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

Posted: Fri Feb 06, 2009 4:43 am Post subject: 


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 82 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 _________________ May the Fourth (dimension) be with you. 

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polymorphic
Joined: 17 Feb 2008 Posts: 26

Posted: Fri Feb 06, 2009 3:24 pm Post subject: 


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 Nchoron is taking a wedge of an 3D (N2).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 _________________ Roger Kaufman
http://www.interocitors.com/polyhedra/ 

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

Posted: Sat Feb 07, 2009 7:10 am Post subject: 


The ones of step 2 have a similar appearance all the way through  at infinity it approaches a curved 4D shape that I call the "bicoiloid"  the bicoiloid is the dual of the convex hull of a curved "2coil" (think of a part of a slinky, but with only two coils (720 degrees all together) then curve this into 4space. 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 62 step tegum does have triangular prisms as cells  it is actually the triangular duoprism itself. 52 is the pentachoron. _________________ May the Fourth (dimension) be with you. 

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polymorphic
Joined: 17 Feb 2008 Posts: 26

Posted: Sun Feb 08, 2009 12:50 pm Post subject: 


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 _________________ Roger Kaufman
http://www.interocitors.com/polyhedra/ 

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