Fibonacci

I’m always on the lookout for an opportunity for a good photo. As it is currently -6 °F at the present time I’m not outside looking for photo opportunities. Instead I’ve been perusing images that I shot within the last year and flowers look good right now. I notice that many of the images show the seeds occur in lines that spiral out from the center. What's even more interesting is that there are two sets, right and left spirals as in the blackeyed susan. The question is why? 

About 800 years ago, give or take, an Italian mathematician named Fibonacci introduced a sequence of numbers to European mathematics. Here is a part of the sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc and it goes on forever. The sequence is calculated in the following way. The first to numbers are 0 & 1. Subsequent numbers of the sequence are the sum of the previous two. So the third number is 0 + 1 = 1, the fourth is 1 + 1 = 2, the fifth is 2 + 1 = 3, the sixth is 2 + 3 = 5, etc. After you get through about 10 numbers the last number in the sequence divided by the second to last number in the sequence begins to converge on 1.618………… The is the golden ration 1.681/1.  Supposedly a rectangle with dimensions where the width divided by the height yields 1.681 is the most pleasing rectangle. My big screen television is 1.77, my laptop is 1.47, both in the ball park.

This number sequence can be used to create a Fibonacci spiral. A square is constructed for each number in the Fibonacci sequence where the length of the side of each square is a Fibonacci number. Each successive square shares a side with the previous square. A curve is drawn in each square that is portion of a circle with a radius equivalent to the length of the side of the square. When linked together the curves become a Fibonacci spiral. Damn, I did a pretty good job drawing that thing. The Fibonacci spiral closely matches many spirals observed in nature; flowers, pine cones, nautilus shells and even galaxies. In many flowers the spiral shapes are Fibonacci spirals and the number of spirals are Fibonacci numbers.

Here are pictures of two examples. In the sunflower there are 34 right-handed spirals and 55 left-handed spirals, notice 34 and 55 are Fibonacci numbers. In the case of the other flower, 13 spiral left and 21 spiral right. You guessed it, both Fibonacci numbers. Why the spiral? My understanding is that it leads to an efficient packing of seeds on a disc. The center doesn’t get to crowded as seeds are added and there is sufficient but not excess space on the outer parts of the disc to accommodate more seeds as they are added.

These two pictures are the best examples of Fibonacci spirals that I have. I hope to take more in the future and they may appear on this blog. However if you are too impatient to wait a Google search will turn up many more images.