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4D Part 4: The 24-Cell and Klein Bottle
Posted:
Jun 18, 1993 8:00 PM
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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
3 the 5 Platonic solids =
tetrahedron, cube,octahedron, dodecahedron, icosahedron =
simplex, cube, cross-polytope, dodecahedron, icosahedron
4 simplex, cube, cross-polytope, 24-cell, 120-cell,600-cell
>=5 simplex, cube, cross-polytope.
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."
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