Fibonacci, the golden ratio, and how things grow
The maths is real. About half of what gets said about it is not.
Start with 1 and 1. Add the last two to get the next. That is the entire rule:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 ...
It turns up in reasoning papers constantly, because it is the classic sequence that resists the usual differencing technique - the differences of a Fibonacci sequence are another Fibonacci sequence, so the standard method appears to fail.
Recurrence relations
Fibonacci is defined by a recurrence: each term is defined in terms of earlier ones rather than by a formula you plug a position into.
That is a different shape of rule from '3n + 2', where you can jump straight to the hundredth term. With a recurrence you normally have to walk there. Recurrences describe processes that build on their own history, which is why they suit growth, populations and compound interest.
The golden ratio falls out
Divide each Fibonacci number by the one before it:
- 3 / 2 = 1.5
- 5 / 3 = 1.667
- 13 / 8 = 1.625
- 89 / 55 = 1.61818
- 144 / 89 = 1.61798
The ratios close in on 1.6180339887..., called phi. It is irrational - its decimal never repeats - and it is the only positive number with the property that phi = 1 + 1/phi.
Nothing mystical is happening. Any sequence built by adding the previous two terms converges to the same ratio, whatever two numbers you start with. Start with 4 and 7 and you still get there.
Three completely different sequences, the same limiting ratio. The ratio is a property of the RULE - add the previous two - not of Fibonacci's particular starting numbers. That is a good example of a mathematical fact being more general than the famous case that introduced it.
Where it genuinely appears in nature
Some of the famous claims are true, and for a reason.
- Sunflower seed spirals very often number 34 and 55, or 55 and 89 - consecutive Fibonacci numbers.
- Pinecone and pineapple spirals do the same.
- Leaf arrangement around many stems advances by roughly 137.5 degrees per leaf, which is a full turn divided by phi.
The explanation is packing efficiency rather than magic. If each new seed or leaf is placed at a fixed angle from the last, any angle that is a simple fraction of a turn makes them line up in rows and waste space. An irrational fraction never lines up, and phi is in a precise sense the hardest number to approximate with fractions - so it packs best. Plants that stumbled onto it out-competed the ones that did not.
Most of the rest is not true. The golden ratio in the Parthenon, in the Mona Lisa, in credit cards, in the 'ideal' human face - these are mostly rectangles drawn after the fact, loosely enough to fit. The nature results are real and well explained. The art and architecture claims usually dissolve under measurement, and it is worth being able to tell the difference.
The tests you will actually meet
Two things to recognise in a puzzle:
- The sequence itself - 1, 1, 2, 3, 5, 8, 13 should be instant recognition.
- The rule in disguise - counts of shapes across a row following 2, 3, 5, 8, or a figure built from the previous two figures combined.
If differencing produces a sequence that looks like the original, stop differencing and check whether each term is the sum of the two before it. That single check converts an impossible-looking question into an obvious one.