July 10, 2021

Cauliflowers grow mathematically.

Consider this article’s claim - that fractals, Fibonacci, and the golden ratio have something to do with cauliflowers:

The branched tips, called meristems, make up a logarithmic spiral, and the number of spirals on the head of Romanesco cauliflower is a Fibonacci number, which in turn is related to what’s known as the “golden ratio.” …

In his 1202 treatise, Book of Calculation, Fibonacci described the numerical sequence that now bears his name: 1, 2, 3, 5, 8, 13, 21… and on into infinity. Divide each number in the sequence into the one that follows, and the answer will be something close to 1.618, an irrational number known as phi, aka the golden ratio.

I tried to prove the division assertion. It’s true:

```
def fibonacci(n):
x = 0
y = 1
while (res := x + y) < n:
x, y = y, res
yield f"{x:5d}, {y:5d}, {y / x:.10f}"
for i in fibonacci(100000):
print(i)
```

Results:

```
1, 1, 1.0000000000
1, 2, 2.0000000000
2, 3, 1.5000000000
3, 5, 1.6666666667
5, 8, 1.6000000000
8, 13, 1.6250000000
13, 21, 1.6153846154
21, 34, 1.6190476190
34, 55, 1.6176470588
55, 89, 1.6181818182
89, 144, 1.6179775281
144, 233, 1.6180555556
233, 377, 1.6180257511
377, 610, 1.6180371353
610, 987, 1.6180327869
987, 1597, 1.6180344478
1597, 2584, 1.6180338134
2584, 4181, 1.6180340557
4181, 6765, 1.6180339632
6765, 10946, 1.6180339985
10946, 17711, 1.6180339850
17711, 28657, 1.6180339902
28657, 46368, 1.6180339882
46368, 75025, 1.6180339890
```

My (likely fallacious) conclusion: cauliflowers are mathematical.