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11-cell
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Jabe



Joined: 12 Jan 2008
Posts: 46
Location: Somewhere between Texas and the Fourth Dimension

PostPosted: Fri Feb 06, 2009 4:43 am    Post subject: Reply with quote

Below is the OFF file that I started with - it generates a polychoron with 8 congruent corners - (the same corners as the 8 - 2 step prism), I used a drawing of an octagon to find sets of tetrahedra (complete tetragons in the octagon) that would fully connect and close - this is how I developped the face and cell list. The polychoron that the off file generates is a starry one - looks quite cool (so does it's dual). The dual of the convex hull is the 8-2 step tegum.

4OFF

8 32 24 16

0.0 1.0 0.0 1.0
0.70710678118654752 0.70710678118654752 1.0 0.0
1.0 0.0 0.0 -1.0
0.70710678118654752 -0.70710678118654752 -1.0 0.0
0.0 -1.0 0.0 1.0
-0.70710678118654752 -0.70710678118654752 1.0 0.0
-1.0 0.0 0.0 -1.0
-0.70710678118654752 0.70710678118654752 -1.0 0.0

3 7 0 1
3 0 1 2
3 1 2 3
3 2 3 4
3 3 4 5
3 4 5 6
3 5 6 7
3 6 7 0

3 2 7 0
3 3 0 1
3 4 1 2
3 5 2 3
3 6 3 4
3 7 4 5
3 0 5 6
3 1 6 7

3 0 1 6
3 1 2 7
3 2 3 0
3 3 4 1
3 4 5 2
3 5 6 3
3 6 7 4
3 7 0 5

3 5 0 3
3 6 1 4
3 7 2 5
3 0 3 6
3 1 4 7
3 2 5 0
3 3 6 1
3 4 7 2

4 0 1 8 17 255 0 0
4 1 2 9 18 255 0 0
4 2 3 10 19 255 0 0
4 3 4 11 20 255 0 0
4 4 5 12 21 255 0 0
4 5 6 13 22 255 0 0
4 6 7 14 23 255 0 0
4 7 0 15 16 255 0 0

4 10 17 28 31 255 255 0
4 11 18 29 24 255 255 0
4 12 19 30 25 255 255 0
4 13 20 31 26 255 255 0
4 14 21 24 27 255 255 0
4 15 22 25 28 255 255 0
4 8 23 26 29 255 255 0
4 9 16 27 30 255 255 0
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polymorphic



Joined: 17 Feb 2008
Posts: 26

PostPosted: Fri Feb 06, 2009 3:24 pm    Post subject: Reply with quote

Very intersting.

I made one observation. Of all of the step tegums you have sent (thanks!) I noticed that for each N, at least one of them has a similar cell type.

For an N-choron is taking a wedge of an 3D (N-2).4.4 prism. e.g. for the Heptachoron the cell is the wedge of a 5.4.4 prism. The Octachoron at a wedge of the 6.4.4. So far, the most interesting one you have made is the Tridecachoron 2, or the one you call Tridecachoron Low Phase. This one has the 11.4.4 prism wedges following each other in succession. Really quite nice!

Could this be the case for 10,11 and 12 cells?

Does it follow there should be a hexachoron which uses the wedge of a 4.4.4 prism (cube) which should be a triangular prism?

Roger
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http://www.interocitors.com/polyhedra/
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Jabe



Joined: 12 Jan 2008
Posts: 46
Location: Somewhere between Texas and the Fourth Dimension

PostPosted: Sat Feb 07, 2009 7:10 am    Post subject: Reply with quote

The ones of step 2 have a similar appearance all the way through - at infinity it approaches a curved 4-D shape that I call the "bicoiloid" - the bicoiloid is the dual of the convex hull of a curved "2-coil" (think of a part of a slinky, but with only two coils (720 degrees all together) then curve this into 4-space. One of these days I'll attempt to "greatstellafy" the bicoiloid which I consider to be one of the many curved dice (has congruent contact regions).

The 6-2 step tegum does have triangular prisms as cells - it is actually the triangular duoprism itself. 5-2 is the pentachoron.
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polymorphic



Joined: 17 Feb 2008
Posts: 26

PostPosted: Sun Feb 08, 2009 12:50 pm    Post subject: Reply with quote

For the step 2's there seems to be a kind of order. Such that it might be possible to find the pattern to make the cells. Then there could be a way to generate the Nth model.

For the other steps it would be another matter.

Roger
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