Stereographic projection of 24-cell...?

Higher-dimensional geometry (previously "Polyshapes").

Stereographic projection of 24-cell...?

Postby Paul » Sun Oct 01, 2006 10:42 am

Hello all,

I posted this at the Math Message board, but no one there seems much interested in higher-dimensional geometry.

It's kinda odd, in that sometimes I get the impression that allot of people consider higher-dimensional geometry 'more' a priori than other areas of mathematics. In epistemology, mathematical truths are almost always the first example given of a priori knowledge, in contrast to a posteriori knowledge. 'Vectors' are in our minds, whether we're talking about them in 3-space, or 24-space... Anyway...

Using these coordinates for the 24-cell in 4-space, and this formula for stereographic projection given in the PlanetMath article, with c=0, I used Cabri 3D to come up with this stereographic projection of the 24-cell:

Image

You'll notice there are only 23 points in the projection above. The fourth coordinate in one of the 4-space coordinates for the 24-cell was one, so the denominator of the transformation multiplier went to zero. I'm guessing that perhaps this point too should also be mapped to (0,0,0). Since another set of coordinates also mapped to (0,0,0), I thinking that perhaps there's a vertex of an octahedron both in the center of the inner cube (in the projection), and also another one that's in the same place in the projection, which is actually in the center of the outer cube. The point mapping to (0,0,0) is -1w and the one mapping to infinity is the 1w along the ana-kata axis. Not much of a surprise, I suppose.

I'm not entirely sure I haven't done anything wrong here...? However, the reason I connected the vertices in the manner I did was because the outer blue-segmented cube and the inner green-segmented cube, along with the yellow segments (in the projection) are precisely the stereographic projection of the tesseract... except for the colors, it's exactly like my Zome model of the tesseract's stereographic projection.

The eight cubes in the tessaract are easy to see, although the shapes of some of them are distorted. There's the green-segmented cube in the center, the outer blue-segmented cube, then there are six other (distorted in shape) cubes which all share a face with the outer blue-segmented cube, the inner green-segmented cube, and four other cubes adjacent to each whose faces are bounded by one blue-segmented edge from the outer cube, one green-segmented edge from the inner cube, and two yellow edges from other adjacent cubes.

In the projection, the purplish-segmented octahedron is precisely the dual of the outer blue-segmented cube... that is, the vertices of this octahedron are right in the center of the faces of the outer cube. I believe the eight vertices of the hexadecachoron (the 16-cell, or dual of the tessaract), which are mapped in the projection to the octahedron, and the two points at (0,0,0), are in the same 3D-realm as the eight cubical cells of the tessaract... so, they're all a distance of one from the origin in 4-space.

Does my interpretation of the projection seem to make sense? This isn't exactly what I thought I'd get... I thought I'd get a small octahedron bounded by an cuboctohedron, which then would be bounded by a outer octahedron. I've seen this projection many times... is this projection just a change of perspective on the one I have?

I haven't tried finding the 24 octahedral cells in the projection yet... does someone else want to describe where they are?
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Postby wendy » Mon Oct 02, 2006 12:05 am

The projection is indeed largely correct.

It shows a vertex-first projection. You only see 23 vertices because the other one is at infinity.

You should see a tesseract in this figure, because the 24choron does contain the tesseract. In the Octa + CO + Octa section, you see the tesseract square first: the diameter-square of the octahedron, and the parallel-opposite faces of the CO.

The 24 purple edges should not be there. They are actually the long diagonals of the octahedra.

The 24 faces lie:

6 faces from central vertex + cube face + octahedron vertex.
12 faces where the purple diagonal + the square it goes through.
6 faces from octahedron through blue cube squares to point at infinity.
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the dream we dream together is reality.
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