(From a paper I wrote for my math 100 class.)
The above photo collage, clockwise from top left, shows the mouth of the Selenga River Delta, an angiogram of a human kidney, a deciduous tree in winter, and a typical graphic of a fractal image. I find the similarity of the four images astounding.
Before watching Nova’s “Hunting the Hidden Dimension,” I had been engaged in a love affair with photographing trees in winter. The first two images below are of the treetops reflecting sunlight on our property in the woods of Vermont, that I took just a few days before watching the documentary. The next two photos were taken in Sacramento a few weeks ago, at sunset. I was fascinated by how the branches looked against the sky. I was fascinated by their intricacy and repeating patterns. I had no idea at the time that this fascination of mine was all about fractals!
“Hunting the Hidden Dimension” takes us on a journey through Benoit Mandelbrot’s discovery (invention?) of fractal geometry in the 1960s and 70s, at first pooh-poohed by mathematicians and scientists as nothing more than pretty patterns that seem to repeat never-endingingly, without much meaning beyond “art.” Mandelbrot eventually applied formulas and geometry to these patterns, especially as the age of computer technology became more and more sophisticated. Scientists soon became fascinated with their application to everything from the way plant life and human bodies grow, live and function, to cell phone technology, cinematography and much more.
In particular, I was drawn to the idea that nature, in both the plant and animal kingdoms, has found a “very simple fractal code” or “mathematical rule” from which to determine when to branch as things grow and get bigger, creating what appear to very complicated organisms from just that simple genetic code: “when to branch.” Think about it. Dr. Brian Enquist, of the University of Arizona, put it this way: “Evolution by natural selection has hit upon a design that appears to give the most bang for the buck.” To think this mathematical genetic code may be common to all living things, as well as the way non-living things in nature change or evolve, creating almost identical patterns in different organisms and natural phenomenon, is nothing short of mind-blowing. A seeming “search for the path of least resistance” is common to our vascular network, tree branches, and water finding its way to a delta – one very simple idea that applies to almost everything, “when to branch,” creating repeating patterns that are the foundation of fractal geometry. The formulas for these patterns work just as miraculously when applied to the technological world. Nature simply knew it all along.
The application of fractal geometry to so many other branches (no pun intended!) of science has been very useful in technological advancements. Antennae have been redesigned into fractal patterns that allow them to be reduced to a size small enough to fit inside your cell phone, with the ability to pick up many more frequencies than their gargantuan rooftop ancestors. Cinematographers use fractal geometry to create amazingly realistic special effects. Doctors are hopeful that the fractal design of healthy blood vessels will eventually lead them to earlier detection of the chaotic nature of blood vessels feeding a cancer, hopefully increasing the cancer cure rate significantly.
I am no mathematician, however, my husband is a math geek (squared) and he has told me many times about the relationship of pi to the constant “e” and the irrational number “i,” which goes like this: e to the power of i x pi + 1 = 0. Pi, e and i were discovered in completely different eras, long before the 20th century, yet have this unique connection. Mandelbrot discovered fractal geometry in the 20th century, and it, too, has a relationship to pi. I watched a video on Numberphile (if you are a math geek like my husband, click here to view) explaining this relationship. Suffice it to say, I am in no way qualified to explain that relationship to you here. That would be a much longer discussion, I am sure! My point is simply that I have a sense of wonder that these pivotal numerical concepts, discovered hundreds – even thousands – of year apart, have such close relationships.
“Hunting the Hidden Dimension” has opened a door for me to a whole new world of awareness.
Balancing a checkbook or measuring ingredients for a recipe are very obviously all about math, but who knew the way trees grow, or a stream finds its way through the woods, or the rhythm of your heartbeat, were all about math, too? Math is everywhere! My husband has been calling me “Fractal Lady” because I can’t seem to get over the utter symmetry and beauty of it all, and what it implies about our world. When we sit outside relaxing in the Vermont evenings, I find my eyes drawn to the treetops, the bushes, or the dead stalks of last year’s flowers, around our property. My husband asks, “Fractaling?” Yup. I haven’t figured out exactly what it all means yet, but I am having a lot of fun observing and pondering this phenomenon all around me.
This fractal image is just fun!