“Clouds are not spheres,
Mountains are not cones,Coastlines are not circles,And bark is not smooth,Nor does lightning travel in a straight line.”
― Benoît B. Mandelbrot
Reality is far less smooth than we’d like to believe. In our quest to represent complex things, like coastlines or the Armenian identity, we often end up oversimplifying them.
Fractals are geometric structures that come closer to capturing the complexity of certain real-world objects. Check out this webpage for a quick tour. “Fractal” comes from the Latin word fractus, meaning “fragmented”- much like the global Armenian existence today. It was coined in 1975 by Benoît B. Mandelbrot, born Benoît Mandelbrot, whose middle initial was added later in life and is rumored to stand for “Benoît B. Mandelbrot”.
From his obituary in the New York Times:
Dr. Mandelbrot traced his work on fractals to a question he first encountered as a young researcher: how long is the coast of Britain? The answer, he was surprised to discover, depends on how closely one looks. On a map an island may appear smooth, but zooming in will reveal jagged edges that add up to a longer coast. Zooming in further will reveal even more coastline.
“Here is a question, a staple of grade-school geometry that, if you think about it, is impossible,” Dr. Mandelbrot told The New York Times earlier this year in an interview. “The length of the coastline, in a sense, is infinite.”
Fractals are difficult to define formally, even for those who study them, and controversy surrounds which natural objects can be considered fractals. Some even say that fractals shouldn’t be strictly defined at all.
I can’t help but think that Armenianness would benefit from such an approach, since far too many Armenians have been alienated from their heritage by rigid and simplistic definitions that shame, judge or enact violence upon those who don’t conform to them. These definitions often coincide with misogyny, homophobia, patriarchy, transphobia, and anti-Blackness, and other things we’d all be better off without. I’m not alone in preferring an Armenianness that protects rather than bullies, recognizes the humanity of each person, prevents people from dissociating and disappearing from their communities, and tells us that yes, we are Armenian enough. We are an inextricable part of this ancient and now global people. Our identity is expansive – to force it into boxes is to call a coastline a circle.
As for fractals, instead of defining them rigorously, we can characterize them by a collection of features.
Fractals are self-similar, which means that each part is like the whole, but smaller. The whole set has miniature versions of itself within itself, and thus appears similar at different levels of magnification. For example, if you break a branch off of a tree and put it upright into the ground, it will look like a tree! And if you repeat the process with that branch, that will also look like a tree! And so on, forever.
In mathematics, fractals are infinitely self-similar, but in nature, physical considerations make the self-similarity end after a certain amount of zooming in. I love thinking of all the different ways that Armenianness is self-similar. Our tensions are many: conformity versus exclusion; leaving versus staying; self versus community; diaspora versus republic; east versus west; community involvement versus the freedom of privacy; “what will they say?” versus what you want; being defined by others versus defining yourself; modernity versus tradition; existence versus oblivion; preservation versus experimentation; personal beliefs versus the Armenian church.
These tensions might manifest totally differently in various Armenian diasporas around the world, but they often manifest at the same levels of zoom: the self, the family, the local community, the global Armenian population. And often, the tension within oneself is similar to the tension within one’s family, which is similar to the tension within one’s local community, which is similar to the tension within Armenians worldwide.
This is not to say that all Armenian communities worldwide face the same problems. When zooming into one area of a fractal, you can encounter new and unique patterns that don’t appear elsewhere. The more we open up to learning about each others’ realities, the more of these patterns we can discover.
Fractals are also detailed. You find more patterns the more you zoom in. Much like Armenianness, more complexity emerges the closer you look. Our community is not monolithic. Each part has its own story and its own issues, even if some of those stories and issues are shared. No matter how closely you zoom in, it doesn’t smoothen. And while its complexity doesn’t diminish, you can often see the same structures appearing again and again. On all scales of magnification, it has details.
Lewis Fry Richardson, a mathematician and pacifist who pioneered modern methods of weather forecasting, discovered something interesting when investigating whether the length of a shared border had any relation to the likelihood of two countries going to war. He saw that the shared Spain-Portugal border had reported lengths that differed by 227 kilometers! This baffling inconsistency led Richardson to discover the coastline paradox: that the length of a coastline depends on the length of the ruler you use. The shorter the ruler, the longer the measured length of coastline- which can in theory be infinite.
Mandelbrot in his famous paper “How long is the coast of Britain?” used Richardson’s findings to explore the idea that fractals exist in-between integer dimensions (in “fractional dimensions”). Mandelbrot reported that the west coast of Britain is not 1-dimensional or 2-dimensional, but 1.25-dimensional. We can also think of Armenianness as existing in-between dimensions. And rather than flattening our complexity to fit into systems too simplistic to contain us, we should just find better systems that represent us. Perhaps we are a 1.25-dimensional people.
My favorite fractal is the boundary of the Mandelbrot set, a beautiful set of complex numbers. When the Mandelbrot equation is given a number (representing a point on the complex plane), the results under repeated iteration either tend to infinity or stay close to zero. These two types of points can then be colored differently, creating a coded map. As Arthur C. Clarke states in The Colors Of Infinity, “What we get from this basic iteration is a kind of map dividing this world into two distinct territories: outside of it are all the numbers that have the freedom of infinity. Inside it, numbers that are prisoners, trapped and doomed to ultimate extinction.” The boundary between these two is a fractal curve, infinitely complicated and revealing ever-finer detail upon magnification. And Armenianness, too, feels similar to this boundary, feet planted firmly in both disappearing and thriving, surviving for thousands of years against all odds. The more we zoom into this boundary, the more there is to see.
I used to wish that that life was more Euclidian. Sometimes I still do. But fractals (and people) have shown me that infinite complexity is breathtaking, even when it’s frightening. It has a pulse. And having the language to discuss that complexity opens hidden doors.
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Coastlines
Mountain ranges
Clouds
Trees
Leaves
River networks
Galaxies
Lung bronchi
Ocean waves
Smoke
Bubbles
The nervous system
Lightning
Fingerprints
Snowflakes
Romanesco broccoli
The vascular system
Queen Anne’s Lace
Earthquakes
Vegetative ecosystems
Ferns
Seashells
Jupiter’s surface
Financial markets
Hurricanes
Crystals
Bacteria
Pressing paint between two surfaces and pulling them apart (Decalcomania)
Algae
Animal coloration patterns
Feathers
Jackson Pollock’s paintings
Mountain goat horns
Heart rates
Canyons
Waterfalls
DNA
Saturn’s rings
Armenianness
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