Geometry in 1997

TAG SEP 1997: Geometry

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.

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.

Read Instructions Above First!


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.

Platonic solids

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.

The Fourth Dimension

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.

The Buckyball

There are many sites about buckyballs (C60), 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 . . .

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)