Thursday, January 10, 2013

Fibonacci numbers in nature.

In yesterday's post, I discussed the Fibonacci sequence, the infinite number sequence that starts

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...

A sequence that generates the next number in the line by adding together the two numbers just before it. Leonardo of Pisa introduced the numbers in a completely whimsical problem allegedly based on the reproduction cycles of rabbits.

It sometimes happens in math that a tool needed to solve a problem will already exist, built to work on another earlier problem that was often created just for the sake of the beauty of it. So it is with the Fibonacci sequence.

We are often shown cell growth as a constant splitting process, the number of cells doubling each time a new splitting occurs. This is not always the case, as some cells split unequally, one of the new cells getting all the materials needed to replicate at the next splitting time while the second cell does not have what it takes and instead has to use that time period to grow into a ready cell.  Here is an example.

Step 1: Cell splits into ready and unready. (2 cells)
Step 2: Ready splits into ready and unready. The earlier unready grows into ready. (3 cells)
Step 3: The two ready cells split, creating two ready cells and two unready cells. The one unready cell from Step 2 grows into ready. (5 cells, 3 ready, 2 unready.)

This kind of growth pattern is seen in many plants. Many flowers naturally grow a Fibonacci number of petals, notably 5, 8, 13 or 21.

This top down view of a pine cone shows the number of spirals emanating from the central point. The number of clockwise spirals is 13, but when counting the counter-clockwise, the pattern is not as tight and the number is only 8. With most browsers, this picture animates to show the two counts. If it doesn't work here, try clicking on this link to go to the original.

Tomorrow, the Fibonacci numbers and the Golden Ratio.

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