Polytope Typology B: Core–Shell, Core–Multi-shell, and Interlayer Configurations of the Polyhedra According to the Separation of Faces

In this paper, I apply my 2.5D cubic schema of polyhedra by the separation of faces (SoF), and rhombic schema of faces (RSoF), to generate core–shell and core–multi-shell geometries for Class II of {2,3,4} symmetry, with respect to the interlayer cells they generate. This morphology of polyhedra by symmetry class with the inclusion of a null element VP recognizes 8 Primary Polyhedra (PPs) (or Primary Polytopes, PTs) for each of 5 classes, 3×3D, and 2×2D. Each PP (or PT) consists of facial polytopes (FTs) that are considered to include 0-dimensional (0D) vertices (1-gons), and 1D edges (2-gons), as well as 2D polygons (n-gons), considering as principial only those PTs that lie normal to (or vertices (VTs) lying on) the axes of symmetry. By locating the smaller PP within the larger PP, core–shell configurations are then developed for pairs of concentric PPs that share an edge of the cubic (zonahedral) schema, where both are of unit edge length, and coaxial, sharing common negative (−ve), neutral (ntrl), and positive (+ve) axes. In the exemplary Class II, these conveniently correspond to the facial, mid-edge, and verticial axes of the cube ((100), (110), & (111)), respectively. Restricting the pairings to the shared edges of the cubic schema abstracted from the SoF reduces the possible cases in each class from 56 to 12, so ensuring their compatibility. The interlayer between inner and outer PPs is then partitioned into radial prismatic (PRS), pyramidal (PYR), and truncated pyramidal ‘frustal’(TFM) (i.e., cupola) elements of (0, α | β, or 2) frequency/orientation according to the FT (Fig. 1), where 0 refers to the VT; α | β in the −ve and +ve cases to facial rotation (truncation), α being the FT of frequency n of the polar (OH or CB), β of the quasi-regular (CO), and in the ntrl case, α | β refer to the PL+–PL– orientations of ntrl edges (EG)s; and 2 refers to the 2n double frequency case. Inner VTs project to outer VTs, NEs, or n-gons to form 0-PRSs, ntrl 2-PYRs, or n-PYRs, respectively; inner NEs project to outer NEs or squares (SQs) to form 2-PRSs or 2-TFMsm respectively; and inner n-gons project to outer n-gons or 2n-gons to form n-PRSs or n-TFMs, respectively, while 2n-gons project to 2n-gons to form n-PRSs. These are all radial, on the main symmetry axes, and combine to fill the interlayer space. The heights of these elements are derivable from the inradii of the concentric PPs, and show constant increase by gender and axis of the cubic schema. Core–multi-shell configurations are also developed, by abstracting 4 or 3 consecutive sequences of coaxially aligned, concentric PPs from the cubic schema, thus using the core VP and/or outer GR, respectively; each of the 3 or 2 interlayers thus formed is completely filled by the corresponding PRS, PYR, and TFM elements. The geometries developed might find application to nanoarchitecture, e.g., of electrode catalysts, and in space structures.