24-cell projection

Higher-dimensional geometry (previously "Polyshapes").

24-cell projection

Postby quickfur » Mon Nov 20, 2006 3:07 am

OK, the ASCII-based 4maze game was a nice diversion, but my heart is always with projections as the "real" way of visualizing 4D, so finally I got off my lazy butt and wrote a program that translates the polytope definitions (with the full lattice meticulously generated by my scripts as I've indicated previously) into something POVRay can render. So here's the first working animation that I made:

http://eusebeia.dyndns.org/~hsteoh/4d/24cell-1.gif

Can anyone guess what this might be? :) That's right, it's everybody's favorite polytope: the 24-cell, vertex-first perspective projection. The center of the projection, where the octahedral cells meet, is the vertex closest to the 4D viewer.

Note that it is the camera rotating, not the object, so the lighting is also rotating with the object. The rotation is only in 3D, not in 4D (the point is to see the structure of the static projected image). Also, obscured facets (from the 4D POV) are culled, as usual. Not culling obscured facets produces a tangled messy image which is hard to understand, culling them makes the image a lot clearer.
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Postby papernuke » Mon Nov 20, 2006 4:08 am

Is that a 24cell? It dosen't look like it. And also, is a 24cell a four dimensional object?
"Civilization is a race between education and catastrophe."
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Postby quickfur » Mon Nov 20, 2006 6:00 am

Icon wrote:Is that a 24cell? It dosen't look like it. And also, is a 24cell a four dimensional object?

That is the projection of the 24-cell, a 4D object, into 3D space. The 24-cell is composed of 24 octahedra folded together in 4D space. In the process of projection, facets of the 24-cell that are obscured to the 4D viewer are culled, so only 6 of the 24 cells are actually visible in the image. This is similar to how, if you look at one of the corners of a cube, you can only see 3 of the 6 faces, because the other 3 faces are obscured behind these 3 faces. Only 6 of the 24 cells are visible to the 4D viewer, because they are "facing" him, whereas the others are behind the 6 cells.

If you look carefully at the image, you will see that it is composed of 6 (slightly flattened) octahedra meeting at the center of the image. These octahedra look flattened because they are slightly bent into the 4th direction, just like how if you look at a square from an angle, it looks like a trapezoid. The octahedra are joined to each other via their triangular faces. Only some of these connections are shown, since the rest of the faces are hidden. They are joined at the "outer" faces of the image.

(Think about looking at the corner of a cube: the image is a hexagon of some sort, with three of the visible faces laid out in the hexagon. The outer edges of the hexagon are where the visible faces are connected to the hidden faces. The analogous 4D->3D situation is similar: the outer faces of the image are where the visible cells are joined to the hidden cells.)
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