding 1 to it do the same thing: 1.618 × 1.618 = 2.618, and 1.618 + 1 = 2.618. The equation has a second solution, -0.618..., but it's negative, and sizes aren't negative. So exactly one r works.

That's the whole proof: three ways to pick the pair, two of them free, one equation. Nobody chose 1.618. It was the only one left.

Now build the ladder. Multiply both sides of r × r = 1 + r by r. The left side becomes r × r × r, which is rung 3. The right side becomes r + r × r, which is rung 1 + rung 2. Multiply again and you get rung 4 = rung 2 + rung 3, and so on.

Every rung is the two below it, added: 1, 1.618, 2.618, 4.236, 6.854, 11.09. Check: 1 + 1.618 = 2.618, and 1.618 + 2.618 = 4.236. Every join on this ladder compares two rungs in ratio 1.618, so each one costs 0.118.

There's a picture of the same equation, and it's the one from art class. Take a rectangle whose long side is r times its short side, so its sides are r and 1. Cut a 1 by 1 square off one end. What's left is a smaller rectangle with sides 1 and r - 1.

For the leftover to be the same shape as the original, its long side divided by its short side must also be r: 1 ÷ (r - 1) = r. Multiply both sides by r - 1 and you get 1 = r × r - r, which rearranges to r × r = 1 + r again. Only 1.618 does it. Check: the leftover is 1 by 0.618, and 1 ÷ 0.618 = 1.618. That's the golden rectangle: cut a square off, and what's left is the same shape, so you can keep cutting forever.

Next, and why: is each rule really needed? Take one away at a time and see what gets in.

Step 4: Each rule earns its place

Take away any one of the four rules and a different ladder gets in.

Here are the four rules in plain words, each with the reason reality's levels follow it:

1. One rule. Every rung is the one below it times the same r. The step from one level to the next does to a stack what it does to each thing in it and keeps every comparison, so it multiplies by one number (Step 1).
2. Earlier. Rung 2 is built from earlier rungs, the ones below it that already exist.
3. A join. Rung 2 is two of those rungs added together. Rules 2 and 3 share a reason: each new level is built by joining levels that already exist.
4. A price. The join costs something, so its two parts are different. A join compares its two parts, so a join of equals isn't a join (Step 2).

Notice what the four leave out: the number 1.618. Only rung 2 has to be a join, and every later rung follows from the one rule.

Now remove them one at a time.

No one rule. Take the rungs 1, 2, 3, 4, 5. Rung 2 is rung 0 + rung 1: 3 = 1 + 2, a real join that costs 0.25. But each step up multiplies by a different number: ×2, ×1.5, ×1.33, ×1.25. There's no single ratio to find.

Not earlier. Let rung 2 be built from later rungs, the ones above it, and the same arithmetic runs downhill: 1, 0.618, 0.382, 0.236, 0.146. Rung 2 is rung 3 + rung 4 (0.236 + 0.146 = 0.382), a costly join. The ratio is 0.618, which is 1 ÷ 1.618. (It's the golden ladder read from the top down: the ladder behind the deep states in Part 3, where each deeper rung is built from the two above it.)

No join. Take the rungs 1, 2, 4, 8. Rung 2 is 4, which isn't 1 + 2. Nothing ties the rungs together, and r = 3 would have done as well as r = 2.

No price. Take 1, 2, 4, 8 again, and say rung 2 is rung 1 + rung 1: 4 = 2 + 2. It's a join, but of equal parts, so it costs 0, and the ladder doubles.

Each rule has one job. The one rule fixes the ratio. "Earlier" gives the ladder its direction. The join ties the rungs together. The price rules out the free join, the doubling. Remove any job and the number isn't forced. Keep all four and there's nothing left to choose.

How do we know? Each ladder above keeps three rules, breaks one, and doesn't climb by 1.618. Every number is on the page, so you can check them yourself.

Next, and why: a natural guess says 1.618 wins because it's the cheapest. It isn't, and seeing why shows what the price is for.

Step 5: Cheapest isn't the point

The golden ratio doesn't win by being cheapest. A cheaper ladder exists, and it isn't the one.

The natural question: if nature is thrifty, shouldn't the winner be the cheapest ladder?

Here's what one join costs on three ladders, priced with J from Step 2:

The doubling ladder costs 0, because its joins are free. The plastic ladder climbs by 1.3247..., a number called the plastic number, and a join on it costs about 0.040. The golden ladder climbs by 1.618..., and a join on it costs about 0.118. So the golden ladder is the priciest of the three, and the only one that passes all four rules.

