If you asked someone to say what mathematicians do, there are, I think, three answers you’d get. One would be “they write out lots of decimal places”. That’s fair enough; that’s what numerical mathematics is about. One would be “they write out complicated problems in calculus”. That’s also fair enough; say “analysis” instead of “calculus” and you’re not far off. The other answer I’d expect is “they draw really complicated shapes”. And that’s geometry. All fair enough; this is stuff real mathematicians do.
Geometry has always been with us. You may hear jokes about never using algebra or calculus or such in real life. You never hear that about geometry, though. The study of shapes and how they fill space is so obviously useful that you sound like a fool saying you never use it. That would be like claiming you never use floors.
There are different kinds of geometry, though. The geometry we learn in school first is usually plane geometry, that is, how shapes on a two-dimensional surface like a sheet of paper or a computer screen work. Here we see squares and triangles and trapezoids and theorems with names like “side-angle-side congruence”. The geometry we learn as infants, and perhaps again in high school, is solid geometry, how shapes in three-dimensional spaces work. Here we see spheres and cubes and cones and something called “ellipsoids”. And there’s spherical geometry, the way shapes on the surface of a sphere work. This gives us great circle routes and loxodromes and tales of land surveyors trying to work out what Vermont’s northern border should be.
For example, we can describe a sphere this way. We start out with a three-dimensional space. Give me a center point, that I’ll name C. Give me a distance, that I’ll name r. I give to you the sphere, centered on C, with radius r, which I say is all the points that are exactly a distance r from the center point C. You probably won’t find that controversial.
And we can describe a circle this way. We start out with a two-dimensional space. Give me a center point, and I’ll name that C again, trusting that we aren’t confusing it with the earlier C. And give me a distance that I’ll again name r. I give to you the circle, centered on C, with radius r. That’s, yes, all the points that are exactly a distance r from the center point C.
But there’s no reason to stop with plane and solid and sphere geometry. Suppose we’ve got fair ideas of what we mean by angles and distances and points and such. Then we can start thinking about things like, what if we had a space that looked like our everyday three-dimensional space, only it had four dimensions of space? Or five? Or a hundred? Or infinitely many? Right away it gets harder to draw pictures of what we’re doing, but that’s all right. We can visualize these complicated shapes in pieces. Or we can work out descriptions of the shapes we talk about and at least understand the descriptions, even if we can’t draw the shapes.
After all, if you give me a center point named C, and a distance named r, I can describe some shape that’s “all the points exactly a distance r from the center point C”. I imagine you have an idea what I mean even if I don’t say whether this is a two-dimensional or a three-dimensional surface. (We’re probably also all right even if we’re talking about the surface of a sphere, at least if the distance isn’t too big.) And this is what gives us the hypersphere. If we’re working out the geometry of four or five or more dimensions of space we can talk about a shape defined by “all the points exactly a distance r from the center point C”.
So we use the word “hypersphere” to talk about the shape that’s in four or five or however many dimensions of space, but that has about the same description that a sphere in three dimensions of space has. We might also use “hypersphere” to talk about a regular old three-dimensional sphere, or a two-dimensional circle. We do this if we don’t care, or don’t want to pin down, how many dimensions of space we’re working in. It may sound casual to not care how many dimensions of space we’re working out some geometry problem in. But sometimes it doesn’t matter. And sometimes if we can work out a geometry problem without having to know whether it’s in two- or three- or 2,038-dimensional space, then we know the answer for every possible space. It’s usually great if we can get away with proving something true that generally.