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.

If you have any questions, ask me. The "Go" menu on the top white 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.**

Remember when we talked about how you could tessllate the plane (that is,
cover the floor) with only three regular polygons: equilateral triangles,
squares and regular hexagons? Well, that is only true if you assume (as we
did) that space is flat, which it always is in Euclidean geometry. If you
allow space to be two-dimensional but *curved*, you get
some new
tessellations. Many of Escher's drawings use these symmetries.

Moving to 3 dimensions, there are only **five**
regular polyhedra. (Remember,
a regular polyhedron has the same regular polygon for each face, meeting
everywhere at equal angles.) These five polyhedra are called the Platonic
solids. We discussed these in class: the tetrahedron with four sides, the
cube with six sides, the octahedron with eight, the dodecahedron with 12, and
the icosahedron with 20. Since it is hard for me to draw them on the board,
here are some
pictures. This site also has a nifty picture of a soccer ball.

Just like shapes in two dimensions with straight edges are called
polygons, and shapes in 3-D with flat faces are called polyhedra, shapes in
4 dimensions with flat hyperfaces (3-D faces on a 4-D object) are called
**polytopes**.

[This name comes from the same root (I believe)
as the word **topology**. Topology is the name for a branch of
geometry in which we study things like
the
Mobius strip (shown here tied into a knot), and the Klein bottle.]

Manipulate the hypercube and other neat-o 4-D polytopes!!! For still pictures, go here and click on "Obligatory Pictures"; for rotating images, go to the same place and click on "Java Examples." Then scroll down until you see "FourD" and read the instructions.

Cross-sections (cuts with a three-dimensional knife) of the 4-dimensional
polytope called the **600-cell** can be found
here.

There are many sites about buckyballs (C_{60}), and the general
category of molecules that it belongs to, called **fullerenes**.

This one is pretty.

Info on fullerenes that have been doped with (had added to them) other atoms

More about fullerenes (written *by* a researcher) than you ever knew
is here. This site has a
good image of a buckyball also.

Truncating the icosahedron to produce a buckyball can be seen here.

Other uniform polyhedra (such as Platonic solids and Archimedean solids)
can be found
here,
and this site also contains (V, E, F) for the buckyball.

Here are a few more things to play with. Don't blame me if it turns your brains to tofu . . .

- Cool knots and the Moebius strip!
- Adventures in Flatland Includes, I think, a trip over the surface of a Klein bottle.
- Weird stuff
- More Klein bottle-based weird stuff
- Vortices and a doughnut
- Some Geometric pictures and generators
- A Visual Dictionary of certain plane curves

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.

(Last updated 7/2/97)