How many sepals does a daisy have?

Answer

Hi Peter,

Your uncle is right that a daisy’s number of sepals has something to do with the golden ratio. Actually they have something to do with the Fibonacci numbers and these have a clear connection with the golden ratio. So to answer your question properly, I must first explain the Fibonacci numbers to you.

What are the Fibonacci numbers? The numbers are:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, …

So in words, each element of the sequence is always the sum of the two preceding elements, starting with 0 and 1.

The interesting thing about this is that this row occurs in many places in nature. Legend has it that Fibonacci, who was an Italian mathematician (c. 1170-1250) drew up this sequence when he was thinking about how rabbits breed. He decided:

You start with a couple of rabbits: 2
This couple is having a baby, and you have three rabbits: 3
These three rabbits in turn have little ones and you end up with 5 rabbits: 5
…

The strange thing, at least to me, is that a lot of things from nature conform to that Fibonacci sequence. Namely, the number of sepals for any flower often turns out to be a Fibonacci number. (I believe this is why it is so hard to find a four leaf clover because four is not a Fibonacci number but 3 is and I usually find 3 leaf clovers in my garden)

Most daisies have 13, 21 or 34 sepals. These are Fibonacci numbers.

The relationship with the golden ratio is the following: if you take two consecutive Fibonacci numbers (far enough in the row) and you divide them together, you get approximately the golden ratio. Strictly mathematically you express this as follows,

lim_n→∞ f_n+1/F_n=(1+√5)/2 (this number is the golden ratio)

To round out the answer, why is the number of petals on a flower a Fibonacci number. You should mainly ask biologists this, but from a mathematical point of view I may be able to come up with an answer.

In the photo you see a sunflower that is not yet in bloom. This flower has 3 layers of petals. The first layer has 5 and the second below has 8 and the bottom 13. These are like 3 consecutive Fibonacci numbers.

If you assume that the positions of the petals are uniformly distributed over 360°, then the Fibonacci numbers are the right numbers such that the different layers of petals below get a maximum amount of incident light.

You can see some nice examples via the linked link.

Have fun further.