This came o=up in another thread: How do you find the convex hull of a general uniform polychoron?
Let's start with the basics: THe pennics are easy - since the only pennic regiments are those of the conex pen truncates, and all polytopes in the same regiment have the same convex hull, those are easy.
THe tessics I've actually thought about. It's not that hard to determine vertex coordinates for the non-convex tessics, based on the cells they have in the 8 positions corresponding to the cells of tes. From there, determining the general shape of the convex hull is easy by just looking at the coordinates - the general rule is, for instance the uniform polychoron prit has vertex coordinates ((1+2sqrt(2))/2, (1+sqrt(2))/2, 1/2, 1/2) all permutations and changes of sign - any thing with al permutations/sign changes of (a,b,c,c) (provided a > b > c > 0) will be a variation of prit. I'll list the coordinates of the non-convex tessic colonels below, assume all permutations/sign changes:
Quitit: ((sqrt(2)-1)/2, (sqrt(2)-1)/2, (sqrt(2)-1)/2, 1/2) - hull is 1/w scaled sidpith
Gittith: ((sqrt(2)-1)/2, 1/2, 1/2, 1/2) - hull is 1/w-scaled tat
Wavitoth: ((sqrt(2)-1)/2, (sqrt(2)-1)/2, 1/2, 1/2) - hull is 1/w-scaled srit
Gaqrit: ((2sqrt(2)-1)/2, (2sqrt(2)-1)/2, (sqrt(2)-1)/2, 1/2) - hull is grit variant
Thatoth: ((1+sqrt(2))/2, (1+sqrt(2))/2, 1/2, (sqrt(2)-1)/2) - hull is grit variant
Thaquitoth: ((1+sqrt(2))/2, 1/2, (sqrt(2)-1)/2, (sqrt(2)-1)/2) - hull is prit variant
Quiproh: ((2sqrt(2)-1)/2, (sqrt(2)-1)/2, (sqrt(2)-1)/2, 1/2) - hull is prit variant
Gichado: ((2sqrt(2)-1)/2 (sqrt(2)-1)/2, 1/2, 1/2) - hull is proh variant
Skiviphado: ((1+sqrt(2))/2, 1/2, 1/2, (sqrt(2)-1)/2) - hull is proh variant
Gaquidpoth: ((3sqrt(2)-1)/2, (2sqrt(2)-1)/2, (sqrt(2)-1)/2, 1/2) - hull is gidpith variant
Thatpath: ((1+2sqrt(2))/2, (1+sqrt(2))/2, 1/2, (sqrt(2)-1)/2) - hull is gidpith variant
Thaquitpath: ((1+sqrt(2))/2, 1/2, (sqrt(2)-1)/2, (2sqrt(2)-1)/2) - hull is gidpith variant
I'm not sure what the exact edge ratios of these hulls are, but there's probably some way to determine them fromt he vertex coordinates.
I don't know how to apply this method to the rest of the uniforms - the icoics and hyics can't be represented like this. And then there's the really strange ones (Bowers's "miscellaneous" category, plus the big snub regiments in categories 27-29 on his website) that I have no clue what their hulls woul look like - at least for the Wythoffian ones I can assume they'll all have Wythoffian hulls. Anyone know how to figure out these convex hulls?