Column: Do you believe in rectangles?

In my previous installment of this column, I wrote about a fundamental difference between mathematics and physics: The former is a deductive endeavor in which the conclusions follow with certainty from the premises, and the latter is grounded in observations, measurements, and inductive reasoning.

Miller’s Diary

Physics Today editor Johanna Miller reflects on the latest Search & Discovery section of the magazine, the editorial process, and life in general.

You could imagine a world where that distinction didn’t exist—where math was just as much an experimental science as physics is. To find the formula for the area of a circle, for example, geometers (“Earth measurers”) would draw a lot of circles, estimate their areas, and look for a pattern. It would never occur to anyone that the pattern should be verified with a proof; the observations would speak for themselves.

That we don’t live in that world has a lot to do with the work of Euclid of Alexandria. Euclid probably didn’t invent the idea of a mathematical proof, but he sure helped it catch on: His magnum opus, the Elements , written around 300 BCE, is arguably the most influential textbook of all time. For more than 2000 years, students learned geometry exactly the way Euclid taught it—as a series of theorems deduced from a modest set of five postulates and five “common notions.”

By today’s standards of mathematical rigor, Euclid leaves a lot to be desired. His proofs are full of implicit assumptions—for example, that for any two points on a line, there are more points between them—that he should have included in his list of postulates but didn’t. To his great credit, though, he did recognize the need to include his infamous parallel postulate.

As learned by today’s geometry students, the parallel postulate states that for any line ℓ and any point P not on ℓ, there is exactly one line through P that’s parallel to ℓ. Euclid’s original version was a bit clunkier, but the statements are logically equivalent.

The parallel postulate bothered a lot of people for a long time. (Those people probably included Euclid himself—tellingly, he put off invoking it in the Elements for as long as he could.) The problem, I suspect, is that it’s just the right mix of obvious and nonobvious, and it can never be empirically confirmed. The Euclidean plane is infinitely large, and some of the lines through P intersect ℓ an incomprehensible distance away. To convince yourself that one and only one of them never intersects ℓ at all, you need to make a conceptual leap from the part of the plane you can visualize to the part that you can’t. And yet, it seemed like the postulate just had to be true; for 2000 years, no one was willing to seriously entertain the possibility that it wasn’t.

If it had to be true but wasn’t self-evident enough to assume, then could it perhaps be a theorem that followed from the other postulates? A proof would settle the matter, and for 2000 years mathematicians tried to find one. They weren’t lightweights, either—some of the brightest mathematical minds of the ages were positively obsessed with proving the parallel postulate. They all failed. Every one of their proofs contained some unjustified assumption.

On the bright side, those assumptions make up a convenient list of statements that are logically equivalent to the parallel postulate. Many of them are familiar to the geometry students of today, such as: Parallel lines are everywhere equidistant.

Or: The angles of a triangle add up to 180°.

Or: Rectangles exist.

Now, if you haven’t heard this story before, you might think it’s getting a bit ridiculous. Isn’t it obvious that rectangles exist? They’re everywhere, after all. Unlike lines that may or may not intersect millions of miles away, rectangles can be drawn, and seen, and studied, right under our noses. Any material rectangle is at best an approximation of the Platonic ideal rectangle—a figure with four perfectly right angles—but surely that’s just because humans are imperfect at drawing rectangles, not because the Platonic form doesn’t exist, right?

If you have heard this story before, then you know what happens next: Over the course of the 19th century, it came to light that the parallel postulate—and the existence of rectangles—could never be proved because there are alternative geometries, just as self-consistent as Euclidean geometry, in which it’s false. There’s hyperbolic geometry, in which there are infinitely many lines (or as mathematicians sometimes put it, “at least two”) through P that are parallel to ℓ. And there’s elliptic geometry, which contains no parallel lines at all.

Elliptic geometry is a bit of a headache to make rigorous, but it’s the easier of the two to visualize: It’s just geometry on the surface of a sphere . Hyperbolic geometry, although beautiful , can be more difficult to intuitively grasp, but you can think of it as the opposite of elliptic geometry. In the elliptic plane, lines get closer together as you extend them; in the hyperbolic plane, they get farther apart. In elliptic geometry, the sum of the angles of a triangle is more than 180°; in hyperbolic geometry, it’s less. In neither geometry do rectangles exist, although in elliptic geometry there are triangles with three right angles, and in hyperbolic geometry there are pentagons with five right angles (and hexagons with six, and so on).

It’s hard to overstate what a paradigm shift the new geometries represented. Euclid’s geometry may not have been motivated by its practical utility—the Elements doesn’t even discuss applications—but it was always meant to be about shapes that exist in the real world. But now there was not just one geometry but three, and they contradicted one another. At most one of them could be the true geometry of physical space. But the other two, whichever ones they were, were just as internally valid and just as “real” in the realm of ideas.

The structure of geometry, therefore, was neither logically inevitable nor constrained by anything in the physical world—it all depended on the assumptions you chose to start with. Mathematicians were free to dream up whatever abstractions they liked, limited only by their imaginations. But with that freedom came responsibility. Because they could no longer fall back on physical intuition, as Euclid had, to fill in the gaps in their lists of axioms, mathematicians needed to take much greater care in declaring their assumptions. As mathematics grew more inventive, it also became more rigorous.

At the same time, the new geometries raised the idea that physical space might not be Euclidean after all. For shapes on a human scale, Euclidean geometry is at least an excellent approximation, but on astrophysical or cosmic scales, space might be curved, and that curvature might mean something. The theory of general relativity—which Albert Einstein said he could never have developed if he hadn’t known about non-Euclidean geometry—holds that spacetime is locally curved in the vicinity of matter or energy. It also allows for the possibility that the universe as a whole might be flat (Euclidean), positively curved (elliptic), or negatively curved (hyperbolic).

So far, it looks like the cosmos is as close to flat as can be measured. But the nature of continuous measurements is that they’re always inexact, and the nature of Euclidean geometry is that it postulates an exactness—there’s exactly one line through P parallel to ℓ, the angles of a triangle add up to exactly 180°—that’s beyond what can ever be observed. Maybe someday cosmological measurements will be good enough to reveal that the universe isn’t Euclidean, but they’ll never be able to conclusively show that it is.

The parallel postulate and the existence of rectangles, which seemed so obvious to so many for so long, will remain forever unverified.