Phi? Phi not!

Each month the Fremont Photographic Society runs a contest that includes a “special topic” category. In August the topic was “curves”. I was at a loss for a subject until my wife, Joy, suggested I photograph a shell. A first I though a shell would be quite boring, but then I got this idea to make it a graphic representation of mathematics in nature. My inspiration was the many graphics I had seen of the “Golden Rectangle”, often shown side-by-side with a Chambered Nautilus shell, like this:

Nested Golden Rectanges create a spiral similar to a Chambered NautilisThe Golden Rectangle has the unique property that, if you lop off a square, the rectangle that remains has the exact same proportions as the original rectangle. Since the new rectangle is just a smaller version of the original, you can imagine continuing loping off squares ad infinitum. Now if you connect the corners of the squares with a smooth curve you get what is known as a “logarithmic spiral”. This “Golden Spiral” bears an uncanny resemblance to the Chambered Nautilus.

The symbol for the ratio of the length of the two sides of a Golden Rectangle is the Greek letter phi, φ, because it is said the renowned Athenian sculptor, Phidias (circa 490 – 430 BC) used it extensively in his work. Mathematically, phi works out to be φ=(1+√ 5 )/2 = 1.6180…

While φ has many fascinating mathematical properties (for example, subtract 1 from φ and you get its reciprocal), what has captured most people’s imagination is the claim that the Golden rectangle is the “most beautiful” rectangle. Phi has been found in everything from the Parthenon and the works of Leonardo da Vinci to the music of Bach. Then there is that Chambered Nautilus. Phi has been found, not just in shells, but in the arrangement of seeds on a  sunflower and the positions of leaves on a stem.

This connection between mathematics, beauty and nature has elevated φ to almost mystical status. In fact, it is often called “the Divine Proportion”.

My project became clear. I would photograph a mollusc shell I had in my possession, overlay the squares and rectangles, float a few equations and capture this mystical connection with an image. Here is the result:

Shell photo submitted to the FPSI think this photo is pretty clever. But there is a problem. Those squares and rectangles don’t look anything like the classic version displayed at the beginning of this blog. And sorry, but there is no way that the ratio AC/AB=φ.

Well, if φ isn’t there in the usual way, it must be there somewhere. Using the measuring tool in Photoshop, I started searching for it. And I found it. Everywhere.

There was still a problem, however.  Every time I thought I found some dimensions whose ratio was φ, I would try to calculate the ratio using the rules of geometry. The ratio never came out to φ. Ever.

I began to worry. Had this mollusc not gotten the memo?

Then I noticed something. It seemed as if every time the shell completed a turn, it doubled in size. More careful measurements confirmed this to be true.

For every turn the radius doublesNow that is pretty interesting. If we take our lead from sports and use “the lap” as the units of measure for the angle θ around the spiral, then the equation for the radius  r of the spiral is r=2θ. What could be more elegant? Not only had this mollusc gotten the memo, but it had gone on to create the simplest possible logarithmic spiral — “2” being an integer, and the smallest one that will generate a spiral.

The Golden Spiral is not the logarithmic spiral, but one of a class of logarithmic spirals. All have the form r=x. For the Golden Spiral x=φ and a=4. For my local mollusc friend, x=2 and a=1. They all have the interesting property of geometric growth: 1 becomes 2, 2 becomes 4, 4 becomes 8 … like cells dividing. Which may explain why logarithmic spirals (although not necessarily golden spirals) are prevalent in nature.

What really happened here? The story of the Divine Proportion is exciting, mysterious and true. These very characteristics started me on this project and without it I would have learned nothing, created nothing. Yet that story also blinded me: as I searched for wonderful examples of how it applied, I could not see broader truths lurking below the surface. I could not see a wonderful diversity of forms, I could not see new forms of beauty.

Revealed truth may be necessary, but deep understanding requires discovery. Discovery involves leaving behind the comfortable and the sure, at least for a time. It is a process of exposing oneself, of lowering defenses, of taking risks. But in the end you not only learn new things, you have a deeper understanding of what you thought you knew all along.


About dougstinson

Doug Stinson enjoys pondering unexpected connections and sharing his discoveries. He is also a physicist, a photographer, a new product realization executive, and a student of history, the environment and religion. You can learn about his other creative ventures at
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2 Responses to Phi? Phi not!

  1. Don says:

    Very nice essay. I suggest you acquire a copy of D’Arcy Thompson’s On Growth and Form. The biology is well outdated but it’ll stimulate lots of photographic and mathematical ideas.

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