What I find most intruiging in this chapter is the topological exploration of the torus, which is a truly remarkable sort of figure. In my youth, I always had a certian contempt for the torus; it seemed to be very irregular, like the ugly stepsister of the beautiful sphere. After all, it has a hole in the middle! But as I consider it more closely, the torus reveals itself to have some wonderful mathematical properties. For one, it is continuous in all directions, like a sphere, so that one can travel in any one direction and end up back at the starting point. As prof. Banchoff mentioned in class, it also has the quality of being "sliceable" into only two parts by any one flat plane, a characteristic shared primarily by objects which are strictly convex.

More than its strange properties of symmetry, what interests me is the relationship of the torus to the sphere. A sphere can be constructed by spinning a circle along the axis of its diameter. If one uses the same axis, but translates the circle within the plane prior to spinning it, one gets a sort of roundish object with a dimple in each end; further translating the circle makes the dimples progressively larger, until a critical point is reached (when the translation equals the length of the radius) and the object begins to fold in on itself. Further translation from this point generates the torus. [If I could remember calculus, I'd try to derive the relationship between the value of this translation and the volume of the resultant object.] When the translation is such that it equals the length of the circle's diameter, the torus generated has the intruiging property that the passage through the middle has the same radius as the body of the torus itself. In four-space, the symmetry of this relationship is revealed, as shown in the figure which is on the cover of B3D.

And, of course, toruses are a great shape for eating, which undoubtedly is the reason that the bagel is so much more popular than the bialy.

The torus is undoubtedly one of my favorite surfaces. It is much more interesting than the sphere from a topological viewpoint, and although it can't be as symmetrical as a sphere, it has quite a few symmetry properties already in three-space. In four-space it has even more, as we will see, or as we can already see by accessing that demo on my home page. I'm glad to see how you made that link--perhaps others will follow suit?

To see some of the other surfaces that can arise from surfaces of revolution of a circle, take a look at another of the images on my project list, by clicking on the word "tight" in the paragraph just ahead of the photograph of Edwin Abbott Abbott.

If you're going to say "Que pasa?" you should probably use "Toro". *[Actually, that was an allusion to a cereal commercial from a couple years ago... "Pasa your cereal, Sugar Bear!" -dave]*

Prof. B.

Dave,

I'm in the process of reading a book called "The Shape of Space" which does a lot of comparisons between different shapes. Spheres and tori are central to the book.

Consider a flatlander on a finite square. Imagine that whenever he exits the square from the right/left, he reappears on the left/right; whenever he exits the square from the top/bottom, he reappears on the bottom/top. This "square" is then a flat torus. The top of the square is connected to the bottom of the square, and the two sides of the square are connected. If you tried to construct such a figure out of a piece of paper, you'd get a torus. As you mentioned, a torus is topologically equivalent to a sphere--so a flatLender (A Square on a bagel) *["FlatLender"- isn't that funny? -dave]*living on a bagel would see his world the same way a flatlander would view his sphere world.

Another way of thinking about a bagel is by calling it a "circle cross cirlcle". It's the product of a circle and a circle. In other words, it's a bunch of circle arranged in a circular pattern.

Tori are very interesting. I plan to play Torus Chess soon (pieces can move off one side and come back on the other). Hopefully, I'll understand them better by the time I'm finished with this great book.

Dan.

-david stanke