1998 Topology class

TAG SEP 1998: Topology

Read Instructions Below First!

There are a number of good Web sites given below. It is a good idea, although not absolutely required, to take them in the order that they are given. For each one, read any text that may go with the link first before going down the link. You may wander far from my site as long as you stick to math, science and certain mathematical art; do not go outside these bounds.

We have not discussed in class every technical topology term you will see in these pages. I want you to see these pages mostly for their pictures. Please feel free to poke around on the Web trying to find definitions for words that you do not understand. You may also ask me while we are still in the lab, or when we are back in class.

If you have any other questions, ask me. The "Go" menu on the top toolbar will let you jump directly back to most places that you have already been, unless you have been too many places since then.

I will be asking you where you went, so it might be a good idea to take some notes.

Read Instructions Above First!

Knots and Tori and the Möbius strip

Take a (straight, unknotted) piece of string. Loop it around any way you want, then tie the two ends together. Mathematicians call this (big surprise!) a knot. From the inside, any knot, such as the trefoil knot, is just like a circle: you travel around, and after a while you come back to your starting place. However, you cannot turn a trefoil knot into a circle just by continuously deforming it in three-dimensional space. A topologist would say that the trefoil knot (or any other knot) is homeomorphic to S1 (the circle), but is embedded differently into R3 (ordinary 3-dimensional space).

A University of Oregon professor (in fact, one of my graduate advisors when I was in graduate school getting my Master's degree) has many pictures of knots and the Möbius strip, plus some other stuff.

For a good starter definition of what I mean when I say 'embedded,' read this definition of the topological term manifold.

A Small Virtual Gallery of Mathematical Art

Some students who have had classes from me before may remember a videotape about the hypercube. It was made by a professor named Thomas Banchoff, who also put together this collection of mathematical pictures. If you wander virtually around in the gallery enough, you will come across a great picture trying to show the Klein bottle in three dimensions. This cannot really be done, so any pictures you ever see of the Klein bottle are really projections or immersions, but this picture gives a good feel for the shape.

Prof. Banchoff was even kind enough to answer my E-mail when I wrote to him in 1995, telling him that I had shown the hypercube video to my Geometry II class, and asking about another video he made concerning the hypersphere. He replied:

	I don't think it will be as interesting to the geometry students
	as is the earlier film, but it does have some good images. Most
	of them are described in my book "Beyond the Third Dimension"
	in the Scientific American Library.
and on the strength of this I decided not to get the hypersphere video, so I have never seen it. I did read the U of O's copy of the book he mentioned, though, and I recommend reading it if you get a chance.

Whoa! This is starting to hurt my brain!

Some of the shapes that are common in topology, such as the Klein bottle, cannot be fully assembled in only three dimensions. They require four (or more!) spatial dimensions. If you want to help improve your ability to visualize more dimensions, there is a program you can play around with for rotating polytopes. To get to it, go here, then scroll down until you see "FourD" and read the instructions.

What's a polytope, you say? Well, just like shapes in two dimensions with straight edges are called polygons, and shapes in 3-D with flat faces are called polyhedra, in geometry we call shapes in 4 dimensions with flat hyperfaces (3-D faces on a 4-D object) polytopes. (This name comes from the same root, I believe, as the word "topology.")

More great weird pictures, including several of the Klein bottle

Tamara Munzner has written about various manifolds, illustrated with many accompanying pictures. I personally really like #12. However, if you want to have some context for these images, start at the Table of Contents for her paper.

The 3-D Torus and the 3-D Klein bottle

The torus and the Klein bottle that I discussed in class are surfaces. That is, they are two-dimensional. They can both be formed by sewing up the edges of a square in a particular fashion. What if we started with a cube and started gluing faces together? What would we get?

If we glue each of the six faces of the cube to its opposite, we get the 3-D torus (often called the 3-torus for short), a three-dimensional manifold that is a "hypertorus" in R4 (4-dimensional space). If instead during the gluing process we flip one of the faces, we get the 3-D Klein bottle, a three-dimensional manifold that is a "hyper Klein bottle" in R4. Ouch! To help understand this, go here.

Jeffrey Weeks and the other people responsible for the page mentioned above have other good stuff here.

Alexander's Horned Sphere

No, this is not named after me. It is difficult to describe Alexander's Horned Sphere. It is a wild embedding of S2 (the ordinary hollow sphere we discussed in class) into R3, sort of like tying it into a knot, but nothing can compare with seeing it for yourself. There are probably other drawings of it out on the Web somewhere. This object just goes to show that you can find plenty of weird topological objects without resorting to higher dimensions.

More on regular 4-D polytopes

If you want to spend some more time in rigid Geometryland, several pictures of the hypercube and other regular polytopes are here; scroll down until you see the words "four dimensions." Cross-sections (cuts with a three-dimensional knife) of the 4-dimensional polytope called the 600-cell can be found here.


Here are a few more things to play with:

The Golden Ratio

I do not know if this number has anything to do with topology, although given the way mathematics usually works, it probably does somehow. I just stuck it in here because this page has more about the Golden Ratio than I ever knew!

If you want to, you can see the page that my geometry class used last year. But be warned: not all those sites may still be operating.

Remind me to check and see if I can bring in my senior thesis, which was written about the topological (and other) properties of a mathematical object called SO(3).

(Last updated 6/27/98)