Once, my husband bought some discounted tickets to a concert of a famed blues/pop/rock guitarist; I was happy to make an evening of it, since I enjoy dates, live music, and discounts. After waiting over two hours for the star to take the stage and witnessing the crowd of thousands that still happily welcomed him by singing along word-perfect with every song, I realized I had dropped in as an outsider. I was a new fan, and I had much to learn. I did not belong at that concert. But as uninitiated as I was, I enjoyed it anyway, and my husband and I continue to listen to his music. That feeling is how I feel offering this little musing on fractals.
I admit here that I never took calculus or passed the mathematics Praxis exam. I am here as a casual observer. I am here because a branching maple grows outside of my window, and in my ribs that same maple, denser and sweeter, is upside down turning air into food for me. When I saw the dizzy image on my screen and thought “what a groovy piece of Hindu art,” I did not realize I would be sneaking chapters of Fractals: A Graphic Guide while I nursed the baby to sleep.
One day you buy discounted tickets for a vaguely familiar musician and the next, you are a super fan. After a two-hour delay, I am pleased to welcome to the stage, the Mandelbrot Set.
Fascinating Fractals
A handy teaching guide from the Fractal Foundation states that fractals “tell the story of the process that created them.”[1] The most familiar example available is that tree I mentioned above. Imagine the sprout that began it all (or go pull a specimen from your weedy back flower bed); the sprout branches; each of the branches branches; those branches then branch again, until you have got the “twig on the bough and the bough on the branch and the branch on the limb and the limb on the tree and the tree in the ground and the green grass growin’ all around.”[2] What you are observing is a pattern of self-similarity. The limb is a smaller version of the tree itself; the branch, a smaller version of the limb.
These self-similar fractal patterns exist all over nature. There are branching patterns, like lungs, maples, and river systems; and spiral patterns, like hurricanes, seashells, and fiddlehead ferns. Mathematically, fractal patterns are created by repeating the same process over and over. This repetition can be done with geometric figures, such as with triangles and cubes, or by plotting repeated calculations, as in the case of the famous Mandelbrot Set.
Benoit Mandelbrot coined the term “fractal” in 1975 to describe a phenomenon that had been known to mathematicians for years. One famed problem that Mandelbrot connected to the world of fractal geometry is the question posed by Lewis Richardson: “How long is the coastline of Britian?” If you drive along the coast of Britian, your car will register a certain distance. If you walk along the coastline, you can measure a greater distance. Then, imagine running your finger along the outermost edge of the coast, taking every detailed curve into your measurement. Continuing in this manner, it turns out that the length of the coastline of Britian approaches infinity.
This phenomenon is an essential characteristic of fractal shapes; as you “zoom in,” or increase the scale of the object, you find more detail of the same pattern that was found at the lower scale. It seems that the deeper you look, the deeper you have to go. This aspect of fractal shapes is called “recursion” and can be visualized in this famous tin of hot cocoa mix or in the last page of a bedtime picture book where the protagonist reads a bedtime story that is the bedtime story you are currently reading to your child.
More on Mandelbrot
Fractal geometry is better at describing the real world of natural shapes than Euclidian geometry. A perfect Euclidian shape—with its smooth curves and surfaces—is not possible in the physical world. It is a mathematical fact but not a physical reality. Fractal geometry uncovers the mathematics operating beneath the rough natural surfaces and patterns that appear to us as both simple and intricate, like clouds and waves and lightning bolts.
As it turns out, much natural phenomena do not fit neatly into our ideas of what an orderly world would look like on paper. Mandelbrot’s methods have helped get a mathematical grip on concepts like “turbulence, noise, clustering, and chaos.”[3] He also applied his concepts to the field of economics, challenging the “efficient market hypothesis” claiming that economics—such as the fluctuations of the stock market—fall into a smooth bell curve. “The very heart of finance,” he claimed, “is fractal.”[4]
Remember the paradox of the infinite coastline? Mandelbrot calculated that fractal shapes also have fractal dimensions; a smooth curve has a dimension of 1, and a smooth surface or plain has a dimension of 2. Fractal shapes occur between dimensions (they have dimensions expressed as fractions). The coast of Britian, for example, has a dimension of 1.25.[5] Lines with dimensions closer to 2 express a greater amount of roughness.
Mandelbrot’s time working for IBM gave him the insight into the use that computing can have to help visualize geometric shapes that were too unwieldy to work with in earlier decades. When he first plotted his equation (zn+1 = zn2 + c) into a computer, he “suspected that the geometric riot he was seeing was due to faulty equipment.”[6] But as he continued to explore the extraordinary shape in greater depth and detail, he realized the world of fractals ran far deeper and with more complexity than mathematicians had realized.
