I was looking at the definition of the duocylinder at the eusebia site, and trying to figure out exactly what this was. I knew about the cubinder (extruded cylinder: surface=2 flat dimensions, 1 curved) and the spherinder (extruded sphere: 2 curved dimensions, 1 flat), but wondered how a duocylinder was formed from a cylinder.
Studying it again, this time I noted the progression from the duoprisms, to the prismic cylinders, finally to the duocylinder. It basically involves taking the values of m and n to infinity (Note, Wendy, this implies that a circle is "the limit of... [the sides of the polygons] taken to infinity").
Well finally coming to understand it now, I realized that this was the "Euclidean [true] 2-torus" discussed by Luminet, Starkman & Weeks in the Scientific American article "Is Space Finite?"
I, like many others always wondered what the global shape of the Asteroid screen was. You would think, at first, it would be a sphere. But then you realize that the top and bottom and left and right sides are joined at lines, not points. Then, Michio Kaku's Hyperspace said that it was a torus or donut shape, formed by curling a square into a cylinder, and then joining the cylinder's circlular ends to each other. He suggests doing that with a piece of paper. I had thought of this, but realized, that it could not be exact, because if you try to do that with paper, you end up scrunching one side, and ripping the other.
So then, I come across the Sciam aticle. It points out that the "true" torus, or Asteroids screen shape could not sit in 3-space at all! "Doughnuts may do so, because they have been bent into a spherical geometry around the outside [i.e. the stretching], and a hyperbolic geometry around the hole [i.e. the crunching] Without this curvature, doughnuts could not be viewed from the outside". The article does not explain how the true torus is formed, then; but I figured out, it was from taking the two ends of the cylinder, and curling and joining them in four dimensional space. The original square you started out with could be curled so easily because it was "flat". But once it becomes a cylinder, it no longer is flat --in 3 space that is. But it is still "flat" in 4-space! A lower dimensional analogy woulf be a belt or ribbon. If you take a circle in 3-space, and step down the dimensions, the circle becomes two points bounding a "hollow" line segment. The curved 2D surface conmecting the circles is then represented by two lines joining the opposite pairs of points. Basically, a belt or ribbon. Try to imagine joining the ends of the ribbon by keeping it flat on the 2D surface. Once again, you scrunch the side that becomes the inside, while stretching what becomes the outside. Now instead, simply lift the two ends up through 3-space and join them. Much easier!
This, basically, is what the duocylinder is in the higher dimension. The curling of the square represents m, and the curling through the higher space represents n. As circles, both would be the limit of polygons at infinity. In a simply extruded polytope, for instance, all lines generate squares, which sets the value of n at 4)
Is all of this right?