Non-CRF facets of CRF duals

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

Non-CRF facets of CRF duals

Postby Keiji » Wed Feb 26, 2014 12:26 am

In case anyone is actually following my findings on this, I've created a wiki page for the non-CRF facets of CRF duals including images for all such polyhedra discovered so far. The total now stands at two self-dual and three non-self-dual pairs, totalling eight polyhedra.

Interestingly, so far the self-dual ones are precisely the ones that are chiral.

The names of all of these figures are entirely tentative and were thought up just to identify them - they're not particularly good/consistent names, so any suggestions will be taken on board.

Calculating incidence matrices is what has forced me to work out these cells in the first place (because my verification program insists that I add all the cells and dual cells first), but I haven't yet successfully worked out what the dual of any 4D CRF itself would look like. I find it fascinating though, how the dual of a CRF can have these bizarre-looking polyhedra, and what the entire 4D dual might actually look like!
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Re: Non-CRF facets of CRF duals

Postby quickfur » Wed Feb 26, 2014 12:53 am

Keiji wrote:In case anyone is actually following my findings on this, I've created a wiki page for the non-CRF facets of CRF duals including images for all such polyhedra discovered so far. The total now stands at two self-dual and three non-self-dual pairs, totalling eight polyhedra.

Interestingly, so far the self-dual ones are precisely the ones that are chiral.

The names of all of these figures are entirely tentative and were thought up just to identify them - they're not particularly good/consistent names, so any suggestions will be taken on board.

Calculating incidence matrices is what has forced me to work out these cells in the first place (because my verification program insists that I add all the cells and dual cells first), but I haven't yet successfully worked out what the dual of any 4D CRF itself would look like. I find it fascinating though, how the dual of a CRF can have these bizarre-looking polyhedra, and what the entire 4D dual might actually look like!

Finding the dual of a 4D CRF (or any polytope, for that matter) is dead easy once you have the Polyview .def file. All you have to do is, at your favorite shell prompt, type:
Code: Select all
/path/to/polyview/dissect /path/to/polytope.def dual | makepoly name_of_dual - /path/to/polytope_dual.def

and bingo, you have the dual all ready to go. :) Render it to your heart's content. :nod:
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Re: Non-CRF facets of CRF duals

Postby quickfur » Wed Feb 26, 2014 1:12 am

Also, if you want to render diagrams of a particular cell shape, first find the cell number (e.g., by using polyview's "select" command to identify it by number of faces, vertices, etc.), then do the following (suppose the cell number is 123):
Code: Select all
load /path/to/polytope.def
translate to center
hide all faces
hide all edges
hide all vertices

lookat cell 123 directly

show vertices in cell 123
show edges in cell 123
show faces in cell 123
render

This will basically produce just (the projection of) a single cell in the povray output, so the final image will basically be just a diagram of that cell. The "directly" in the lookat command basically causes polyview to move the viewpoint so that the line from the camera to the origin crosses the centroid of the cell (it also works for any other polytope element). If this line is parallel to the normal vector of the cell's hyperplane, then the projection will basically just be the cell itself, undistorted by the projection. (It may be rescaled by perspective projection, but you could always just say "parallel" at the top of the script and that should take care of it.)

(Originally "lookat ... directly" was intended to always give you the undistorted cell, but that hasn't been implemented correctly yet. Please feel free to bug me to do it if you think it will be a useful feature. ;) )
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