Core–Shell and Core–Multi-shell Configurations of the Polyhedra According to the Separation of Faces, and their Interlayer Polytopes

Abstract

In this paper, again inspired by Critchlow, and Grünbaum & Shephard, I apply the 2.5D cubic schema of polyhedra according to the separation of faces and rhombic schema of faces that I have earlier developed to suggest core–shell and core–multi-shell geometries, using Class II of {2,3,4} symmetry as an exemplary case. These might find application in a variety of fields, particularly in nanoarchitecture. The morphology of polyhedra by symmetry class and inclusion of a null element  recognize that each of the 8 Primary Polyhedra (s) of each class consists of facial polytopes (s) that include 0-dimensional (0-D) vertices (1-gons), and 1-D edges (2-gons), as well as 2-D polygons (n-gons), where only those s that are normal to the axes of symmetry are considered principial. Core–shell configurations are developed for pairs of s that share an edge of the cubic schema, by locating the smaller  within the larger , where both are concentric, of unit edge length, and share coaxial negative (−ve), neutral (ntrl), and positive (+ve) axes; in Class II, these consist of facial, edge, and verticial axes of the cube, respectively. Restricting the pairings to those of the shared edges of the cubic schema that is abstracted from the separation of faces reduces the possible cases in each class from 56 to 12, while ensuring their compatibility. The interlayer formed between inner and outer s can then be partitioned into radial prismatic (), pyramidal (), and truncated pyramidal frustum () (cupola) elements of (0, α | β, or 2) frequency/orientation according to the rhombic schema of faces (Fig. 1), where 0 refers to the ;  α | β in the −ve and +ve cases refer to facial rotation (truncation), α  being the facial  of frequency n of the polar ( or ), β being that of the quasi-regular (), and in the ntrl case, α | β refer to the – orientations of ntrl s; and 2 referring to the 2n double frequency case. Inner vertices project to outer vertices, ntrl edges, or n-gons to form 0-s, ntrl 2-s, or n-s, respectively; inner ntrl edges project to outer ntrl edges or squares to form 2-s or 2-s, respectively; and inner n-gons project to outer n-gons or 2n-gons to form n-s or n-s, respectively, while 2n-gons project to 2n-gons to form n-s. These are all radial, on the main symmetry axes, and together fill the interlayer space. The heights of these elements can be derived from the inradii of the concentric PPs, and importantly, show constant increase by gender and axis of the cubic schema. Core-–multi-shell configurations are developed by abstracting 4 or 3 consecutive sequences of s from the cubic schema, thus utilising the core  and/or outer , respectively, and similarly aligning them coaxially and concentrically; each of the 3 or 2 interlayers thus formed being completely filled by the corresponding , , &  elements. The architectures developed might find application to nanoarchitecture, molecular engineering, biochemistry, protein folding, biomedical scaffolds, catalysts, water filters, crystallography, metamaterials, and deployable space structures, quantum theory, and the structure of empirical space.