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When dealing with sets of polytopes and esp. therefrom derived further polytopes, then some notions are useful. J. Bowers once introduced the army concept:
This concept e.g. is being used for vertex regulars. Or is being used for facetings, when considering some given dimensional embedding space. – Shortly thereafter W. Krieger brought up the dual concept. While the first was based on military terms of land forces, the second similarily got based on navy terms.
That concept intentionally was designed for the dual process of faceting, the stellation.
But when considering facetings, there are occasions, where the navy concept proves suitable too. E.g. the sets of uniform polytopes of Ddimensional space each can be grouped into classes of figures with the same ridge (i.e. their D2dimensional elements) skeletons. In the above terms those would be members of the same ship. For each class one representative (i.e. its comodore) can be chosen as the most convex one possible. Each such class then can be investigated further for other polytopes with uniform facets only, following the constraint that the ridges remain a nonempty subset of the skeleton under consideration. – For D=3 this investigation was subject of the article AxialSymmetrical EdgeFacetings of Uniform Polyhedra of the author in 2002. The then being found polyhedra will be detailed below. So far that work of the author had been for display only on this site, hosted by U. Mikloweit.
So the process of investigation can be outlined as the following:
Within the cited article and within most of the below linked subpages the mentioned further constraint was being chosen to get polyhedra which provide at least an ngonal axial rotation symmetry, where n > 2. The article moreover provides an algorhythm on how to process the above steps in an automated way and even discusses its efficiency.
Before going ahead, we first have to get more explicit on the used terms. A truncation in general chops off some vertex of a given polytope (or does so in a globally symmetrical manner all over at all vertices at the same time) and replaces that very vertex by some sectioning facet underneath. That one happens to be an instance of the corresponding vertex figure. The depth of truncation as such in general remains undefined, if no further restrictions are being considered (such as resulting in a further uniform polytope, i.e. especially one with equally sized edges only). But, even more general, also the tilt of that intersecting hyperplane needs not to be understood as orthogonal to the vertex ray. Now, if the depth and tilt of a truncation is being chosen such that the cutting hyperplane runs through at least some of the vertices of the polytope (that count surely has to be large enough to define the cutting hyperplane uniquely), then this special instance of truncation is called a faceting. If a faceting plane of a polyhedron cuts within edges only, then it is called an edgefaceting. More generally, if a faceting hyperplane of a polytope cuts within ridges only, then it is called a ridgefaceting.
As it is obvious, not all polytopes ensure that the set F(S) is indeed larger than the facet set of the comodore itself. For uniform polyhedra this surplus requires at least that the vertex figure is nontrigonal. But furthermore, some nonnext turning edgecircuit has to exist, which happens to be planar. The uniform polyhedra, which pass both requirements, are given in the following table, grouped by their according ship (here equivalent to: their regiment). Their most convex representant each, i.e. their comodore (here equivalent to: their colonel) is presented on the lefthand side together with the naming codes. These thus give a clue on the possible facet types of that ship. Both furthermore link onto the respective subpage, which then enlists all the possible ridgefacetings. The individual ridgefacetings, as already seen in the central column, then provide instead of those #{p} bits the according counts each (optionally followed by some suffix to distinguish different cases with the same counts).
The picture triples in the central column of the following table (and within all the subpages) for axial polyhedra always provide a top, a side, and a bottomview. For full symmetrical polyhedra those provide a view onto the different symmetry axes each.
Comodore of Ship (linking to subpage)  Example of RidgeFaceting  Counts & Symmetries 
Contained Uniform Polytopes *) already uses a subskeleton only 

oct#{3}#{4} 
oct43 
 
co#{3}#{4}#{6} 
co804 
 
sirco#{3}#{4}#{8} 
sirco495a 
 
gocco#{8/3}#{3}#{4} 
gocco349a 
 
ike#{3}#{5} 
ike106 
 
id#{3}#{5}#{10} 
id2006 
 
srid#{3}#{4}#{5}#{10} 
srid51599 
 
sidtid#{5/2}#{3}#{4}#{5} 
sidtid55150b 
 
siid#{5/2}#{3}#{6}#{10} 
siid610106d 
 
sissid#{5/2}#{3} 
sissid610 
 
did#{5/2}#{5}#{6} 
did610 
 
raded#{5/2}#{4}#{5}#{6} 
raded610610a 
 
gidditdid#{3}#{10/3}#{5}#{6} 
gidditdid106610a 
 
gid#{5/2}#{3}#{10/3} 
gid6103a 
 
gaddid#{5/2}#{3}#{10/3}#{4} 
gaddid151125 
 
gidrid#{5/2}#{3}#{4} 
gidrid244060 

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