I really enjoy your articles, and your recent one on relativity and noneuclidean geometry was no exception. Since I actually know a little something about this topic (unlike most of the topics you write about), I thought I'd offer up a few comments of my own. I hope you find them interesting and/or edifying.
In the article you say: "Euclidean geometry is famously based on five axioms, but there's an unspoken axiom of uniformity, which assumes that the universe is geometrically the same at every location and at all scales." And you go on to observe that a spherical manifold is nonuniform in a way that violates this assumption. In fact, the assumption in Euclidean geometry is not uniformity, but flatness; the two are not the same, and your example of a sphere is a good example of why. A spherical manifold is indeed uniform; by definition it has uniform, positive curvature, and it is this property that causes all of the other effects you mention, including the absence of parallel lines. Spherical geometry does not seem not too weird to us because it can be embedded in a higher dimensional flat space. The surface of our planet is a good example; we have to take account of curvature in planning things like great-circle air routes, but because we can look at the whole thing as being embedded in Euclidean 3-space, everything fits into our intuitive concept of geometry nicely.
One can also do geometry on a manifold with constant negative curvature. As you observe in your article, a saddle shape is an example of negative curvature in two dimensions, but it is not *constant* negative curvature (it took me a long time to realize this because the point is typically muddled in nonmathematical descriptions of the subject). You won't find a drawing of the constant negative curvature 2-D manifold anywhere because it can't be embedded in Euclidean 3-space the way the sphere can. This makes it seem a little weirder than the sphere because we can't visualize what is going on by stepping out into the 3-D (flat) space we are comfortable with, but really it's not fundamentally different from spherical geometry.
[SCDB note: One way to demonstrate the difference between "flat", "positive curvature" and "negative curvature" is to describe its effect on a circle. If a circle is drawn on a flat plane, the ratio of its circumference to its diameter is exactly pi. If it is drawn on a surface with positive curvature, that ratio will be less than pi, as a function of how curved it is (and other factors). In spherical geometry, the diameter can exceed the circumference, making the ratio less than 1. That's because the area in the middle of the circle bulges out, and the diameter follows a path from one side to the other which is longer than a Euclidean straight line would be. On the other hand, that ratio for a circle on a surface with negative curvature is greater than pi. As Robert points out, that's harder to explain.]
Now, in the case of the hyperbolic (i.e. uniform, negative curvature) we are deprived of our crutch of using higher dimensional flat spaces to describe the curvature of a manifold. That turns out to be ok because it turns out that curvature can be described entirely in terms of properties measurable within the manifold. That means that you don't need to make reference to the higher dimension that the manifold is "curved into"; in fact, the higher dimension need not exist at all. In other words, to the question "What does space curve into?", a legitimate answer is "None at all; it's just plain curved." Note that this applies even to geometries that could be embedded in a higher dimensional space. Just because a 2-sphere could exist in Euclidean 3-space doesn't mean that it *requires* Euclidean 3-space to exist.
[SCDB note: I was mistaken about that.]
The manifold that general relativity operates in is not any of these; it is something called a Riemann manifold, and the version used in GR really is a truly 3-dimensional spacetime, not just a 3-space with a time coordinate tacked on. However, a 4-D spacetime is also different from a 4-D space, but it's difficult to explain why without throwing some equations into the mix. The geometrical properties of a manifold can be encapsulated in a tensor called the "metric", which tells you how to compute the distance between two infinitessimally separated points. Integrate along the metric, and you can calculate the distance or "interval" between two points of finite separation.
[SCDB: Tensor calculus. Moan...]
For special relativity the metric is pretty simple; it's: ds² = -dt² + dx² + dy² + dz². It's that minus in front of the dt² that makes all of the difference between a "space" and a "spacetime". Note that the metric of SR (the "Minkowski metric") is flat; the spacetime of SR has no curvature.
Einstein's stroke of genius was to postulate two things. First, that freely falling objects move along geodesics in spacetime. You can calculate a geodesic from the metric (although I confess I'd have to dig out my old notes to recall how), and if you do it for the Minkowski metric of SR you find that you get back Newton's first law; i.e. dx/dt, dy/dt, and dz/dt are all consta