The plastic number is the positive solution of q × q × q = q + 1. The plastic ladder does add rungs together: rung 3 is rung 0 + rung 1. But look at rung 2, the first new rung. It's 1.755, and no two earlier rungs add up to that: 1 + 1 is 2, 1 + 1.325 is 2.325, and 1.325 + 1.325 is 2.649. The plastic ladder skips the first join, and the join rule says the first new rung is a join. So it's out, even though it's cheaper.

If cost alone picked the ratio, it would pick one that costs nothing. J is 0 at ratio 1, which is a ladder that never climbs, and doubling's joins are free too. But a free join isn't a join.

So what is the price for? In the proof it has one job: to be zero for equal parts and more than zero for different ones. That throws out the free join. Any price that did the same job would leave the same answer. The golden ratio comes from how levels join, not from a price tag.

Next, and why: the question we started with. If adding the last two is the rule, why does 1.618 turn up in so many places?

Step 6: Start anywhere, land there

Any list of positive numbers that adds its last two settles to 1.618..., whatever numbers it starts with.

Start with the most famous list of this kind. Begin with 1 and 1, and add the last two each time: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89. (That's the Fibonacci sequence, and 8 and 13 are in it.) Now divide each number by the one before it:

1, 2, 1.5, 1.667, 1.6, 1.625, 1.615, 1.619, 1.6176, 1.6182

The sixth one, 1.625, is 13 ÷ 8, the pinecone's number. As arithmetic, it's one hop on the way to 1.618.

Start somewhere else. Begin with 3 and 4: 3, 4, 7, 11, 18, 29, 47, 76. The ratios: 1.333, 1.75, 1.571, 1.636, 1.611, 1.621, 1.617. Same place.

Begin with 1 and 100: 1, 100, 101, 201, 302, 503, 805, 1308. The ratios: 100, 1.01, 1.99, 1.50, 1.67, 1.60, 1.62. Same place again.

Why does every start land there? Look at how one ratio makes the next. If the last two numbers are 5 and 8, the ratio is 8 ÷ 5 = 1.6. The next number is 13, which is 8 + 5, so its ratio to 8 is (8 + 5) ÷ 8 = 1 + 5/8. And 5/8 is 1 ÷ 1.6, which is 1 divided by the old ratio. So the new ratio is 1 + 1 ÷ 1.6 = 1.625. In general:

new ratio = 1 + 1 ÷ old ratio

That's a little machine, and 1.618... is the one positive number it leaves alone. Feed it 1.618 and out comes 1 + 1/1.618 = 1 + 0.618 = 1.618. (Multiply x = 1 + 1/x through by x and you get x × x = x + 1, the same equation as before.)

Every other positive number gets pushed toward it. After the first few rounds, each round cuts the gap to 1.618 down to about 38% of what it was. The new ratio also lands on the other side of 1.618. (That 0.382 is 1 ÷ 1.618 ÷ 1.618. The golden ratio even sets the speed of its own pull.)

You can run the machine on your phone. Type any positive number. Divide 1 by it, then add 1. Take the answer and do it again. Start with 7 and you get 1.143, 1.875, 1.533, 1.652, 1.605, 1.623, 1.616, 1.619, 1.618, 1.618. Start with 0.001, or a million. Same place.

That's why 1.618 keeps showing up. Wherever something grows by adding its last two amounts, the ratio between neighbors drifts to 1.618, whatever its start and its units. Nothing in the process has to aim at it.

So is that why a pinecone has 8 and 13? That's botany's question, and this post doesn't answer it. What this post answers is where 1.618 comes from when a ladder adds its last two rungs.

Next, and why: what this says about reality's own levels, with real numbers.

Where this goes: reality's own levels

What a level is. This post calls reality's permanent record the ledger. Every event leaves a complete trace in it, called an entry, and nothing is ever erased. Physics gives the reason: its basic laws never destroy information. In quantum mechanics this is unitarity: the present fixes the past exactly. Part 1 gives the full account of the record. A level is a stretch of the ledger's entries, and the levels are the stretches you reach from the simplest one by stepping up again and again.

Reality's levels climb by 1.618. Reality's levels obey all four rules:

• Each step up does to a stack what it does to each thing in it and keeps every comparison, which gives the one rule.
• Each new level is built by joining levels that already exist, which gives the earlier and join rules.
• A join compares its two parts, so a join of equals isn't a join, which gives the price.

