I want to make a correction and an addition to my article on hoops. First, the correction: the torus of revolution should revolve around the y axis, not the z axis. I also did not include any background information on the hoops researcher Dan Asimov; please see below.
This previous article was based on an interview with Dan Asimov during his recent visit to the Center. Asimov works in the area of visualization at NASA Ames Research Center. The previous article focused on his current research in the area of "hoops" in three dimensions; this is a subject which some readers may have remembered from a puzzle that Asimov posted a few weeks ago in geometry.puzzles.
In addition to the hoops research, I would like to briefly describe some observations that Asimov made about 4D. I wrote to him for details, and I have directly quoted large portions of his response (in quotation marks). For more information, also see Asimov's posted description of his n dimensional visualization program, called the "grand tour."
In 3D, there are five regular solids (polytopes). In four dimensions there are six regular polytopes. In all other dimensions, there are only three basic types of polytopes. This means that three and four dimensions are exceptional; they have the three standard types of polytopes seen in higher dimensions, but on top of those they have additional polytopes.
In four dimensions, one of the extra polytopes is the 24-cell, consisting of 24 octahedra. These octahedra are the "faces" of the four dimensional polytope, six meeting at each vertex. The object is self-dual. This means that the new polytope formed by connecting the centers of all the faces is again a 24-cell. In what sense is the 24-cell distinguishable from the three standard polytopes, and why does the self-dual property make the 24-cell special? In Asimov's words:
"The 24-cell is best understood in the context of the classification of all regular polytopes in all dimensions:
dim polytopes --- --------- 2 regular n-gons for n >= 3
The only self-dual polytopes in dimensions >= 3 are the simplex (which occurs in all dimensions) and the 24-cell (which occurs only in dimension 4). This gives the 24-cell a kind of symmetry unique to itself, with no analogue in any other dimensions."
Asimov also observed that in 4D, it is possible to immerse the Klein bottle in a symmetric manner in the three-sphere (S^3). Remember that an immersion is only locally injective. In the case of this immersion, there is self-intersection. He used a stereographic projection of the three-sphere to get a surface in three dimensions. I was quite impressed with the beauty of this surface. I asked Asimov to give a more detailed explanation, which follows:
"As for the Klein bottle, the surface I showed you is the stereographic projection of a particularly beautiful Klein bottle K that is immersed in S^3 as a minimal surface, and has a great circle in S^3 as its set of self-intersection. This surface K in S^3 has a great deal of symmetry: its group of isometries is a 1-dimensional Lie group.
This Klein bottle can be described as those points (x,y,z,w) of S^3 which satisfy the polynomial equation
w(x^2 -y^2) = 2xyz.
In addition, K is the union of two embedded Mobius bands in S^3 whose intersection is two linked great circles in S^3, one of which is their common boundary.
For the stereographic projection (used to send K from S^3 into R^3 where it could be viewed fairly well), its projection point was chosen to lie on the self-intersection circle of K, in order that as much as possible of K's symmetry be preserved after it is projected. After projection by stereographic projection S: S^3 -> R^3, the image S(K) represents the Klein bottle with two points removed, immersed in R^3 with the z-axis as its set of self-intersection. Unlike K in S^3, this image S(K) in R^3 is not a minimal surface. It is, however, conjectured by R. Kusner that it minimizes the integral of squared mean curvature.
One of the Mobius bands that constitute half of K--call it M--can also be stereographically projected in the same way, of course, and the result is a surface S(M) in R^3 which depicts a Mobius band in R^3 whose boundary is not just a topological circle, but a perfect geometric circle. 'Depicts' is more appropriate than "is" here, since the projection point chosen to preserve symmetry lies *on* M in S^3, so it is missing from S(M). As a result, S(M) has to spread out to infinity in R^3. This Mobius band (actually a Mobius band minus a point) S(M) is the subject of a short computer graphics film, 'The Sudanese Mobius Band,' made by myself and Douglas Lerner for the 1984 Siggraph Film and Video Show."