# Multi-shell Polyhedral Clusters

When studying materials at a nanoscale, some well structured lattices can be observed. The hexagonal structure of graphene is a well known two-dimensional carbon structure that can live in a three-dimensional world in the form of a nanocone or a nanotube. The Buckminsterfullerene or C${}_{60}$ which is a dodecahedron is a simple example of a closed surface. Also three- or higher-dimensional structures with strict topological geometry have practical applications. Think of a cube which has 8 atoms on its vertices and add atoms at the body center and at the center of the 6 faces and the midpoints of the 8 edges and this will give a hyper-structure with 27 atoms. The cube is divided into 8 sub-cubes but this is easily generalized to a structure consisting on $n^3$ sub-cubes. Hence in mathematical chemistry, an extensive literature emerged that investigated the topological properties of these nanostructures. The simplest ones are the Platonic solids: tetrahedron (T), cube (C), octahedron (O), dodecahedron (D), and icosahedron (I). They are represented by undirected three-dimensional graphs, assuming atoms at the place of the vertices and the edges representing chemical bonds. Plato identifying earth, air, water, fire, and ether with cube, octahedron, icosahedron, tetrahedron, and dodecahedron respectively. Kepler in his *Mysterium Cosmographicum* used a nesting of the Platonic solids to model the position of the planets in the solar system. These five polyhedra are now revived as basic building blocks in these nanostructures. All kinds of maps can be applied to them to form more complex blocks from which much more complicated constellations can be composed. This book wants to describe some of the structures that can be obtained and tabulate their topological properties. It provides some kind of atlas for particular sets of these structures.

To describe all the complex structures, some introductory chapters are needed to give the necessary definitions from graph theory and of the topological indices of these graphs. The second chapter introduces operations on the elementary structures which will be the main tools to construct the more complicated ones. Some examples of these transforms: the *dual* of a graph exchanges the role of faces and vertices; the *median* of a graph takes as vertices the midpoints of the original edges and connects a pair if they belong to originally adjacent edges; and a *truncation* cuts off the vertices of the polytope by a plane that intersects all its incident edges; and there are more complicated operations possible like stellation, snub, leapfrog, etc. . The result is an atlas of single shell structures.

The third chapter defines how more complex constellations can be formed by adding more shells to the structure. For example, one could add a vertex at the body center of a polyhedron and connect it to the surrounding vertices (the P-centered clusters). Other examples are the cell-in-cell clusters that place a polyhedron inside another polyhedron and connect the nearby vertices of inner and outer cell. Or there can be abstract structures like the 24-cell (a four-dimensional generalization of a Platonic solid).

Depending on what property one is interested in, different notations exist in the literature (a Schläfli symbol like {*p,q*} or {*p,q,r*}, Conway's notation, Coxeter diagrams etc.), this can already be confusing, but unfortunately the author has to add another one for the more complex structures. The author is of course not a mathematician, but it is somewhat regrettable that there is not a strict formal definition of the notation in its most general form. One should try to grasp the meaning from the many examples. Another unfortunate fact is that the operations and the more complicated structures get different names in the literature, although median, snub, and stellation are pretty standard. When these are used as abbreviations in the formal notation this can be confusing and so some familiarity with the different nomenclatures is advisable.

Chapter 4 is the last of the "introductory" chapters and introduces symmetry and (structural) complexity, which can be measured by several indices like Euler characteristic, centrality and ring signature. Also for translational and spongy structures and other structure generating techniques such parameters can be computed.

Chapters 5-11 form the main part of the book and describe collections (they form an atlas) of several clusters that are based on the icosahedron, octahedron, tetrahedron, dodecahedron, or constellations like multi-tori and spongy hypercubes. The last chapter 12 requires a bit more (carbon based) chemistry and considers structures with C${}_{20}$ (dodecahedron), or C${}_{60}$ (truncated icosahedron) or D${}_5$ configurations.

