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This complex polyhedron can be considered to be the mutualStott expansioneither ofx₃-3-o₃-4-o₂ byo₃-3-o₃-4-x₂, or the other way round.Accordingly the new vertex count is just the product of the former vertex count with the facet (in here: edge) count of its vertex figure.(This clearly will be independent of the chosen starting figure.)Faces here are both the faces of either starting polyhedron (shifted a bit out each) plus the mereprism product of the respective edges. The count of the formers clearly gets maintained in this expansion.The count for the additional facet type equates to the count of either of the former's edges.

In order to derive the respective edge counts, one observes that the respective edge counts per vertex are alsothe corresponding vertex counts of thevertex figure. Thus these are the vertex counts of the polytope according to that subsymbol of the current one, when both currently ringed nodes are omitted and then either the first (for the first edge type) or the last one (for the second edge type) gets ringed. (Within the current case of a polyhedron this simply amounts just in the index of the medial node for both cases.)Or, stated differently, its vertex figure happens to be (up to some scalings) just thelace prismx₃|| x₃.

Alternatively this polyhedron also could be viewed as the once morerectified version ofo₃-3-x₃-4-o₂. Then it becomes clear, that itscomplex faces are the duals each of those of the pre-image, and the here additional one then is just the former's vertex figure.Likewise are their total counts each obtained from those of the faces and vertices of the pre-image. Furthermore, as usual for rectifications, the total vertex count is again being obtained from the edge count of the pre-image.Hence this derivation thus requires the further consideration of the previous paragraph as well.

Incidence matrix according toDynkin symbol

x3-3-o3-4-x2.   .   . | 432 |   3   3 |  1   3  1-------------+-----+---------+----------x3   .   . |   3 | 432   * |  1   1  0.   .   x2 |   2 |   * 648 |  0   1  1-------------+-----+---------+----------x3-3-o3   .♦   8 |   8   0 | 54   *  *x3   .   x2♦   6 |   2   3 |  * 216  *.   o3-4-x2♦   6 |   0   9 |  *   * 72