4D Duals

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

4D Duals

Postby BClaw » Fri Mar 12, 2004 11:28 pm

I was trying to animate the 4-simplex (5-cell) and it's dual, which in the 3D world would be the tetrahedron and it's dual (itself). The resulting figure is called a Stella Octangula, and is also a stellation of the tetrahedron. The 2D version would be the figure called the Star of David. OK, so I figured that since the lines cross on both the 2D and 3D versions, that they must cross in the 4D version as well. When I tried to render it though, this did not seem to be the case. Am I wrong in my assumption that the lines would intersect in the 4th dimensional version? If so, then my math is wrong.

I ended up with a figure that would be a 4D pyramidal shape with the Stella Octangula as it's base:

Pyramidal 4-Stella Octangula (DivX)
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Re: 4D Duals

Postby pat » Sun Mar 14, 2004 5:22 am

BClaw wrote:When I tried to render it though, this did not seem to be the case. Am I wrong in my assumption that the lines would intersect in the 4th dimensional version? If so, then my math is wrong.


They look to me like they cross in your animation. I'm confused.
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Re: 4D Duals

Postby BClaw » Sun Mar 14, 2004 7:00 am

pat wrote:
BClaw wrote:When I tried to render it though, this did not seem to be the case. Am I wrong in my assumption that the lines would intersect in the 4th dimensional version? If so, then my math is wrong.


They look to me like they cross in your animation. I'm confused.


Well no wonder you're confused with my lousy explanation!! :)

The figure I have is not what I was going for, but I thought it would be of interest, so I posted it anyway. This is the 4D equivalent of taking a star of david (6/2 polygon) and making a pyramid out of it. Like rotating a second tetrahedron 60 degrees around the z axis only. Does that make sense? That's what the animation shows, and since the base object really is the stella octangula, the lines show crosing as I expected. I'll render a still image of what I came up with that I thought was right mathematically, but didn't *look* right, so you can see that I mean.

http://home.rmci.net/tuvel/private/4-stoct.jpg

Sorry for the confusion!
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Postby Polyhedron Dude » Sat Apr 17, 2004 7:57 am

Lets call the 4-D version of the stella octangula - the stella decangula - call it stad for short. Cross sections of the stad would be described as the following:

a point grows into a tetrahedron (first tet)
Then all of a sudden a large tetrahedron in dual position appears, it covers most of the first tet, but allows only the corners of it to peep through.
The first tet continues to grow while the second tet shrinks, eventually the two tets are the same size and you have a stella octangula (this is the center section).
The first tet keeps growing while the second keeps shrinking until the second one is only peeping out of the first with it's corners, then the first one vanishes.
The second tet keeps on shrinking to a point and vanishes.

The compound of 2 icoes (ico=24-cell) is similar but with 48 corners and 48-sides.

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