Square Pegs in Round Holes

It began I suppose with Plato (c. 375 BCE)…like most everything else. His Timaeus, which Alfred North Whitehead dubbed the foundation of all Western philosophy, sought to explain the World in terms of the 5 aptly named ‘Platonic Solids’. The faces of each of these solids can be constructed from planar triangles.  

Plato’s concept of a triangle involved a flat construction, consisting of three straight line segments, intersecting at three nodes (vertices), and forming three internal angles, the sum of which is always 180°. No surprise here: Plato was ‘bookended’ by Pythagoras (Pythagorean Theorem) and Euclid (The Elements).   

Even today, when folks refer to a ‘triangle’, they most likely mean exactly what Plato meant 2450 years ago. Most likely…but not necessarily. The concept of ‘triangle’ inherited uncritically by, and then from, Plato has been expanded by more recent ‘discoveries’ in geometry and topology.  

Plato’s triangles are defined in the context of what is now known as Euclidean Geometry. However, certain ‘non-Euclidean’ geometries are internally consistent and externally relevant, at least in some applications. 

Generally speaking, non-Euclidean geometries apply to curved rather than flat surfaces. As a result, either parallel lines do not exist or, if they do, they intersect. The internal angles of a non-Euclidean triangle will not necessarily total 180° (may be more or less depending on the curvature of the surface) and the shortest distance between two points may be an arc. Most importantly for our purposes, a non-Euclidean cube will have curved edges. 

These newly discovered geometries still describe intuitively recognizable configurations of reality. We are all familiar, for example, with the surface of a ball. Whether your geometry is Euclidean or ‘non’, whether your topology is level or curved, whether you’re a flat-earther or a round-earther, certain generally accepted principles still apply; for example: 

If you’re walking from Boston to Detroit, you can expect to have many adventures along the way; keyword: along the way! There is one kind of adventure you do not expect. You’re not concerned, like Stephen Dedalus in Joyce’s Ulysses, that you are about to step off into Eternity, or what amounts to the same thing, that you’ll suddenly and unexpectedly find yourself in Memphis. 

That’s because any path along any surface will be continuous. I call it the One Foot in Front of the Other property; Mathematicians call it the ‘Archimedean Property’. Translation: no one just ‘suddenly shows up’ in Memphis. It turns out that there are interesting and internally consistent geometries that don’t include the Archimedean Property and where people do suddenly ‘materialize’ on Beale Street. But not today; keep a look out for future TWS on that subject.   

Euclidean geometry is just a special, degenerate case (zero curvature) of more general non-Euclidean geometry. Just as we use the positive term ‘Entropic’ to represent a loss of actual order so we can use the positive term ‘Euclidean’ to represent a loss of potential variety.  

Why limit yourself to boring old 180° triangles when the angles of non-Euclidean triangles can add to any positive number up to 540°, depending on the curvature of the surface? Ok, I can smugly pose that question today…prior to c. 1800 CE, it was assumed that there was only one geometry… and that it was Euclidean.   

Imagine. For more than 2 millennia, Euclid was the only game in town. That’s 2000 years in a row that Euclid won all 4 PGA majors (the Masters, the US Open, the PGA Championship, and the ‘British’ Open)…and he wasn’t even a very good golfer. (You’d be embarrassed to show up at your posh country club with his handicap.) 

How’d he do it? Simple. He was the only golfer on the course. He was the GOAT, albeit a very lonely goat. Today, Euclid is just one entrant in a crowded field…and he struggles every week just to make the cut. We should keep Euclid in mind whenever we’re tempted to think ‘we know it all’. Remember: the thing you’re most sure of in all the world may turn out to be completely wrong! 

I am reminded of Sir Roger Penrose. A precocious lad, he was middle school age when his math teacher presented the class with a classic problem in geometry: “There is only one solution to this problem. No one has been able to prove that there’s only one but we’re sure that’s true.”  

Never say ‘sure’ to a rebellious teenager. Them’s fightin’ words, especially to a kid who would one day become ‘the smartest man on earth’. Next day, he arrived at school with a second solution to the problem. Then he discovered other solutions. Today, we know that there are an infinite number of solutions to the so-called ‘tiling problem’.  

So what? Who cares? Euclid was a genius with the best of intentions and his ideas no doubt did a lot of good; but he also took Western metaphysics down a rabbit hole from which it is still struggling to emerge. For two millennia we assumed that we lived in a ‘flatland’ made up entirely of straight lines; we don’t. On the contrary, everything curves! 

You’re 14 years old and you’ve just enrolled in your first high school physics class. Day one, your teacher (you go to a progressive school) takes the class to a billiards parlor (pool hall to you) so that you can observe first hand ‘cause and effect’ in action. Balls collide, momentum is transferred, angles of incidence determine angles of reflection, etc. It’s all very neat and clean…and linear. 

The real world is rarely neat and clean and it’s never linear. In fact, it is massively non-linear and recursive. Everything affects everything else. Worse, everything that ‘acts’ acts on itself at the same time. Think karma. Whatever you’re doing to another, you’re doing the same to yourself. “Do unto others as you would do unto yourself.”  Ultimately, the foundation of all ethics will turn out to be self-interest…once we understand what those interests are. This gives the meme, ‘whoever lives by the sword dies by the sword’, a whole new meaning.    

Unfortunately, Euclidean linearity has infected every corner of our thought process. We are determined to describe the world in terms of discrete entities (subjects) acting on (verbs) other discrete entities (objects); we are committed to viewing all ‘transmissions of influence’ as straight lines. Welcome to the modern Indo-European language family. As a result, what we know, or at least what we knew before 1800, conforms to the straight-edged geometry of a Euclidean cube. 

Let’s suppose we know everything we can know, not everything there is to know, but everything we can know. Suppose we model everything there is to know as a sphere, dense with information; everything we can know would be a subset of that sphere.  

But because of our Euclidean blinders, whatever we do (or can) know must conform to the geometry of a straight edged cube. Suppose you say, “We already know everything we can know…or close to it.” Ok, I’ll spot you that. As a Harvard professor (c. 1896) told graduating seniors re physics, we know virtually everything that it is possible to know. Whatever we don’t know isn’t worth knowing. 

According to this model, the cube of what we can (and do) know must fit inside the sphere of what is knowable; it’s a subset after all. Obviously, we want as much of the volume of the sphere as possible to lie inside the inscribed cube. But we are ultimately limited by the fact that our sphere is round while our cube must have straight edges. 

So, the 8 vertices of the cube, and only those 8 points, will lie on the surface of the sphere. Still as long as the volume of what we can know is close to the volume of the sphere of what’s knowable, we’re ok for most purposes, right? I mean, nobody likes a know-it-all. A little bit of uncertainty (aka ignorance) keeps life interesting, don’t you think? 

I’m fine with this as long as the cubic volume of knowable information includes most of the spherical volume of all information. So how close is it? Is it a 90% match? Surely it can’t be less than 80%, right?  

Would you believe it’s slightly less than 37%? By confining our thinking to phenomena that can be modeled via Euclidean geometry, we limit what we can know to less than 3/8ths of what is available to be known. A dismal performance! 

On the other hand, putting the same problem in a non-Euclidean context allows the edges of the cube to bow and extend so that they template the inner surface of the sphere. By analogy, our insistence on treating the non-linear universe as if it were an assemblage of linear forms leaves us 63% blind.  

2450 years after Plato we still don’t have a clue…and we won’t, until we free our thought process from its Euclidean play pen and stop trying to fill round holes with square pegs.  

Image: M. C. Escher. Relativity. 1953. Lithograph print. 27.7 cm × 29.2 cm.