# Geometry [in college?]

As I was trudging through calculus homework, I had a thought about geometry. Do colleges offer any courses in geometry? if so, what level of math is that considered?
if they don’t, then why isn’t geometry offered as a college course?
It’s odd to see that geometry is pretty wide with euclidean geometry, non-euclidean geometry, and topology along with other stuff. this has puzzled me for a while.

Sure, of course they do. I think my university offers, for example, differential geometry at the 400 level. Topology is a 2-part (six-credit) sequence available at the 400 or 500 levels. Geometric concepts are also pervasive through undergrad calculus.

Of course there are many geometry courses, and they’re on many different levels. Looking at my own school’s schedule, this semester there’s M329: Modern Geometry, M512: Geometric and Algebraic Topology, and M527: Geometry for Teachers, and next semester there’s M420: Geom, Measure, and Data Mid Gr [not sure what that stands for], plus plenty of courses that would probably include some geometry, like M135: Math for K-8 Teachers, and M242: Methods of Proof.

And of course when you get into the really advanced classes, the boundaries between geometry, algebra, calculus, and whatever else all start to get blurry. A matrix is a bunch of numbers, but the set of matrices with certain relationships between the numbers will map out an abstract space with certain properties: Is that algebra or geometry?

The math courses offered in high school are those that are simple enough and need little enough background in other math courses that they can be understood at that level. Thus a year or two of algebra, plane geometry, trigonometry, and first-year calculus are basic enough that a high school student can handle them. (Although calculus has become standard only in the past fifty years or so.) I think that it would be useful to have a course with some simple combinatorics and basic probability and statistics in high school. Other math courses are harder and need more mathematical background, so they aren’t offered until college.

Not only that, but, of course, much (indeed, I daresay most) of the interest in studying matrices is because they can be seen as representing maps between certain kinds of spaces preserving certain kinds of spatial properties. So, yeah, is that algebra or geometry?

Though you don’t have to wait until the advanced classes for the concepts to blur. You just have to notice the blurring of the concepts at whatever point you like. Even the now very old, very simple, very pervasive idea of using coordinates to describe points in space (i.e., analytic geometry) is all about bringing the methods of algebra to bear on geometry, demonstrating that there is no clear separation between the subjects.

As for what “level of math” geometry is at: it doesn’t really work like that; any kind of abstract investigation of spatial ideas is, in some sense, geometry, whether very simple or very complex; the parts which are very simple and which you can jump right into are at what you might call introductory-level, and the parts which are more sophisticated are what you might consider to be at a more advanced level, but geometry as a whole isn’t on any particular level, any more than, say, chemistry as a whole could be classified as at some particular level of science, or even organized into any kind of natural linear sequence of progression, or even cleanly isolated from all other areas of study. It’s a big wide field with lots of ideas within it, rubbing shoulders with lots of other ideas connecting out to lots of other fields.

All this is true, but it’s unusual for geometry courses to be taught between and high school and advanced undergrad/beginning grad because you need a pretty strong background in linear and abstract algebra to do much more than what you’ve covered in high school.

Just for general interest, here’s the course website for a more advanced general geometry class aimed at computer scientists.

Yes, but it’s not the “a rhombus has four sides, pull out your protractor and compass” geometry that you have in high school.

That’s the geometry that one typically has in elementary school. My high school geometry course was an entire year of inductive proofs that angle ABC is the same as angle XYZ.

Edited title to better indicate subject.

Colibri
General Questions Moderator

I hope you mean “deductive proofs”?

Yeah, probably. I can never remember which is which. Like rhombuses and trapezoids.

If you mean HS Euclidean Geometry - there usually is a course for future math teachers - but you don’t usually get ‘math’ credit for it.

Of course there’s college Geometry courses, same as there’s college Algebra courses and even remedial math courses (as needed). However, some of these more basic math courses may not satisfy mathematical requirements, if needed for a given major.

The IB standard level math class has probability and statistics, but it doesn’t offer much in the way of combinatorics. I’m pretty sure the higher level one does, though.

I don’t think most HS students make it calculus before graduation, in the first place. Of the ones that do, I don’t think they can do it without the second year of HS algebra–which, of course, they can’t do without that first year of algebra. So isn’t it fair to say that most HS math courses build on one another? OTOH, HS geometry does seem to be an animal apart from the other typical classes, and one can do well in it even without having done exceedingly well at first year algebra, which usually comes before geometry.

In college math courses, would it not be fair to say that calculus, in its applications, replaced much of what used to be done by advanced geometry, e.g. Archimedes?

That’s about what my HS geometry class was. But we certainly used compasses; some of the basic theorems are that you can construct certain figures, or bisect angles, using only a straightedge and compass.

We had a great teacher who laced his lectures with a patter of jokes and gentle ribbing. In demonstrating a construction at the board, he would often say, “So now you take out your little compass…”, as he got out his great big chalkboard compass and brought the fulcrum end emphatically up against the board.

ES geometry was just a chapter in most of the arithmetic books, which explained about points, lines, planes, and solids. There were no proofs, nor even any mention of postulates or theorems to be accepted at face value. It seemed to have nothing whatsoever to do with the rest of mathematics.

I don’t know if it is taught any more, but when I was a student (this goes back more than 50 years) I took a course in Euclidean geometry. The basic text was Artin’s Geometric Algebra. Some of the topics include Hilbert’s 22 axioms (or whatever the number was–Euclid missed all the axioms involving betweenness and his “principle of superposition” badly begs the question) and a completeness axiom. We then did things like construct the real numbers. We also studied the relation between Pappus’s theorem and associtiavity, Desargue’s theorem and commutativity. Quite a fine course, all in all.

I took a course in transformation geometry in college. An upper division course. It was actually quite fun.

As I am currently teaching a Geometry course at a college, the answer is yes.

And it’s a real math course! The basic idea is that it covers Euclidean, Hyperbolic and Projective Geometry, all from an axiomatic point of view. We show that Hyperbolic Geometry is relatively consistent with Euclidean by constructing a model of HG in EG, for example.

As for the one I took in college (way back in 2001), it covered Euclidean, Spherical, Projective, and Hyperbolic, but these were from a completely algebraic point of view. Thus the very first definition was SO3®, and we proved that there were only 5 platonic solids within a month (I think). Then since the sphere is embedded in R3, tw’aint no thing to’ve look’d at the surface of a sphere. Then all those theorems about Spherical Geometry were translated to Projective Geometry once we identified antipodal points. I remember thinking this was pretty slick.