I am sitting on the shore of Lake Michigan, and I trace my finger through the sand. My sunburnt skin is pink like the sky; waves lap at my feet. I am trying to make a perfect circle — flawlessly round, no bumps, not oblong.

And I am struggling.

You have probably seen the videos of professors drawing perfect circles on a chalkboard. Or maybe, you have heard of Giotto, the artist who proved his worth to Pope Benedict XI by freehanding a perfect circle. We seek out circles in nature and the world around us. There seems to be a very human obsession with the circle, the perfection of this simple geometric entity. We like the idea of this perfection, even if we admit it’s unattainable. Despite math being a notoriously daunting and inaccessible subject, a circle is something we can all understand. But, does the perfect circle even exist?

To begin to explore that question, I first had to define what it means to be a perfect circle. I knew my doodles in the sand did not meet the criteria, but I wondered whether they were attainable at all. This inquiry quickly thrusted me into a speculative and conceptual realm. Apparently, it’s not enough to say a circle is … round.

In an interview with The Michigan Daily, mathematics professor Daniel Burns explained that the concept of the perfect circle can be defined in many ways.

“You can write an equation (for a circle), which is more abstract, but more precise,” Burns said, “There’s some ideal circle somewhere, and it has various definitions. Those definitions … are perfect.”

Dr. Steve Cousins, the executive director of the Stanford Robotics Center, concisely stated one of these definitions in an interview with The Daily.

“A perfect circle is a set of points that are all the same distance from a center point,” Cousins said.

The equation for a circle is simple: r2=x2+y2. It only needs two components — a center (conveyed through the *x* and *y* components) and a radius (*r*, the distance from the center of a circle to the edge). These two components align with the two parts of Cousins’s initial definition.

It seems the more abstract you get — and the more your circle looks like an equation — the more perfection you uncover. So, in some regards, I could make a perfect circle in the sand. All I would have to do is arbitrarily choose a center point and assign it a coordinate (0,0). Then, I would choose a radius, such as 2. I could scrawl in the sand, as messily as I wanted, x2+y2=22, and I would have a perfect circle. But, the more grounded in reality I become — and the more my circle looks like a twist in the sand — the more I am forced to confront imperfections. No matter how many times my finger spins through the sand, I will always be infinitely far from the perfect circle I strive for.

Burns defined the mathematical world and physical world as two separate realms, where the perfect circle exists in the first and not in the second. There exists a mathematical world full of immutable perfections that do not exist in the physical world.

“Talking with (physicists), it’s so clear that the mathematical world is much bigger than the physical world … there’s so much there that doesn’t have physical interpretation,” Burns said.

He described how many theoretical mathematicians are content floating in this mathematical world without ever touching the ground. This “Platonic realm,” full of the perfect forms of imperfect realities, has value in and of itself without being confined to actuality. “You can’t make mathematics fully grounded in some finite sense,” Burns said.

However, Burns made it clear that one of these realms isn’t better than the other. He described using circles to teach his classes. He joked about how sometimes those circles “look more like eggs or sausages and things.” But students do not have a problem interpreting what he’s saying; they don’t think he’s trying to represent anything other than a circle. “I tell them I’m drawing a circle and we think of it as a circle,” he said. The perfection of these circles isn’t important. Instead, the importance lies in what is communicated or conveyed.

Beyond communication, Burns surprised me when he began to articulate a different notion of a perfect circle. He first referenced the same definition of a circle as Cousins — a set of points the same distance from a center point — but further built upon it and said, “In a different geometry, maybe one of my eggs was a circle for some other notion of distance.” If you define distance differently but still measure from a center, you end up with a very different perfect circle. For example, using the taxicab geometry, your circle ends up looking like a square rotated by 45°. A perfect circle doesn’t even have to be round, like I initially thought. In this way, any given circle is only perfect when confined to a specific type of geometry, and the lines begin to blur between the perfect and imperfect.

Frances Flannery, a professor of religion at James Madison University, described how there are many beliefs which challenge this same distinction between perfection and imperfection. Speaking generally, she described the concept of Samsara in Buddhism and Hinduism.

“It is the cycle of death and rebirth, the cycle of delusion you want to get out of,” she said.

Breaking from this cycle means realizing there are no distinctions between anything — perfection and imperfection are one. Just like how Burns’s blackboard eggs could be perfect by changing the notion of distance, anything could be considered perfect, imperfect or neither, by breaking from this cycle.

“Perfection is not the framework, and the idea of perfection versus imperfection within those religions is part of the confusion or the illusion we’re trying to overcome,” Flannery said. The distinction breaks down.

When the lines blur between perfect mathematical and imperfect physical realities, travel can occur between them. Suddenly, the simple profoundness of the circle is applicable; its perfection is not a hindrance to imperfect reality, but an asset. Robotics is built upon this translation; a robot thinks in perfect, mathematical terms, but simultaneously has to exist and function in the physical world.

Cousins described a simulation: a sushi restaurant where a robot needs to pick something up from a rotating table. This appears to be a highly complicated scenario — accurately predicting the intersection of the robot and the target’s path seemed challenging, and I pictured the robot chasing its mark in circles. But, Cousins described how the equation of the circle was simply extended by a dimension to include the factor of time. I was mesmerized as he explained this process. Over the course of my exploration, I had seen the distinction between perfection and imperfection become less sharp, less rigid. And here, Cousins completely transgressed the divide between math and physicality.

“Very elegantly, the robot was able to watch and pick something up from the table perfectly as it came in, because you could model (the situation) so beautifully … It takes a little bit of working, it’s not the way our minds work in everyday usage, but it’s very beautiful once you use it,” Cousins said. It doesn’t seem right that we could force a perfect concept onto a volatile and imperfect reality, but this imposition allowed the robot to navigate our imperfect world.

“The real world is messy, and mathematics is really clean and pure, Cousins said. “So in mathematics, you have an infinite set of points at exactly the same distance from the center of the circle, which is an infinitely small point..

The translation from mathematics to reality involves the introduction of blemishes. Cousins described the construction of circular components for projects in the real world: the material is attached to a lathe, where it is spun and shaved down to be circular. But, when attaching it to the lathe, that center point is not infinitely small — it has a diameter. The perfect mathematical model has already been broken, and that circle is not going to be perfect.

This lack of perfection has to be accounted for in robotics. “When you engineer things you have tolerances,” Cousins said. “There is some kind of gap that’s built-in, it’s plus or minus something. Maybe that something is in microns and maybe it’s in millimeters, and that’s going to define how perfect it is.”

Again, the definition of the perfection of a circle is variable and dependent on the definitions and constraints placed upon it. As long as a circle is within the defined tolerance, it might as well be perfect.

I look down the shoreline, and there is a trail of circles in the sand. The sun is setting and I am having trouble discerning one from another. Some have been washed away by the waves. None of them are perfect, I don’t think. Although, I suppose that perfection depends on how I am choosing to define it, and I’m unsure whether the distinction even matters. Because as they disappear into the darkness, they are simply another aspect of this world. And maybe, they remind us that wonder can be found in the simplest of forms. Maybe, they are just beautiful.

*Statement Columnist Eleanor Barrett can be reached at egbarr@umich.edu. *

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