Interlude about geometry

Before I try to explain what’s up with three-dimensional manifolds, I need to talk a bit about geometry, specifically spherical vs. Euclidean vs. hyperbolic geometry.

Euclidean geometry is “high-school” geometry, the geometry of the plane put into axiomatic form by Euclid of Alexandria. By which I mean, Euclid replaced the study of actual physical points and lines with the study of formal mathematical objects called “points” and “lines”. Euclid’s “points” and “lines” were inspired by real-world points and lines, and have properties based on the properties of real-world points and lines. For example, there’s a single, unique “line” through any two distinct “points”. But Euclid’s “points” and “lines” are just a formalism. They have the same relationship to real-world points and lines that poems have to roses.

And like poems, Euclid’s formalism permits analogies.

What if, instead of a plane, we consider the surface of the sphere? And instead of straight lines in the plane, what if we define a “line” to be a great circle on the surface of the sphere, such as the equator? Then we get spherical geometry instead of Euclidean geometry. The “lines” in spherical geometry don’t look like the “lines” in Euclidean geometry, but they act like lines in lots of important ways: they intersect in points, they subtend angles with one another, their segments have well-defined lengths, and so on. The formalisms of Euclid’s geometry are just a language, and by applying that same language to spherical geometry we can highlight the similarities between the two.

At the same time, we discover the differences. Triangles, for example: in Euclidean geometry, the angles of a triangle always add up to 180 degrees, but in spherical geometry they always add up to more than 180 degrees. Spherical geometry is finite: no shape can have an area greater than the area of the entire sphere. There’s no such thing as similar triangles in spherical geometry. Also, most famously (if that’s the word) there’s no such thing as parallel lines.

Hyperbolic geometry, like spherical geometry, uses the language of “points” and “lines” to describe a completely different space from the Euclidean geometry. But hyperbolic geometry is much stranger and weirder than spherical geometry. It’s hard to visualize: it doesn’t live on any surface in Euclidean space the way that spherical geometry lives on the sphere. And it turns the properties of spherical geometry on their heads. The angles of a triangle in hyperbolic geometry always add up to less than 180 degrees. Hyperbolic geometry is infinite…but bizarrely, no triangle can have area greater than pi. (Circles, on the other hand, grow as large as they want, exponentially with respect to their radius.) There are no similar triangles in hyperbolic geometry, but there area infinitely many lines “parallel” to any given line. And so on.

It’s just bizarre. It’s beautiful and it’s frustrating and it’s bizarre, and I’ve been studying it for so long that I’m more comfortable with it than I am with Euclidean geometry most of the time. I live in a Euclidean world, but I tend to think in a hyperbolic one.

Next time I’ll try to explain what this has to do with manifolds.

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