Discovering the Magic of the Fibonacci Sequence
Take two numbers. Add them to get the next one. Then add the last two again, and keep going. That is the whole rule, and it is short enough to memorise in one go:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…
Each number is the two before it added together. 3 + 5 = 8. 5 + 8 = 13. 8 + 13 = 21. The list never ends, and it was written down in Europe in 1202 by a mathematician called Leonardo of Pisa, later nicknamed Fibonacci.
Build the spiral
The sequence has a shape hiding in it. Draw a square for each number, fitting each new square against the ones already there, and the squares curl round.
- Squared paper if you have it, which makes this far easier, or plain paper and a ruler
- A pencil
Step through the build, then copy each stage onto your own paper. Count squares on the paper rather than measuring in centimetres.
- Draw two squares side by side, each one square big.
- The next square has to be 1 + 1 = 2 across. It fits exactly underneath the pair.
- Next is 1 + 2 = 3. Notice it fits the left-hand side exactly, because 1 + 2 is the height you already had.
- Next is 2 + 3 = 5, and it fits across the top for the same reason.
- Next is 3 + 5 = 8. Every new square is exactly as long as the two before it, which is why there is never a gap.
- Now draw a quarter circle in each square, corner to corner, smallest first. The separate curves join into one spiral.
The squares never leave a gap and never overlap. That is not luck — it happens because each new side is the sum of the two it has to sit against, which is exactly what the sequence rule says.
Where it really does turn up
Some of the claims made about this sequence are true and worth checking yourself. Others get repeated a lot without much behind them, so here are the ones you can actually go and count.
- Pinecones. The scales spiral two ways at once. Count the spirals going clockwise, then the ones going anticlockwise. You will usually get 8 and 13.
- Pineapples. The same two-way pattern on the skin, usually 8, 13 and 21.
- Sunflower heads. The seeds spiral both ways, often 34 and 55, and on a big head 55 and 89.
- Petals. Buttercups almost always have 5. Lilies have 3. Plenty of other flowers have 4 or 6 and are not Fibonacci at all, so a flower with 4 petals has not broken any rule.
Why plants land on it
Plants are not counting. The pattern falls out of something much simpler.
A growing tip adds each new seed or leaf at a fixed turn from the one before — a bit over a third of a full circle, close to 137.5 degrees. That turn never repeats itself neatly, so new seeds keep dropping into the gaps left by earlier ones instead of lining up in spokes. The result is a tightly packed head with no wasted space, and when you count the spirals that packing makes, they come out as neighbors in the Fibonacci list.
A sunflower's seeds spiral in Fibonacci numbers. Which explanation is right?
- The plant counts its seeds and picks Fibonacci numbers
- Each new seed grows at the same fixed turn from the last, and that spacing produces those spiral counts
- It is pure coincidence and has no cause at all
Go and count something
The best part of this pattern is that you do not have to take anyone's word for it. Find a pinecone. Put a pencil mark on one scale so you know where you started. Count the spirals one way, then the other, and see what you get.
If the numbers come out as 8 and 13, you have just measured something about how that tree grows.