# Pentagonal chains and annuli as models for designing nanostructures from cages

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## Abstract

Carbon is the most versatile of chemical elements in combining with itself or other elements to form chains, rings, sheets, cages, and periodic 3D structures. One of the perspective trends for creating new molecules of nanotechnological interest deals with constructs which may be formed by chemically linking of cage molecules. The growing interest in fullerene polyhedra and other molecules with pentagonal rings raises also a question about geometrically consistent in \({\mathbb {E}}^{3}\) nanoarchitectures which may be obtained by aggregating many such molecules. Simple examples are chains and rings assembled from pyramidal (car)borane subunits. Adequate geometrical models of such objects are a chain and an annulus built from regular pentagons wherein any two adjacent pentagons share an edge. Among arising combinatorial problems may be both analytical and constructive enumeration of such chains and annuli drawn in plane with no two edges crossing each other. This may also employ several mathematical disciplines, such as geometry, (spectral) graph theory, semigroup theory, theory of fractals, and others. We discuss some practical approaches for solving the mentioned mathematical problem.

## Keywords

Cage molecule (Car)borane Nanoarchitecture Pentagonal chains and annuli Constructive and analytical enumeration Quasicrystal## Notes

### Acknowledgments

We thank very much the reviewers for their helpful remarks made on this manuscript. VRR acknowledges the support of the Ministry of Absorption of the State Israel (through fellowship “Shapiro”). This work was partially supported by Russian Foundation for Basic Research (grant 16-01-00499). JMO acknowledges support from CSIC through project COOPB20040. JR is supported by the FP7-PEOPLE-2013-CIG project CountGraph (ref. 630749) and the Spanish MICINN projects MTM2014-54745-P and MTM2014-56350-P.

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