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European Journal of Applied Sciences – Vol. 11, No. 1

Publication Date: January 25, 2023

DOI:10.14738/aivp.111.13783.

Meurant, R. C. (2023). Core-Shell and Core–Multi-shell Configurations of the Polyhedra According to the Separation of Faces, and

their Interlayer Polytopes. European Journal of Applied Sciences, 11(1). 94-120.

.

Services for Science and Education – United Kingdom

Core–Shell and Core–Multi-shell Configurations of the Polyhedra

According to the Separation of Faces, and their Interlayer

Polytopes

Robert C. Meurant

Director Emeritus, Institute of Traditional Studies; Adjunct Professor, Seoul National

University PG College of Eng.; Exec. Director, Research & Education, Harrisco Enco

4/1108 Shin-Seung Apt, ShinGok-Dong 685 Bungi, Uijeongbu-Si, Gyeonggi-Do 11741,

Republic of Korea

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 VP

recognize that each of the 8 Primary Polyhedra (PPs) of each class consists of facial

polytopes (PTs) 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 PTs that are normal to

the axes of symmetry are considered principial. Core–shell configurations are

developed for pairs of PPs that share an edge of the cubic schema, by locating the

smaller PP within the larger PP, 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 PPs

can then be partitioned into radial prismatic ( PRS ), pyramidal ( PYR ), and

truncated pyramidal frustum ( TFM ) (cupola) elements of (0, α | β, or 2)

frequency/orientation according to the rhombic schema of faces (Fig. 1), where 0

refers to the VT; α | β in the −ve and +ve cases refer to facial rotation (truncation),

α being the facial PT of frequency n of the polar (OH or CB), β being that of the

quasi-regular (CO), and in the ntrl case, α | β refer to the PL+–PL– orientations of

ntrl EGs; and 2 referring to the 2n double frequency case. Inner vertices project to

outer vertices, ntrl edges, or n-gons to form 0-PRSs, ntrl 2-PYRs, or n-PYRs,

respectively; inner ntrl edges project to outer ntrl edges or squares to form 2-PRSs

or 2-TFMs, 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 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 PPs from the cubic schema, thus utilising the core VP

and/or outer GR , respectively, and similarly aligning them coaxially and

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Meurant, R. C. (2023). Core–Shell and Core–Multi-shell Configurations of the Polyhedra According to the Separation of Faces, and their Interlayer

Polytopes. European Journal of Applied Sciences, 11(1). 94-120.

URL: http://dx.doi.org/10.14738/aivp.111.13783

concentrically; each of the 3 or 2 interlayers thus formed being completely filled by

the corresponding PRS, PYR, & TFM 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.

Keywords: separation of faces, structural morphology, order of polyhedra, core-shell,

core-multi-shell, nanostructure

THE RHOMBIC SCHEMA AND THE SEPARATION OF FACES IN THE CUBIC SCHEMA

Inspired by the work of Critchlow [1], and Grünbaum & Shephard [2], and following my much earlier

research [3], the rhombic schema that I have earlier developed [4] shows the progression that

faces undergo as the steps of the cubic schema progress from VP to GP (Fig. 1). The 0-faces are

vertices (VTs); they first progress to α or β faces; −ve & +ve α faces are the (non-verticial) faces

of the correspondingly gendered PL, while −ve & +ve β faces are the corresponding faces of the

QR, and are rotated (truncated) versions of either α face; ntrl α & β faces are the −ve & +ve ntrl

faces (edges) of the −ve & +ve PL, respectively.

In Class II, the −ve face is the SQ of the CB, the +ve is the TR of the OH; the ntrl α & β are the

edges (EGs) of the OH and CB, respectively. The 2-faces are the double frequency case of the

α|β face; in Class II, the −ve, ntrl, & +ve OG, NS, & HX of GR (or of TC, SR, & TO), respectively.

Figure 2 shows the cubic schema that I have earlier developed [4] in horizontal (upper) and

bird’s eye view (lower) for Class II of {2,3,4} symmetry; it generalizes to all 5 symmetry classes.