Although the geometry of fractals and roughness seemed to contradict the classical understanding of mathematics, Mandelbrot’s discoveries actually fit into a classical understanding of mathematics, where from “an unlikely assortment of particulars” (cauliflowers, lightning bolts, coastlines, the stock market) a Platonic essence emerges that can be interpreted mathematically.[7] Mandelbrot showed that patterns that earlier mathematicians saw as too rough and broken actually held intricate beauty.
Beauty in Brokenness
I do not pretend I have understood even half of what I have tried to learn about fractal geometry. But I do know that the nuggets I have panned have led me to greater awe for the amazing world we inhabit and the God Whose imaginative and creative power spoke that world into existence.
I was first gob-smacked by fractals when I was actually led to believe that a finite shape can be proven to have infinite surface area. At first, I was tempted to think it was just a bunch of modern nonsense—mathematicians disconnected from the real world in ivory towers playing with silly calculations to make pretty shapes. But then I remembered Ecclesiastes 3:11: “he has put eternity into man’s heart” (ESV) When I look out over the ripening grasses in the back field, the sunrise pinkening the sky as fog still hangs heavy on the tree line, it does feel like eternity is bursting in my heart. I am a finite creature trying somehow to take in the beauty of an infinite God.
Augustine wondered about the infinity tucked away in the finite as well. He begins his Confessions with questions and reflections about God’s transcendence and imminence: “In filling all things, you fill them all with the whole of yourself. . . . [I]s the whole of you everywhere, yet without anything that contains you entire?”[8] If that was not enough, things really come to a head in the incarnation—God made man—made baby—tucked into that young lady named Mary: “Thou’hast light in darke; and shutst in little roome,/ Immensity cloistered in thy deare wombe.”[9] Suddenly, diving through Mandelbrot’s Set feels like only one small part of the great world of mystery we have been plopped into; “Great is the Lord, and greatly to be praised, and his greatness is unsearchable!” (Psalm 145:3).
Then, as I started to learn more about how many seemingly chaotic or unpatterned phenomena actually do exhibit a fractal pattern quality, I realized again that I was seeing another picture of a spiritual truth. Paul succinctly conveys this truth in one of the most famous chapters of the Bible, Romans 8, that all things “work together for good, for those who are called according to his purpose” (v. 28). Joseph recognizes that same pattern emerge from the broken and jagged pieces of his own life: “you meant evil against me, but God meant it for good” (Gen. 50:20). There are no unwoven strands in the history of humanity.
Mathematicians are still exploring the Mandelbrot Set, and other shapes like it. Undoubtably future mathematicians and thinkers will uncover new mysteries of creation. But we will have all eternity, beginning now, to wonder in amazement at the mystery of the gospel.
“Eye hath not seen, nor ear heard, neither have entered into the heart of man, the things which God hath prepared for them that love him” (1 Cor. 2:9, KJV).
[1] Fractal Foundation, Educators Guide, https://fractalfoundation.org/fractivities/FractalPacks-EducatorsGuide.pdf.
[2] Sam Hinton, “Green Grass Grows All Around” from Classic Folk Songs for Kids from Smithsonian Folkways (Smithsonian Folkways Recordings, 2016).
[3] Jim Holt, When Einstein Walked with Gödel: Excursions to the Edge of Thought (New York: Firrar, Straus and Giroux, 2018), 91.
[4] Ibid., 98–99.
[5] Ibid., 101.
[6] Ibid., 100.
[7] Ibid., 101.
[8] Augustine of Hippo, Confessions, trans. Henry Chadwick (New York: Oxford University Press, 1992, reissued 2008), § I. iii, p. 4
[9] John Donne, “La Corona,” from John Donne Poetry and Prose, ed. Frank J. Warnke (New York: Random House, 1967), 250.
June 11, 2024
I first learned of fractals and Mandelbrot in the 1990’s, via screensaver programs for the Mac. Fascinating indeed! Only recently have I looked into the mathematics behind it all. As you so masterfully describe it, we encounter fractals everywhere we look, it seems. Sometimes, I’ve wondered whether or not time and/or space may even have a fractal connection. But that is all beyond me. Thank you so much for this excursion!
June 19, 2024
How interesting! That could possibly explain why the universe appears so old by certain dating estimates? Something to think about.