The Math Behind That Spiral

1, 1, 2, 3, 5, 8, 13, 21... 196418, 317811...
Many may recognize this this as the Fibonacci Seqence. Leanardo Pisano Bogollo or Fibonacci first came across this pattern when theorizing about rabbits. If two original rabbits are bred, they produce one more rabbits. Each pair of rabbits then produces another rabbit. Continuing this pattern, you end up with the sequence above. Although this problem was theoretical the Fibonacci sequence can be seen in many places in nature. 
If you have ever looked at a pinecone, you can quickly notice the spiral looking pattern in the placement of the scales. The same for seeds in sunflowers seeds, leaves on a plant, and the shape of a snail. But what do they all have in common? 
Next time you pick up a pinecone, count the number of spirals going down from the top in both directions. You should find 5 in one direction and 8 in another, or maybe 8 in one and 13 in another, or maybe 13 in one and 21 in another.

Spirals on a Pineapple

You might notice that these numbers are in the same list above. They are all Fibonacci numbers, but why?
Well to answer this we have to look at another relationship of these numbers and that is the Golden Ratio. Now you could look up this number and find it to be 1.61803398875... or you can look at the list above. 
Let's start with dividing 3/2:
3/2 = 1.5
now
5/3 = 1.667
8/5 = 1.6
13/8 = 1.625
21/13 =1.61538
...
317811/196418 = 1.61803398874
As you can see, the larger and larger the numbers get, the closer they get to this Golden Ratio number or Phi. Now back to our plants. The reason the leafs spiral is because they need light to grow. If the leaves didn't spiral, the upper leaves would block the lower ones, and if it rotated half, then the third leaf would block the first. Nature has found a way to minimize this layer of one leaf onto the next, and that magic lies in Phi. The location of leaves can be determined by dividing 360 degrees by Phi, this gets you about 222.5 degrees for each rotation (or 137.5). Placing each new leave 222.5 degrees around from the previous leads to the right amount of space for each leaf to still have access to light. As the leaves spiral around, there is always room for the next leaf. 
Now going back to the Fibonacci spirals. How does the plant know how to spiral like this? This all has to do with how a plant grows. Plant leaves push out from the top of the plant called the shoot meristem. As each new leave grows, it grows to be the furthest away from the other plant so that it doesn't compete for sunlight or growth hormones. The first leaf grows at any point, then the next leaf directly opposite the nest, then the third leaf pushes to be furthest away from both of the other leaves. As we already discussed above, each leaves repelling away from the rest results in the magic angle measure of 222.5 degrees between each leaf. Of course, there are exceptions to this rule depending on how the leaf first grows. Even with an abnormal start, the leaves will generally still fall into a pattern and will continue on a different spiral. In things like a snail shell, it also comes back to how new cells and proteins are added to a growing strand. In fact, scientists in Japan have been able to change the direction of the snail's twist by switching the location of the initial 4 cells. 
This basic sequence found in 1202AD can be seen governing the world all around you. What one may first think is completely mystifying comes down to a simple mathematical and biological answer.