This schema, which is abstracted from the separation of faces, is critical to exploring the

morphology of the polyhedra as they stand in relation to one another, and is central to evincing

the various core–shell and core–multi-shell configurations of this paper. Each (gendered) face

appears twice in the cubic schema progression; it first appears at spacing of adjoining faces of

d=0; then, by the separation of faces, appears at spacing of adjacent faces of d=1.

Pairs of the same kind of faces, if adjoining, share a common ntrl VT (NV) or EG (NE); if

adjacent, are separated by a NE or ntrl square face (NS). When a PP transforms to another PP

in the vertical progression from VP to GP, along the edges of the cubic schema, of the three

genders (−ve / ntrl / +ve), the face of one gender undergoes the separation of faces, so those

faces remain of the same kind (0, α or β, 2), but the distance apart of adjoining pairs increases

by unit (edge) distance to become adjacent pairs.

Meanwhile, the faces of the other two genders transform to the next higher state on the rhombic

schema. Thus the cubic schema is developed from two rhombic schema (Fig. 1, right), a lower

d=0 case of adjoining faces, and an upper d=1 case of adjacent faces. This corresponds to a front

horizontal view of the cubic schema.

The cubic schema of two overlaid rhombic schema represents the case for the ntrl faces,

presenting the standard horizontal view of the cluster of PPs. Figure 3 shows that rotating the

PP cluster by +2π/3 about the vertical VP– GR axis, where individual PPs need also to rotate

individually in unison to present the proper face, presents the corresponding schema for the

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European Journal of Applied Sciences (EJAS) Vol. 11, Issue 1, January-2023

Services for Science and Education – United Kingdom

+ve faces; rotating by −2π/3 about the vertical VP–GR axis, it presents the schema for the −ve

faces.

Therefore, all cases of concentric aligned inner PP and outer PP are accommodated, as the

corresponding inner and outer faces accord with the schema; and thus the interlayer cells

described between inner and outer face are either prisms for the separation of faces case, of

VT→VT, α→α or β→β, or 2→2 PRS; or they are transformations of NV→NE, VT→α, VT→β;

NE→NS; or α →2, β →2, which form 2-PYR, α-PYR, β-PYR; 2-TFM; or α-TFM, β-TFM,

respectively.

Therefore the only kinds of interlayer cells are PRS, PYR, and TFM, in which the 0-PRS is a

virtual spike, 2-PYR is an isosceles TR, 2-TFM is a gable, and ntrl α→α and β→β are 2-PRSs, i.e.,

2-gon edge prisms (of different orientation).

All cells are radially oriented, coaxial to the symmetry axes. This presumes that the inner and

outer PP pairs are directly related (share an edge) on the cubic schema. Figure 1 shows these

various cases in plan at the mid-points of the edges of the rhombic schema for each gender

(left), see also Figs. 3 & 4.

Neutral interlayer cells α-PYR & β -PYR are 2D isosceles TRs that span between inner NV and

outer NE, while α-TFM & β-TFM are gables spanning between NE & NS.

Angled edges are associated α|β PYRs from 0 faces (VT), or α|β TFMs to 2 faces (OG, NS, HX by

gender). Faces of each gender migrate thru cubic schema from VP to GR through PRS, PYR, &

TFM in various orders.

20 node lies concealed behind 01 node at center; 2- PRS spans between inner

20 and outer 21 faces.

Interlayer cells α & β PYR & TFM are inserted at mid-points of the rhombic schema between

inner and outer faces; meanwhile, PRS can be imagined at each node of the generic rhombic

schema.

Interlayer cells prs, pyr, tfm start from vp with prs or pyr, End with prs or tfm to gr.

d=1 rhomb adjacent

d=0 rhomb adjoining

Vertical edges show separation of faces, with associated (0, α | β, and 2) PRS s; angled

edges are corresponding transformations of faces.