Final Exam Info

The final exam for this course will take place at 1:30 PM on Saturday, May 10, in our regular classroom. Like the first two tests, it will count for 17% of the total grade for the course. It will be of the same general format as the two previous tests, but it will probably be five pages long instead of four.

You should expect one or two general essay questions about mathematics or the way people do mathematics. It will be useful to remember some of the themes and ideas from earlier in the course while you are answering those questions, and you might want to draw some examples from that material. Other than that, the exam will cover only material from the third part of the course, since the second test.

The major reading in the final segment of the course was *The Poincaré
Conjecture*, Chapters 1 through 8. I also talked in class about what
goes on in the rest of this book, although I didn't require you to read it.

You should be familiar with the historical sections of this book. You
should at least understand the roles of Pythagorus, Euclid, the non-Euclideans
(Gauss, Lobachevsky, and Bolyai), Riemann, Poincaré, and Perelman.
You should have some idea of what is in Euclid's *Elements*.
You should know about the fifth postulate (or parallel postulate) and
attempts to prove it, and how those attempts failed. You should know
what is meant by non-Euclidean geometry, and you should be familiar with
the two types of non-Euclidean geometry: spherical and hyperbolic.
You should know how this relates to the sum of the angles in a triangle
and the area of a circle. You should have some understanding of Riemann's
ideas about geometry,
including the notions of geodesic and curvature (and you should know, for
example, that a cylinder is not curved in this sense). You should know
what the Poincaré conjecture says, and you should know that there
was a long history leading up to Perelman's solution.

You should know the difference between geometry and topology. You should be familiar with manifolds, atlases, and the classification of surfaces. In addition to surfaces, you should know about one-dimensional and three-dimensional manifolds, and you should understand the possibility of even higher dimensions. You should know what it means for a surface to be orientable, finite, and with no boundary. You should know about Poincaré's Disk Model for Hyperbolic Geometry.

And you should know what all this has to do with the shape of the earth and the shape of the universe.

We covered several things in addition to the reading. We encountered the
movie *Flatland* and the idea of two-dimensional beings as well as the
possibility of higher dimensions. This led to the idea of imagining what
a manifold looks like to someone living inside of it -- that is, looking at
the intrinsic properties of a manifold.

We looked at a few non-orientable surfaces that are not mentioned in the book: the Moebius strip and the Klein bottle. You should know something about their properties. We did some "cut-and-paste geometry" on these and other surfaces. There might be a relatively simple exercise of that sort on the exam.

We looked at symmetries of patterns in the plane, including translation,
rotation, reflection, and glide reflection. We looked at the different
types of symmetry that a pattern can display and how those symmetries
form one of only a few types of groups: the rotation groups, the dihedral
groups, the seven frieze groups, and the seventeen wallpaper groups.
You should know how to recognize rotational and dihedral symmetry in a
pattern. You are **not** responsible for memorizing the
frieze and wallpaper groups; however, I might ask you to use the classification
table from the handout "About Symmetries of the Plane"
to classify a wallpaper pattern, or I might give you the description of the
frieze groups from that handout and ask you to use them to classify a
frieze pattern.