Phi in the natural world
"Eadem mutato resurgo"-although changed, I rise again the same
We are all familiar with strikingly beautiful structures of many shells of molluscs such as the many chambered nautilus (Nautilus pompilus, the very image we see on this site).
In this page we will see that the growth of the spiral in these shells is governed by the Golden Ration.There us a great beauty in the spirals or curves, known as the logarithmic spiral.
The motto in the heading of this piece describes the fundamental property unique to the logarithmic spiral, it does not alter its shape as its size increases. This feature is known as self-similarity.
Many a piece has been written about this spiral that I wanted to share with you the beauty of the curve and how it features in nature of times gone by.
The ammonite and Phi
In 2004 the family and I went to Lime Regis in Dorset , England where we were able to collect a large number of ammonite fossils. These had been sitting in the garage since then. In August 2006 I was reading about the nautilus and shells fitting in with the Phi theory when I wanted to prove it to myself. In 2005 I decided to create an Excel spreadsheet that plotted the Golden Section curve. This showed the curve in all its glory. Goldensection.zip (49 Kb file). This sat around for a year until August 2006 when I pondered the fossil problem. Having retrieved the fossils from the garage I photographed a few of them and set about revising my Excel workbook ammonite.zip (578Kb file)
On playing around with the numbers I finally came across an amazing find. Now, I stress here that this has been proved by countless mathematicians through the ages but it so enjoyable when you discover something without the use of a book or the internet, just thinking it through, experimenting and discovering, the ethos of the budding Kitchen Table Scientist. I finally worked out that the growth rate of two of my fossils was Phi1/5 or 1.10102.
Amazed how I was, I took a screen dump of the excel workbook and converted the spiral into a graphic. All proportions were kept the same and I super-imposed this over the image of two of my fossils, below:
I created an animation of the ammonite and spiral combo using a different fossil and it shows it off quite nicely
and finally a diagram which shows that if you draw a line acroos the diameter of the spiral and call it 1 unit, the ratio of the long radius to the shorter is 0.618:0.382
Another Spiral
Wanting to test this theory a little further I downloaded an image from Wikipedia to see if my spiral graphic would work on other samples. Here is another animation to compare, a very satisfactory result.
