Upper divsion undergrad course in geometry?

[MathWarrior]

I was just wondering they have a course at my university and I am trying to figure out what exactly it covers. Most people learn basic geometry in high-school so I was surprised to find a course for mathematics majors that is an upper division course in geometry at the college level.

The course description is as follows:
Congruence, area, parallelism, similarity and volume, and Euclidean and non-Euclidean geometry.

Which is rather vague. Has anyone here taken such a course and if so what type of things do you learn? Does it eventually branch into topology?

 

[Klungo]

A wikipedia search of euclidean and non-euclidean geometry roughly reveals the entire course. Think of it as planar and none planar geometry where say, on the surface of a sphere, a triangle's total angles don't add up to 180 as they do in planar.

No. The the geometry course doesn't branch into topology. They are taught separately at least in my university. (Geometry has prereq of calc 3, topology typically requires proof based courses like real analysis or linear algebra at my university.)

Edit: Topics may include: fractals, polygons, etc.

 

[micromass]

The course looks very similar to a high school course in geometry. The only difference is likely that this college course is much more rigorous (and hence much more interesting). A very good book that treats the topics in your course is Hartshorne: https://www.amazon.com/dp/1441931457/?tag=pfamazon01-20

 

[MarneMath]

It might be a course designed for math education major to help polish up and give more insight into geometry for future math teachers. At least, that is the intent of the course at my university.

 

[HallsofIvy]

If this is a course at your university then ask the mathematics department or, if you know who, the person who is teaching the course.

 

[homeomorphic]

I took a class like that. There's some overlap with high school geometry, but the class I took studied the foundations more. In Euclidean geometry, there are certain postulates, one of which is the parallel postulate, which says that there is a unique parallel line through a point not on a given line. For a long time, mathematicians thought that it was the only possible geometry, but after a lot of people tried to prove it and failed, a few people, like Gauss realized that other geometries were possible. Today, with 20/20 hindsight, it seems almost silly that people thought that the parallel postulate could be deduced from the other axioms of geometry. The reason why it seems silly is, in part, because there are simple examples of geometries where the parallel postulate is false (models).

It doesn't branch into topology. More like differential geometry, to some extent. But you probably won't see any in the class. There are models of hyperbolic geometry that are set up using differential geometry ideas. One way to think of hyperbolic geometry, at least locally, is that it is like you are doing geometry on a certain surface of constant curvature, called a pseudo-sphere. Another differential geometry idea that enters the picture is the idea of a metric, which is just a way of measuring distances and angles. Another realization of hyperbolic plane geometry is just a disk, where you measure distances and angles differently from usual.