Bruteforce searching for deltachorons

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

Bruteforce searching for deltachorons

Postby PythagorasReincarnated » Sat Jul 25, 2015 10:00 am

First, by deltachoron I mean a (convex, in this case) polychoron that has only (regular) tetrahedrons as facets (analogous to deltahedrons). The idea I have is that we could easily try to search all topologicaly possible convex deltachoron, imposing some simple rules before starting the search (no more than 20 tetrahedrons per vertex and 5 per edge, and the maximun total facets would be 600).
The search would probably be done by "gluing" tetrahedrons and eliminating absurd results, and then trying to generate the model, to see if it's convex.
I'm not good enough at programming to do it myself, but I would like to help.
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Re: Bruteforce searching for deltachorons

Postby Klitzing » Sun Jul 26, 2015 6:57 am

This research, in fact, was the first idea on how to extrapolate the Johnson solids idea onto higher dimensions: Considering all convex polytopes in N dimensions, the (N-1) dimensional facets of which would be regular ones in turn. And that research already has been completed way back around 1979. The couple Roswitha and Gerd Blind then published the complete enumeration - outside the combinatorically huge amount of possible diminishings of the 600-cell. But in the meantime even that number has been computer-counted. - So nothing new in here.

In 4D we have (beyond the well-known archimedean ones):
- the dipyramid of the tetrahedron
- the dipyramid of the icosahedron
- the pyramid of the octahedron
- the pyramid of the icosahedron
- the pyramid of the octahedron attached onto a rectified pentachoron
- and all those numerous diminishings of the hexacosachoron

in all spaces beyond:
- the dipyramid of the simplex
- the pyramid of the cross-polytope

You then just need to select the ones which have tetrahedra solely from that above shortlist:
- the 2 mentioned dipyramids
- plus the 3 known regular ones
That's it.

Btw., a still on-going research is a different extrapolation of Johnson's idea: Considering all convex polytopes in N dimensions, the 2-dimensional faces of which would be regular ones. This is what is called CRF (convex regular faced). Lots of individual findings with respect to N=4 already can be found in this forum. In fact, even for N=4 no complete enumeration has been achieved so far.

--- rk
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Re: Bruteforce searching for deltachorons

Postby quickfur » Mon Jul 27, 2015 5:56 pm

The set of all possible (4D) CRFs seems to be very large, much larger than I had initially expected, and they also admit a large number of parameters, due to the large number of 3D CRFs (i.e. Johnson solids). That makes a brute force search for them quite difficult, since you have to account for so many parameters. I wonder if it might be possible to make a brute-force search more manageable if we narrowed it down to the subset of CRFs that only contain triangular 2-faces? I.e., non-tetrahedral cells are permitted, as long as they are themselves deltahedra.

Or has that also been done before already?
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Re: Bruteforce searching for deltachorons

Postby quickfur » Mon Jul 27, 2015 6:03 pm

P.S. This would narrow down the set of permissible cells to 8: (the regulars) tetrahedron, octahedron, icosahedron; (the Johnsons) triangular bipyramid J12, the pentagonal bipyramid J13, the snub disphenoid J84, triaugmented triangular prism J51, and the gyroelongated square bipyramid J17.
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