Each chapter starts with a short introduction, with some hints on the notation and tables that contain all the so-called figure counts (number of vertices, edges, and faces of successive order, the rank of the structure and the Euler characteristic. Then enlarged pictures of the graphs, one per page, visualize the structure, but they become quickly hopelessly complicated when the structure is a bit more complex, even when they are multi-coloured, it is often hard to distinguish the nested layers of edges inside the cage.

Each chapter has a long list of references, many of which are by the author. Some chapters correspond for a large amount with one of his papers. For example the discussion of the ring structure index in chapter 4 is largely overlapping with the paper C.L. Nagy, M.V. Diudea, Ring Signature Index, in *MATCH Commun. Math. Comput. Chem.* **77** (2017) 479-492. It may in fact help to look up some of these papers because the text is really telegraphic, and is clearly a compilation of previous results, and thus not always explaining all the details. Therefore, I consider it is a reference text for the specialist, but I would not recommend it as a first reading on the subject of nanoclusters.

The book is number 10 in the Springer series *Carbon Materials: Chemistry and Physics*. Diudea is a very prolific writer in this area. He was co-editor of two other books in the series: *Diamond and Related Nanostructures* (vol. 6, 2013) and *Distance, Symmetry, and Topology in Carbon Nanomaterials* (vol. 9, 2016). Perhaps because of the pressure to publish on the very quickly evolving subject, the quality of English and mathematical editing of this book could have been much better. For example at several places articles are missing (the structure of entire polytope, p,45) or in excess (however at the Plato's time, p.125) or typos produce words that are just wrong (inconsistences, p.39, convex hall, p.42); names are misspelled (Platon p, 77; Hässe, p.45); references are wrong (reference to graph 1.2.4 should be 1.2.2, p.7, Fig. 4.4 refers to top and bottom figures but there is no bottom figure, p.67). For a chemist, this might be nitpicking but it will irritate many a mathematician that variable names are inconsistent ($s_1$ and $S_1$ for the same operation on one line, p.29, and on p.34: $p_4T,P_4(C),p_4(D)$ are three different notations for the same operation $p_4$ on one line); roman and math font are mixed (header of two successive tables 4.3 and 4.4: once in roman, once in italic, p.66); symbols and terms are not defined or used before they are defined (I do not find the meaning of $f_n$ defined, but it probably means an $n$-gon, if so, then the headers $f_5$ and $f_6$ in table 3.2 should be $f_4$ and $f_5$, p.47, in the same table the meaning of $c_n$ and $M$ are not explained, chapter 1 uses RS and CS for row-sum and column-sum, without telling, while in chapter 4, RS is ring signature chapter 2 uses P for polytope or platonic solid, but in the atlas (p.32) it is a prism, in the atlas symbols for rhombic (Rh), antiprism (A), pyramid (Py) were not explained before. Notations like $(3.4)^2$ (fig.2.3), $5.6^2$ (Thm. 2.1) and similar ones on page 33 were used without explanation, since the definition of ring signature follows only in chapter 4); sentences like "Computations at a higher level of theory: Hartree-Fock and DFT have been performed with the HF/6-31G(d,p), B3LYP/6-31G(d), B3LYP/6-31G(d,p) and LDA/3-21G(d) sets, on Gaussian 09 (Frisch et al. 2009). PM6 computations were done with the VSTO-6G(5D;7F) set." (p.400) are very cryptic when the abbreviations are not explained; and most unfortunately, the figures, a prominent feature of this book, are not always helpful (the atlas is like a picture book of all these clusters, and while the graphics are very useful for simple structures, they soon have too many edges in the more complex ones to make anything clear); another typesetting glitch: the text explaining the figure at the bottom of page 31 is on top of page 32.

Even though this may not be the best place to start, I think the subject is a very interesting one where there is work for mathematicians. It is the resurrection of a subtopic of crystallography 2.0 and graph theory requiring somewhat more geometrical (and chemical) insight than just studying the symmetry groups, but it is simpler in a sense because only topological properties are needed, which means that the structure is completely characterized by the 0-1 adjacency matrix.

**Submitted by Adhemar Bultheel |

**19 / Mar / 2018