What is a Geometry?

This might sound like an odd question — most students seem to think so when I ask it.

The reason it sounds a little strange is that most students — even when I taught at a magnet STEM high school — think there’s just one type of geometry:  Euclidean geometry.  This isn’t surprising given the typical K-12 mathematics curriculum, and it’s not necessarily problematic, either.  But by the time students graduate from high school, they really should be aware that Euclidean geometry is just one among many different possible geometries.

Why should students be aware of other geometries?  Most will never need to use any geometry other than Euclidean in their day-to-day lives.  In my opinion, one compelling reason is that  you can get at a really big idea (the concept of a geometry) with very little effort, really.

Let’s look at a simple example for a moment:  spherical geometry.  Let’s define Point and Line in this geometrical framework, where I use capital letters to distinguish from the ideas of points and lines in our usual Euclidean geometry.

Imagine confining our attention to the surface of a sphere, and say that a “Point” is a pair of opposite points on a sphere (like the North pole and South pole), and that a “Line” is a great circle on the sphere, like the Equator or a line of longitude.  (Note how we use the term “line of longitude,” even though this line is in fact a circle!)

With these definitions, we offer our first Thereom of spherical geometry:  Any two Lines intersect in exactly one Point.  Now if you’ve got a globe in your room, this is easy to demonstrate.  Any two Lines of longitude intersect in one Point (consisting of the North and South poles), and it is also easy to see that the Equator intersects any Line of longitude in exactly one Point.

But, you might object, you just defined Points and Lines in some arbitrary way!  How can you just do that?  The simple answer is that this is precisely how you define a geometry.  The concepts of “point” and “line” depend on the geometrical context.  And since we’re looking at the surface of a sphere, a “line” just can’t mean the same thing as it does in Euclidean geometry.

OK, you might respond, but what about your definition of “Point”?  Surely that’s a bit off!  How can a “Point” be two “points”?  And why are your Lines only great circles?  What about other circles on a sphere, like lines of latitude?  Why don’t you include these as well?

This is a valid objection, and there is nothing preventing it.  But when you try to actually study the geometry defined in this way, it’s hard to make progress.  For example, two lines defined in this way could intersect in 0, 1, or 2 points.  And given any two points, there would be infinitely many lines passing through these points.

But with the definition of Point and Line as given earlier, we have two very nice results:  Any two Lines intersect in exactly one Point, and there is exactly one Line passing through any two distinct Points.  No ambiguity here — 0, 1, 2, or infinity.  All we need is 1.

So while in a technical sense, you can define basic ideas like “Point” and “Line” any way you like, in a practical, mathematical sense, your definitions should be useful.

Yes, useful.  We usually don’t think of definitions in this way, but mathematically, it really is necessary.  With your definitions and axioms, you want to prove interesting theorems.  If your definition of lines allows two distinct lines to intersect in any number of points, well, you haven’t really said very much.  Your definitions don’t get you anywhwere.

So what makes a definition useful?  Though perhaps stating the obvious, a definition is useful if other people actually use it.  Do other mathematicians think your construct allows for interesting results?  Can it be applied in other ways as an aid to learning about other areas of mathematics?  Are others interested enough in your mathematical system that they want to see what results they can come up with?

In a real sense, anyone can write down a few definitions and maybe state a simple theorem or two.  But you’ve got to convince others to care about what you’ve created.

So what has this got to do with Points and Lines?  These definitions are just the beginning of spherical geometry, which leads to spherical trigonometry and other areas as well.

And why do we care?  Among applications in studying spherical triangles and polyhedra, we think of spherical geometry when we ask a practical question such as, “What is the shortest flight path between two cities?”  It turns out the that shortest flight path is along a Line on the sphere, just as the shortest distance between two points is along a line in the Euclidean plane.

If you’ve been paying attention, you will also have noticed one important consequence of the theorem that any two Lines intersect in exactly one Point:  there are no parallel Lines on a sphere.  Simply put, any two lines intersect.

This means that spherical geometry is in some fundamental way different from Euclidean geometry, where it is possible for two lines to avoid intersecting.  In other words, we’ve created an entire world of Points and Lines distinct from the one we are familiar with, together with many interesting properties — some of which are just like those in the Euclidean world, while others are quite different.  We’ve created a new geometry.

What I’ll be doing in the posts in this series is gradually introducing you to different non-Euclidean geometries.  In this way, I hope you’ll develop an intuition for what a geometry is.  How concepts like point, line, angle, and circle seem fairly natural in various contexts, even though they may not be exactly like what you’re familiar with in Euclidean geometry.  But why, nonetheless, we still call these systems geometries.

Now the question has been thoroughly asked:  What is a geometry?   Hopefully, by the end of the series, you’ll be able to formulate your own answer to this new question — one you may never have asked yourself before.  But one question that is well worth answering.