Sunflower seeds are distributed in the following way. A seed is created in a circle of cells and that circle then expands outwards. Now how do you most efficiently fill the space in so doing? If you don’t rotate the angle at which you place seeds in each concentric circle, you end up with a straight line of seeds (the most inefficient use of space)
Recall that there are 360 degrees in a circle. Suppose we rotated through 180 degrees each time we placed a seed.
Great now we’ve created a straight line on both sides of the circle, still a very inefficient use of space. So what values should we try? How about 90 degrees. That is sow a seed, rotate by 90 degrees and so on…
Now we’ve managed to create straight lines in all four directions of the circle. So, 0 degrees produces 1 line, 180, produces 2, 90 produces 4. The average of 0 and 180 is 90 degrees and in so doing we doubled the number of lines. Could the average of 0 and 90 and 90 and 180 produce 8 lines?
Yes! we get 8 lines in both cases. The problem is that the straight lines aren’t a good use of space in the circle. It turns out that rational numbers such as 1/2 or 1/4 times the possible 360 degrees of rotation are NOT good choices because they produce these lines as opposed to spirals.
So let’s try an irrational number. Irrational numbers are numbers which can not be expressed in the form p/q where p and q are both whole numbers. Let’s try pi for example. Only the decimal part of pi is relevant. (if you rotate 3 full times in a circle, you end up where you started in the circle, just a little dizzy). So 0.14159 * 360 =
50.9724 degrees. Here’s what we get:
We start to see spirals, which is a much more effective use of space. Just the same, we see 7 arms because 0.14159 is very close to 1/7. So irrational numbers are better than rational numbers, but what are the ‘most’ irrational numbers?
A branch of mathematics called continued fractions answer this. A continued fraction is a recursive fraction. For example the continued fraction for pi is:
Now hold your breath through this part of the explanation because it’s a bit tricky. All continued fractions ‘converge’ which is to say, arrive at some irrational number such as 0.4829209384… . The rate of convergence is determined by the amount of ‘progress’ made on each step. The slower the convergence of the continued fraction, the harder it is to come up with an approximation. The simplest continued fraction has the slowest convergence.
This number is known as Phi, the golden ratio. So let’s say that we approximate Phi with 8/5 (one of the approximations) the slow convergence means that this approximation isn’t good for very long; it diverges from the value of 8/5 quickly. The approximations for Phi are given by
Where Fn is the nth Fibonacci number.
If you’ve turned blue from the previous explanation, suffice it to say that Phi is the ‘most’ irrational number possible, the ‘most’ random, and the most difficult to approximate.
So using an irrational number, the easier it is to approximate, the easier it will be for us to see the number of arms in the spiral of seeds. If we see arms, the distribution isn’t maximally efficient because to see the arms, we need to see spaces between seeds. So using the ‘most’ irrational number Phi which is approximately 1.61803… and multiplying that through 360 degrees we get 222.492 degrees. However, in circles we ‘see’ the smaller angle = 137.50776 degrees.
Using 137.5 degrees, we get this distribution of seeds:
Look familiar?
The moral of the story is just eat sunflower seeds and don’t study them; the math is too difficult. Another take on things is that nature is fortunately a much better mathematician than most of us.
As an interesting postscript, creationists use the fact that Fibonacci numbers (hence phi) exist in sunflowers as a proof for God’s hand in nature. If only God can understand Fibonacci numbers, does that make all mathematicians saints? Have you met many mathematicians? Have you met many creationists? I just answered all my questions come to think of it.
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more reading and sources:
http://demonstrations.wolfram.com/PhyllotaxisSpirals/
http://mathworld.wolfram.com/GoldenRatio.html
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat2.html
