You stick with your triangles
In Charlottesville and Mosul, Berkeley and Sirsa—and in any number of other places in between—the fault lines are clearer than they have ever been. It’s hard to consider what’s been happening in those places and not feel a certain despondency. That is, if the thought that humans seem more divided than ever, more willing to expose, observe and widen those fault lines, makes you despondent. Others’ mileage may vary, of course: I realize that the very nature of divides means there are those who provoke them, those who wallow in them.
But there’s despondency, and maybe there’s rationalization, and then there’s also the realization that this is a phenomenon worth studying and thinking about. Certainly I can imagine sociologists and anthropologists doing so. But trying to understand an intriguing phenomenon is also what scientists and mathematicians do. It’s why we have possible explanations about the connection between cicadas and prime numbers, or why some ants choose to turn left.
And so it should be no surprise to find that there are mathematicians who have put their minds to trying to understand how and why social divides happen.
Vi Hart (@vihartvihart on Twitter) is easily one of the world’s more entertaining mathematicians. She studied music too, and she actually calls herself a mathemusician. She has a fascinating presence on YouTube: check her clips on why pi should actually be 6.2831…, not 3.1415… (yes, there are excellent reasons); or her examination of Mobius strips using the services of the estimable gents Wind and Mr Ug; or her Christmath Special (“On the first day of Christmas, my true love gave to me … the multiplicative identity.” Delightful.) She is also responsible for several clips in which she counts down the clock on a microwave (“eight twenty-two, eight twenty-one, eight twenty …”), for no reason I can fathom.
Given her quirky mathematical interests, it hardly surprised me to find that Hart is the co-author, with Nicky Case, of a mathematical experiment that makes it easy to visualize and understand certain ideas about human divides. The experiment is called Parable of the Polygons: A Playable Post on the Shape of Society, and it is no academic paper. Instead, they refer to it as a “playable post” (see it here: bit.ly/12ncSpc). That means that there’s some text, but there are also illustrations that you, the reader, can play with and change—and what happens as you make changes carries some lessons.
It goes something like this. You have a grid—let’s say 10x10—to simulate a neighbourhood of homes. It is largely, but not completely, filled with squares and triangles positioned at random. They are all what Hart and Case call “slightly shapist”, meaning that while the shapes generally “prefer being in a diverse crowd”, that’s subject to an upper bound on such diversity. That upper bound is expressed in a rule like this: “I wanna move if less than 1/3 of my neighbours are like me.”
That is, I don’t want to be in a totally mixed and integrated neighbourhood. I want at least a few of my neighbours to be like me. Not a lot—a third will do, in this case— but at least some. Seems like a perfectly reasonable desire, and one that a lot of us feel, to some degree or other, all through our lives. Right?
Well, the Parable of the Polygons then lets you simulate what happens to the neighbourhood as this perfectly reasonable desire plays itself out. It starts with the grid and the randomly placed squares and triangles—the sharp-cornered denizens of Polygonland. The fellows who are satisfied with where they are located—because a third or more of their neighbours are like them—have a happy face. The ones who are unhappy—because less than a third are like them—have, yes, an unhappy face. Your job is to move each disgruntled denizen to an empty spot in the grid, and keep doing so until nobody in Polygonland is unhappy. As Hart and Case say, “Just move them to random empty spots. Don’t think too much about it.”
So you do just that, or I did just that. It didn’t take too long. I started with a pretty mixed city. There were some stretches of three squares or four triangles, but that’s what random placing will produce anyway. (A totally mixed pattern is more than likely produced by careful planning and placing). One-by-one, I moved the sad faces to random empty spots (I didn’t think too much). The move would sometimes make them happy, sometimes keep them sad, sometimes turn a neighbour unhappy and she would need to be moved in turn. I stuck manfully to my task.
Eventually, everyone was happy.
But me, I was startled, and maybe a little sad. For what began as a relatively mixed city had turned into one with extensive mono-shape areas. A precinct of squares here; adjoining it a district of triangles; beyond it, another swathe of squares. Wondering if I had made some mistake, or if this was just a one-off occurrence, I tried again, and then again. Result: the same every time. Simply by moving a few unhappy individuals around until everyone was happy, I was producing exactly the ghettoes I find so hard to take in cities.
Later in the parable, you can experiment with a larger layout. The software automatically moves unhappy dudes to random empty places for you. Concurrently with the simulation, a graph reports how ghettoized, or segregated, your model city is. Following this one-third rule, most simulations end up with a level of segregation close to 50%(100% would be if all the squares were together in one block, and all the triangles in another block).
It’s all rather depressing. The implication is that lightly held prejudices, if acted on, can lead to pretty marked divides. Or as Hart and Case remark: “Though every individual only has a slight bias, the entire shape society cracks and splits. Small individual bias can lead to large collective bias.”
The parable also allows you to specify different levels of bias. Instead of a third, I tried the simulation at a 50%preference level; meaning that if less than half of their neighbours are like them, the shapes are unhappy and want to move. As you might guess, this led to consistently higher levels of segregation, usually about 80%. A 60% level produced even more segregation, now touching 90%.
But something interesting happened as I nudged the preference levels higher still. At 80%—meaning unless at least four of every five neighbours of a given shape is the same as him, he will want to move—the simulation never ends. That is, you never come to a point where everyone in Polygonland is happy. What’s more, the segregation level rarely rises above the 20% level. The takeaway here is, in a perverse way, rather satisfying. The more prejudiced everyone in a community is, the harder it is for them to find a way to live among their own kind. The harder it is, as the parable has it, to find happiness.
The ideas behind the parable experiment come from economist Thomas Schelling. In a 1971 paper (Dynamic Models of Segregation, The Journal of Mathematical Sociology), Schelling says that this kind of analysis “can be used to explore the phenomenon of ‘neighbourhood tipping’”, in which an influx of “different” people into a largely homogeneous area “causes the earlier residents to begin evacuating”. There are estimates that in some American cities, mostly white areas needed only a 20% influx of blacks for tipping to happen. (Thus 20% is a “tipping point”).
There’s plenty more to mull over in Schelling’s work, as well as plenty to fiddle with in Hart and Case’s Parable of the Polygons. There are lessons about how lines in our minds translate into often destructive realities in the world around us. But Hart and Case end on an optimistic note.
Instead of acting on lightly held individual biases towards our own kind, what if we ignore them and actively seek out the “other”? You guessed it: segregation declines dramatically—and everyone ends up happy. Here’s how they describe that:
“If small biases created the mess we’re in, small anti-biases might fix it. Look around you. Your friends, your colleagues, that conference you’re attending. If you’re all triangles, you’re missing out on some amazing squares in your life—that’s unfair to everyone. Reach out, beyond your immediate neighbours.”
In the time of Charlottesville and Mosul, Berkeley and Sirsa: Amen to that.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His latest book is Jukebox Mathemagic: Always One More Dance.
His Twitter handle is @DeathEndsFun