Demipentacross?

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Demipentacross?

Postby kingmaz » Fri Oct 30, 2009 11:30 am

Given I have enough trouble with tetraspace, this may be a silly question:

In 5d, the demipenteract appears to have an unknown dual (according to Wikipedia). If there were none, I'd assume this to be stated.
So I had a look and it appears this is the case for all demihypercubes for 5d and over.

In 4d, it's interesting that the demitesseract is the directly analogous to the orthoplex / 16-cell. Therefore the tesseract appears to be dual of its own 'half' :\

As this doesn't appear to be the case in 5d, can we start at a hypothesis that a demipentacross must exist as the dual of the demipenteract? Can a demipentacross exist, or is there some hypertopological constraint (dihedral angle?) that makes such a figure impossible?
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Re: Demipentacross?

Postby Keiji » Sat Oct 31, 2009 6:48 pm

The demipenteract has elements 26, 120, 160, 80, 16 (ordered tera to vertices). So if it had a dual (which should be the case; every convex polytope should have a dual, as far as I know), the dual would have elements 16, 80, 160, 120, 26.

Now the pentacross has only 10 vertices, so even if it were possible to construct a "demipentacross" (which it is not, as the pentacross contains triangular faces; alternation can only be performed on polytopes with no faces that have an odd number of sides on them), that object would have only 5 vertices, so it would definitely not be the dual of the demipenteract.

The dual of a demipenteract would have 26 vertices, so if this was a demi-tope itself, the original polytope would have 52 vertices. I could find no polytopes with either 26 or 52 vertices in Category:5-polytopes, so the dual is yet to be found.
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Re: Demipentacross?

Postby wendy » Mon Nov 02, 2009 8:15 am

The dual of this class of figure (half-cube), derives from the cross-polytope in the corresponding dimension.

One raises pyramids on alternate faces of the cross-polytope, such that the new sloping faces align with those of the remaining faces. This gives 2^(n-1) faces, each of form a simplex, with pyramids.

The examples in 5d, 6d, 7d can be used as faces for polytopes in the next dimension, giving polytopes with 27, 126 and 1920 demu-cross faces. The demicross in eight dimensions tiles space, as does the polytope with 27 demicross faces.
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