Posts Tagged ‘mathematics’

When a certain physicist I know tells me excitedly that he picked up something really cool for me to photograph from the farmer’s market, I know that it will undoubtedly be an edible object that in some way visualizes some fundamental principle about how the world works. In this case it was a head of romanesco broccoli with its beautiful repeating pattern that is a “natural representation of the Fibonacci … a logarithmic spiral where every quarter turn is farther from the origin by a factor of phi, the golden ratio.” (source)

Indeed! Well, I did have fun photographing it. When I thought I was done, I put away my camera and picked up a knife. It was time for dinner, you see. But then at the look on the physicist’s face, I put the knife down and said, “Uhm, would you like the honors?” And so he gently broke it apart revealing and reveling in the ever smaller yet repeated pattern of the larger broccoli.

In the end he sauteed the little bits in garlic and olive oil and topped it with a bit of cheese. Quite good. And there remained just enough of the veggie to place in a little ramekin. “Like a little Christmas tree,” he said. “We could decorate it with baby capers!” I don’t think so but it looks like I will have the opportunity to photograph this tasty mathematical subject a while longer.

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The story behind the image:  Steve and I were taking a short walk along Revere Beach.  The tide had receded quite a bit.  He followed the water. I stayed on shore searching out seashells and stones and wishing I’d worn a thicker sweater.  As he returned to me, he suddenly paused and shouted, “Come here. You have to see this.” I raced over and looked down at where he was pointing.  Lines and curves in the sand?  “Bifurcation diagrams in nature,” he exclaimed.  I peered more closely, frowning.  He tried explaining the mathematics of what he saw for me. “It’s like the multiplication of little streams leading to chaos.” “Well,” I said slowly, “I’m reminded of those Asian landscape paintings of mountains with cascading waterfalls over the rocks.”  We studied the sand for a bit longer, he helping to point out different ways to frame photographs of the bifurcation he was seeing, and we both appreciating our different perspectives of the world.

A poster print of this “mountainous” scene is available online here.




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As I prepare this post, I sit in a hotel room in Dublin, Ireland.  Rain falls pretty steadily.  The air is chill and the sky is the color of smoke.  I should be cold and grumpy and yet I am warmed and made cheerful by the fractal images of Robert Grzybinski.

I was first introduced to Mr. Grzybinski at a company picnic.  Somewhere in the course of our brief conversation, as I talked about my photography, he shared that he produced fractals.  Well, if you follow this blog at all, you know how much I love shapes and colors.  I asked if he’d share some of his images with me, and thankfully, he agreed.  He also shared the creative process and inspiration behind his work.  It is my pleasure to share his words and images with you.  Enjoy! 😉

How do you create these images?  I use an ancient MS-DOS program to make them. I give the program a bunch of input parameters, and it generates some output, which usually doesn’t look like much. After that the process is a lot like looking at a microscope slide — zooming in, moving around, looking for the interesting bits. You never know what you will find; it just continually amazes me what is hidden in that space of pure mathematics. Then I compose the image and assign the colors, which is sometimes the hardest part.

What’s the difference between these two images?The first image (“emboss”) is kind of a classic fractal – curvy, self-similar, spirally (spirals are very common in fractal patterns).  It has a kind of sculptural quality.  The second image (“treez”) has a spirally character too, but is made up of angular shapes and is completely flat, like something made out of cut-out paper.  I especially love the confetti-like background.

How did you choose the basic algorithm for each? The fractal program has a bunch of built-in functions.  From experimenting, I know very roughly what kind of fractal each one will produce.  “emboss” was made from one of the built-in functions.  The program also gives you the ability to write your own functions, and I have had more fun and mostly more interesting results doing that.  The functions are not very complicated, but it is just amazing to see the complexity that results from a few simple lines of code.  “treez” was made from one of my own functions.

How many free parameters do the functions have? Depending on the function, there can be up to four or five numerical parameters.  It’s usually not obvious or predictable what these parameters do.  You have to just stick in some numbers and see the results.  There are also many other settings that change the way the image is calculated.  Again, you need to play with these to get a feel for what they do.

How did you choose the colors? The programs uses an indexed color system, where each region of the images is represented by a number.  You then apply a palette which maps a particular color to each number – so to change the coloration, you just apply a different palette.  I created a lot of different palettes with different characteristics (cool, warm, subtle, contrasty, etc.).  Sometimes I know what effect I am going for, but sometimes I just try a lot of different palettes and hope something serendipitous happens.  “emboss” is an example of that.  It was an interesting pattern, and I knew there was something there, but it didn’t really work until I hit on the red/gold palette.  Then it just popped out, like something embossed in gold foil.

What inspires you to create new images?  What inspires me most is the sense of exploration.  It’s a lot like looking through a microscope at a drop of pond water, or maybe exploring the depths of the ocean in a submarine.  You just never know what weird and beautiful things will show up next.  In a sense, these images already exist somewhere in a mathematical space, and I am just using the computer as a tool to discover them.

View an expansive gallery of Grzybinski Fractals via this link.  For more information about Mr. Grzybinski’s fractals, you can contact him directly at cha.otic[at]earthlink.net.

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