Every schoolchild has run into the brick wall of pi, and been told that the beauty of circles contains the mystery of it. Scale a circle large or small and the ratio between its circumference and its diameter will always be constant. Mirabile dictu! Not only that, but in our decimal system of numbers, the mysterious pi rolls on into infinity without repeating itself or terminating.
This, the schoolchild is told, opens the doors! This is how we learned to crack geometry, agriculture, architecture, wheels, gears, and fluid dynamics—the first mystery of the Universe!
P’raps it is. It raises a lot of questions, though, which although they may lurk at the edges of the child’s understanding, are rarely well-seen enough to articulate, and if they were, are not understood enough to be answered.
Why is the circle so compelling to human vision, for example? There’s a thought for humanists.
More important, what is it about our very space that generates this compelling form from a fixed distance and a single point? It works only in Flatland, where two dimensions define the nature of existence. Given three or four or five dimensions, the circle could go almost anywhere. For a perfect sphere, 4 πr2 will tell you the surface, but if the surface is dimpled, distorted, stretched or dinged, all bets are off. Old pi can work ‘til his digits start to burn, but he won’t manage irregularity. And irregularity is the nature of real-world circles, spheres, and mangled milk-cartons. Things don’t come in perfect forms, and yet, wondrous though it is, we make an assertion that holds true for ideal circles, spheres and other curvaceous things when we contemplate them in perfection. And there’s the interesting thing. Because we can reflect on an abstract and perfect circle, and can use our numbers to find its circumference, we can build airplanes and alfalfa sprinklers and machine tools. We can work back from these idealistic, abstract considerations and forge applications that are close enough for our purposes.
Of course, what that close-enough notion is depends on what we’re doing and why. The circle of a cylinder-head may only tolerate a variance of a few ten-thousandths of an inch. The circle of a crude ox-cart wheel, hand-hewed from old trees, may be as rough as a half-inch in some places and work well enough for the ox. Neither of these objects will prove π, but the proof of π in the abstract is strong enough to make their building possible.
This capacity for idealism reveals something powerful about the human spirit that never gets mentioned in fifth grade. We can override the existing in favor of the ideal, enough to navigate our way toward an envisioned end. This capability lies behind every important analysis human minds have accomplished, from garden-plots to skyscrapers. Somehow, that energetic capacity to envision and calculate on the ideal—which is done purely in the imaginative realm—gets often forgotten in favor of the hard and rough dimensions of the physical result. No skyscraper is a straight line, nor any rose-window an exemplary presentation of π. In the real world, poor old π gets missed, causing those who come afterwards to measure with most of its digits cut off. But in its proper realm, it operates perfectly in space as we envision it. If the psychological sciences, as they are laughingly called, could find access to the agency that drives that picture-show of perfect pi, and re-invigorate it, we would become a more perfect union by far.
None of this speaks to yet another odd question, which is why our particular concept of numeracy produces this gallant result when married to our peculiar version of spatiality. To some degree we buy the version of space that happens to match up to our sensory systems in the meat packages we ride around in. The body’s perception of light, heat, mass, gravity, force, and the space in which they play becomes very much our primary concern when those forces draw us into a battle for survival.
This is a considerable distraction from the Elysian fields where π is born, but tigers and bears may have their own version of ideals, including eating our children or stealing our groceries. Effort and counter-effort become the medium of exchange, and form the casbah of our days. This is an entertaining situation, to be sure, with all kinds of colorful sideshows to attract the attention. But that doesn’t make it true, and certainly not exclusively true.
The dictates of physical occupation inform us that space, for example, is a three-dimensional matrix in which solids play; that it is contiguous, continuous, and if you reach the end of it and look out you will see more of it. The agency that produces and uses ideals like π and i, for another example, may have a much wider and richer repertory of space-designs to play with, but this is a secret of the hearts and minds who treasure that capability as a refreshing escape from the compression of ordinary mass-centric spacetime.
Our shared numeracy was born in this commons of spacetime, so that we could count cattle, shekels, and stones. But because it was born in this realm of solid objects, it shares certain of its quirks. The illusion of unity is one such quirk. While a stone or a shekel are obviously single and alone things from the perspective of meat perceptions, and thus can be counted and added up, this unity, the characteristic of being one thing, is a convenient and apparently necessary illusion.
Absent that illusion, the matrix of space and the things in it might seem very different—a dance of melting starlight, a torrent of occasional solitons in a fury of birth and death and instantaneous re-creation—there are many possibilities. Spiritual gurus and wild-eyes artists like Dali have all tripped over this bridge between realms, where the counting up of things falls by the wayside in a heartfelt explosion of rising and falling being driven by very different concerns. This is not recommended for civil engineering tasks, to be sure, which have an obligation to support the framework of meat movements and meat needs.
But as any software engineer knows, putting something in the system requirements of a project does not make it any more true.