So the proof from Steps 2 and 3 applies. Reality's levels are 1, 1.618, 2.618, 4.236 and on up, each one the two below it joined, and each one 1.618 times the one below. That's the ladder in the Step 3 picture.³

Here's the one rule with real numbers. Step 1 said that a step that does to a stack what it does to each thing in it, and keeps every comparison, multiplies everything by the same number. Now watch it happen.

The ledger counts a stretch of entries with two numbers. That count lets you watch the one rule appear with real numbers, without assuming it. Take a small stretch of three entries: two independent steps, the steps that carry a choice, and one kept record. Write it (2, 1). A stretch's size is a count: each independent step counts r and each kept record counts 1, so this one has size 2 × r + 1. In general, a stretch with a independent steps and b kept records is written (a, b), and its size is the count a × r + b. In the picture below, an independent step is a tall shaded block and a kept record is a short plain one.

The base level is the simplest stretch, one kept record: (0, 1), size 1. To step up, the ledger joins. Take (2, 1) again: its three entries each become an independent step, and its two independent steps are also kept as records, so it becomes (3, 2). In general, every entry of the current stretch becomes an independent step of the next one, and the current stretch's independent steps are also kept as records. So (a, b) becomes (a + b, a). On the levels themselves, this rule is the join written as a count. From the base the levels run (0, 1), (1, 0), (1, 1), (2, 1), (3, 2), and from (1, 1) on, each is the two before it added, count by count: (3, 2) is (2, 1) plus (1, 1).

Now compare two other stretches. A is (2, 0): two independent steps. B is (0, 3): three kept records. Does the step up keep their comparison?

Say r = 2. Then A is 2 × 2 = 4 and B is 3, so A is ahead. Step both up: A becomes (2, 2) and B becomes (3, 0). A is now 2 × 2 + 2 = 6, and B is 3 × 2 = 6. A tie. The step erased the comparison.

Say r = 1.618... Then A is 3.236 and B is 3, so A is ahead. Step up: A becomes (2, 2), which is 2 × 1.618 + 2 = 5.236. B becomes (3, 0), which is 3 × 1.618 = 4.854. A is still ahead, and both are exactly 1.618 times what they were: 3.236 × 1.618 = 5.236, and 3 × 1.618 = 4.854.

Run the step on the base level over and over. At r = 1.618 the sizes are 1, 1.618, 2.618, 4.236, 6.854, the ladder from Step 3. At r = 2 they're 1, 2, 3, 5, 8, and there's no single ratio.

That's no accident. The step turns a stretch of size a × r + b into one of size (a + b) × r + a. When r × r = r + 1, that new size is exactly r times the old one. And 1.618 × 1.618 = 1.618 + 1, so at r = 1.618 every stretch in the ledger is multiplied by the same number. At any other positive r, some pair of stretches ends up tied or reversed.

And notice what we did this time: we never assumed one rule. We asked only that the step keep every comparison, and the one rule came out, with 1.618 as its ratio.

Reality has two kinds of ladder. The golden ladder is the costly kind: each rung is two different rungs joined. The other kind is the doubling ladder, 1, 2, 4, 8. Each rung is two equal copies of the one below, and that join is free. The eight in Part 1 is the doubling kind. Space has three directions, each with two sides, so each direction doubles the number of combinations: 1, 2, 4, 8. That's 2 × 2 × 2 = 8, the eight corners of a cube. Reality's levels are the golden kind. One question tells the two kinds apart: does comparing the parts of the join cost anything?

One step is still being worked out. You can check every number above by hand, and the ladder follows from its rules by proof. What isn't finished is the reason underneath the rules themselves: why each step up does to a stack what it does to each thing in it and keeps every comparison, and why each new level is a join of two different levels that already exist. We're still tracing those to something deeper.

The whole trip, in one breath

Reality's levels sit on a ladder with one rule: every rung is the one below it times r. A new rung is two earlier rungs joined. Joining a rung to itself costs nothing and records nothing, so the real join pairs rung 0 with rung 1. Then r × r = 1 + r leaves one positive answer: 1.618..., with every rung the two below it. Take away any of the ladder's rules and other ladders get in. Cost doesn't pick the number, because a cheaper ladder breaks the join rule. Any list of positive numbers that adds its last two settles to 1.618... from any start, which is why the number keeps showing up. And in reality's ledger, only 1.618... keeps every comparison when a level steps up. So reality's levels climb by 1.618..., and that's not magic and not luck. It's a number you can build with a ladder and some addition.