Santiago Canez

12-8-99

UNVR 195A-011

The Mathematics of Natural Patterns: Logarithmic Spirals

It is well known that many patterns in nature can be described using mathematics. Mathematical models are often compared to actual formations in nature through experimentation. By making direct measurements and hypothesizing models in nature, it can be seen whether certain formations actually do follow mathematical patterns. One example of such a formation is the logarithmic spiral. Logarithmic spirals occur in many natural settings, two of the most common being sea shells and sunflowers. These are the formations that I considered when searching for the occurrence of logarithmic spirals in nature. By making direct measurements on sunflowers and by observing sea shells, I was able to verify that these formations do exhibit logarithmic spirals.

The use of models is very important when carrying out such an experiment. I decided to use the idea of Fibonacci numbers to test these formations for logarithmic spirals. I used a transparent piece of plastic to make measurements on the sunflowers and to compare to spirals on the sea shells. On this piece of plastic, I constructed logarithmic spirals using Fibonacci squares: I drew a square whose sides were of length 1 cm.; then another one next to that; then another one whose sides were of length 2 cm. next to those; then a square whose sides were of length 3 cm.; and I continued drawing squares whose sides' lengths matched the Fibonacci sequence. After the squares were drawn, I drew an arc connecting opposite corners of the first square and then continued this arc, connecting corners' of squares, until I had drawn a logarithmic spiral. This was just a basic logarithmic spiral but I thought that it would be sufficient to carry out my comparison.

I began by comparing seed arrangements on sunflowers to the logarithmic spiral I had drawn. The sunflowers I was planning on measuring had to fit the scale I used to draw the spiral so I had to make sure I used small-enough flowers. I measured five flowers using this method. Since the logarithmic spirals of sunflowers is found in the arrangement of the seeds, I just basically placed that transparent piece of plastic, with the logarithmic spiral drawn on it, on top of the flowers. Moving the plastic around a bit, I was able to match the logarithmic spiral I had drawn to those on the flowers. The squares I drew were also useful because they allowed me to make estimates as to how different the spirals on the flowers were to the one I had drawn. The seed arrangements in the flowers almost perfectly fit the spiral I had drawn. A few arrangements had about a less than 1 cm. discrepancy but overall my model was an almost exact replica of the natural formations of the flowers. I did find it interesting that my model did in fact fit the seed arrangements so well, but, from what I learned about this topic, it was not too surprising and had been noticed by others before.

Pictured here is a sunflower which illustrates the occurrence of logarithmic spirals. The red lines outline seed arrangements and match my models of logarithmic spirals. This certain sunflower is not one that I observed; my measurements on this sunflower were made directly on the image above.

This sunflower is one that I actually observed. The red lines once again outline seed arrangements and match my models of logarithmic spirals.

Then I compared formations on sea shells to certain models. To measure the shells, I mainly used a simple shell generating applet I found on the internet. I compared five shells to the images drawn by the shell generator. My comparisons were purely observational because I felt I didn't have adequate equipment to make direct measurements on the shell. Overall, the images gave an almost exact model of the shells. The 3D-images drawn by the shell generator looked very similar to the actual shells I used. One of the shells did differ more from its image than the other shells but I attribute this to defects in the actual shell itself. Another observation I made came from looking at the shells from above. The spiral I had drawn to measure the sunflowers was too small to be used on the shells so I just had to make a comparison solely from visual observation. Looking at the shells from above, I noticed the spiral formations that outlined them. These spiral formations did match the general shape of the spiral I had drawn. So I concluded that the spiral formations on the seashells were in fact logarithmic spirals.

Logarithmic spirals certainly do occur naturally in certain formations. As I have already stated, I observed the formations in sunflowers and sea shells. Using the model I had drawn using Fibonacci squares, I found that the seed arrangements in the sunflowers.

The image on the left is that of a computer generated logarithmic nautilus shell; the image on the left is that of an actual nautilus shell. As you can see, the model almost exactly matches the actual shell.

I observed were in fact logarithmic spirals. Using this method, and also images from shell generators, I was also able to confirm that the sea shells I observed also exhibited logarithmic spirals. These results not only support the idea that logarithmic spirals occur in nature, they also support the idea that many pattern formations in nature can be described through mathematical approaches.