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In their search for abstract polytopes even the restriction to dyadicity was set apart by Shephard (1952) and Coxeter (1974). Today the outcome of their finds usually is connoted as complex polytopes. This is because e.g. an n-edge, i.e. a somehow 1-dimensional object (edge) with n 0-dimensional subelements (vertices) can easily be represented within the 1-dimensional complex "line" as the convex hull of the n roots of unities, or, in its 2-dimensional real equivalent, the Argand plane, as a convex regular n-gon. Thus indeed there is a realization of these more general abstract polytopes which is dimension respecting.
Despite the formal geniality of the complex description, there is a much more visual approach to these figures too, which moreover is quite close to Coxeter's original description of these figures. It is more in the spirit of deriving maps or skew polytopes from other polytopes. In fact the above already mentioned identification of n-dimensional complex space and the 2n-dimensional real space are being used here. Thus those complex polytopes also can be described as some selection of even dimensional elements only in a corresponding (real space) polytope (of twice the total dimension). Both, the connection to these describing (real space) polytopes and the to be applied selection of even dimensional elements here becomes of interest on its own right.
Zero-dimensional complex polytopes (or according elements of larger polytopes) clearly are zero-dimensional real elements too (as two times zero still equates to zero). Thus the realization of those vertices remain mere classical geometric points, both within complex and within real space considerations. Nothing special here. 1-dimensional elements (edges) however start to fall apart. Those have been described already in the introductory lines above. Thus the remainder will be devoted to higher dimensional complex polytopes.
Within Coxeter's description these are denoted p{q}r and represent an abstract "q-gon" (the understanding of that number still has to be given) which uses p-edges, i.e. "edges" with p incident vertices each, and having r such edges incident to every vertex. So, within the usual realm of real polygons, where dyadicity always holds, we clearly have p = r = 2 and therefore 2{q}2 is nothing else as the usual polygon {q}.
This polytope can be understood as the selection of the p-gons from the p-gonal duoprism. In fact, these (real space) p-gons represent the (complex polytopal) p-edges, i.e. "edges" with p incident vertices, and moreover there are indeed exactly 2 such p-edges incident to every vertex.
Obviously its incidence matrix thus is
p² | 2 ---+--- p | 2p
Note that this realization supresses the toroidal skin of the p-gonal duoprism, i.e. all its squares. So we have (real) 4-gonal pseudo faces all over.
This polytope, aka the Möbius-Kantor polygon, can be understood as the selection of 8 out of the 32 triangles of hex. Within the projection of hex, given at the right, these mentioned 8 triangles are being represented as those 8 which are projected undeformed (up to mere total scalings), i.e. the small ones with red and the large ones with blue outlines. In fact, these triangles represent the 3-edges, i.e. with 3 incident vertices, and moreover there are indeed exactly 3 such 3-edges incident to every vertex.
Obviously its incidence matrix thus is
8 | 3 --+-- 3 | 8
But how is the above mentioned projected selection being lifted back into the pre-image? – The vertex figure of hex, i.e. oct, can be considered as trigonal antiprism, having the bases red (r), and additionally the lateral up-edges of the girthing zigzag colored red too. Being rolled onto an other face, the same can be done for yellow (y), green (g), and blue (b), thus coloring all edges and faces exactly once. Any face, together with all its 3 edges, thus encompasses all 4 colors each. – Being a vertex figure, the oct-coloring of edges represents the colors of vertex incident faces of hex, whereas that of oct faces represents that of the opposite faces of the vertex incident cells (tet). Thus all these vertex incident tets are 4-colored. Further, this scheme can be extended to all cells of hex consistently.
Any single color of this coloring scheme would provide the above mentioned subset of 8 out of 32. Indeed, the 3 lateral up-edges of the girthing zigzag on the vertex figure represent the 3 vertex-incident triangles (3-edges) of the selected color. Note that this realization clearly supresses all the triangles of all the not selected colors. So we have (real) 3-gonal pseudo faces all over.
This polytope can be understood as the selection of 24 out of the 72 squares of gico, i.e. the compound of 3 teses within an ico. Note that both use the same (real) edge skeleton. In fact, the squares of the former are nothing but the diametrals of the oct cells of the latter. Within the projection of ico, given at the right, these mentioned 24 squares are being represented as those 24 which are projected undeformed (up to mere total scalings), i.e. the ones with red, with yellow, with green, and with blue outlines. In fact, these squares represent the 4-edges, i.e. with 4 incident vertices, and moreover there are indeed exactly 4 such 4-edges incident to every vertex.
Obviously its incidence matrix thus is
24 | 4 ---+--- 4 | 24
In order to understand the lifting of this selection we present at the right both the projection of any of the inscribed teses (1^{st} pic) and the (real) 4-dimensional realization of the complex 2-dimensional polygon 4{4}2, i.e. the 4-gonal duoprism or tes (2^{nd} pic). As any vertex of ico belongs to exactly 2 teses of gico we thus have 4 colored (real) squares (i.e. complex 4-edges) per vertex.
Note furthermore that the holes in this encasing ico are all its (real) 3-gonal pseudo faces.
This polytope can be understood as the selection of 24 out of the 96 triangles of ico. Within the projection of ico, given at the right, these mentioned 24 triangles are being represented as those 24 which are projected undeformed (up to mere total scalings), i.e. the ones with red, with green, and with blue outlines. The latter ones even are shown filled. (Note that this coloring is applied only for the ease of recognition here and will have nothing to do with the 4-coloring mentioned below!) It should be emphasized here, that the thus to be selected faces only provide 3 times 24, i.e. just 72 out of the 96 (real) edges. Only those have been shown in this projection pic. – These 24 triangles then represent the 3-edges, i.e. with 3 incident vertices, and moreover there are indeed exactly 3 such 3-edges incident to every vertex.
Obviously its incidence matrix thus is
24 | 3 ---+--- 3 | 24
The lifting of this selection can be understood by a 4-coloring of ico. We start with the 4-coloring of oct: simply color opposite faces alike. This scheme can be extended to all cells of ico consistently using the obvious color matching rule of rotated copies only. In fact, diametrically opposite octs at a vertex figure then will be nothing but a shifted copy with an axially skrew quarter turn. – The required subset then is the selection of just one of those colors from these subsets of triangles.
Further we note that 3 such octs and thus 3 differently colored triangles will be incident at each edge of ico. Therefore each edge could be colored as well easily by applying the missing 4^{th} color. And when turning to the vertex figure, which is a cube, then it becomes evident that the edges of such a cube correspond to the vertex-incident triangles of the ico. So those 12 edges each are colored accordingly, 3 from each color. In this coloring of the cube the 3 of either single color then happen to be the alternating ones of its Petrie polygon. (The remaining ones of that very Petrie polygon are all the other colors, one each.)
The comparision of the here being displayed projection pictures each, of 4{3}4 (all edges) and that of 3{4}3, exhibits that in the latter case exactly 1 (real) edge of each selected square of the former has been omitted. Moreover it appears that those omitted edges form 4 complete hexagons, which have no vertex in common, and thus in total visit each vertex exactly once. And indeed such hexagons occur as flat equatorial pseudo faces within each co, which in turn is the flat equatorial pseudo facet of ico. In fact there are 16 hexagonal great circles in the edge set of ico. These fall into 4 sets of 4 with disjoint vertices. These simply correspond to the 4-coloring of the edges mentioned in the former paragraph.
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