Polyhedron Dude wrote:I decided to make this thread to display some of these by releasing a few polytera each week.
Klitzing wrote:Polyhedron Dude wrote:I decided to make this thread to display some of these by releasing a few polytera each week.
Very good idea!
I'm tuned to see them all.
Or, taken in fb-jargon: like!
--- rk
Klitzing wrote:Wow, that one Looks dramatic!
Btw. both garpop and ripdip lack linked pictures at your 4D website...
--- rk
wendy wrote:I think these are gui lace cities? I don't completely follow...
o3x4o o3o4q o3x4o
o3o4q o3o4q
o3x4o o3o4q o3x4o
quickfur wrote:I like spix. It's nice and simple, yet intricate in its own way.
Of course, I'm partial to convex polytopes... but the others are really nice too. I've been busy and haven't been able to work on my projection-based renders recently, but I've still been thinking at the back of my mind about how to deal with rendering projections of non-convex polytopes. Well, in principle I know what needs to be done, but it's just a matter of how to implement it. I do find the projection method more intuitive (for me, anyway) than sectioning.
Klitzing wrote:The last one, i.e. tin, looks best to me. Esp. because of its chessboard-like framework pattern ...
wendy wrote:I think these are gui lace cities? I don't completely follow...
Polyhedron Dude wrote:quickfur wrote:I like spix. It's nice and simple, yet intricate in its own way.
Of course, I'm partial to convex polytopes... but the others are really nice too. I've been busy and haven't been able to work on my projection-based renders recently, but I've still been thinking at the back of my mind about how to deal with rendering projections of non-convex polytopes. Well, in principle I know what needs to be done, but it's just a matter of how to implement it. I do find the projection method more intuitive (for me, anyway) than sectioning.
I find your renders quite fascinating, they really help bring out the 4-D-ness of the polychora.
Sectioning works well with star polytopes to bring out the intersection details, but you do lose that 4-D effect.
[...]
Now for our next polyteron, this one will noqyapants off, its Noquapant. Noquapant is the penteractiquasiprismated penteracti32teron. Its symbol is (x'x"x)xx = Cxx for short. It is a lone operative and is the most amazing looking of the simplex-verfed polytera. Its facets are 10 thaquitpaths (yellow), 10 gaquidpoths (cyan), 32 gippids (green), 80 hodips (red-orange), and 40 cotcopes (lavender).
http://pages.suddenlink.net/hedrondude/noquapant.png
quickfur wrote:Just out of curiosity, how do you handle coordinates for these polytopes? Do you use full Cartesian coordinates, or just a point and a generating symmetry (via a Wythoff construction perhaps)? Currently, my renderer uses explicit coordinates for vertices, and vertex sets for j-faces. To avoid having to manually specify vertex sets (which is impractical beyond the simplest polychora -- the 24-cell, for example, took me 2 days to write out in full), I use a convex hull algorithm to generate the j-faces. Unfortunately, this means it can't handle non-convex polytopes.
I've been thinking about some kind of automated faceting algorithm for generating non-convex polytopes, though. Is it true that the hyperplanes of all cells in a non-convex uniform polytope always corresponds with (i.e. parallel to) some element of a convex uniform polytope? Or are there cases where the hyperplanes are non-parallel to all j-faces of any convex uniform polytope? If the former, I may have an easy way of generating non-convex polytopes; if the latter, then how in general do you construct these things?
(In any case, even if I can generate coordinates for non-convex polytopes, it will still take a while before the renderer can actually handle them correctly, since to produce the correct projection images I'd need full 3D volumetric clipping, and right now I don't have a good way of doing that yet.)
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