From the Fibonacci Icosagrid to E8 (Part I): The Fibonacci Icosagrid, an H3 Quasicrystal

Abstract

This paper introduces a new kind of quasicrystal by Fibonacci-spacing a multigrid of a certain symmetry, like H2, H3, T3, etc. Multigrids of a certain symmetry can be used to generate quasicrystals, but multigrid vertices are not a quasicrystal due to arbitrary closeness. By Fibonacci-spacing the grids, the structure transit into an aperiodic order becomes a quasicrystal itself. Unlike the quasicrystal generated by the dual-grid method, this kind of quasicrystal does not live in the dual space of the grid space. It is the grid space itself and possesses quasicrystal properties, except that its total number of vertex types are not finite and fixed for the infinite size of the quasicrystal but bounded by a slowly logarithmic growing number. A 2D example, the Fibonacci pentagrid, is given. A 3D example, the Fibonacci icosagrid (FIG), is also introduced, as well as its subsets, the Fibonacci tetragrid (FTG). The FIG can be thought of as a golden composition of five sets of FTGs. The golden composition procedure is another way to transit a random structure into aperiodic order, and the associated rotational angle is the same as the angle that resolves the geometric frustration for the H3 tetrahedral clusters. The FIG resembles another quasicrystal that is the same golden composition of five quasicrystals that are cut and projected and sliced from the E8 lattice. This leads to further exploration in mapping the FIG to the E8 lattice, and the results will be published following this paper.

From the Fibonacci Icosagrid to E8 (Part II): The Composite Mapping of the Cores

“…For non-crystallographic point symmetries such as H 2 (pentagonal) and H 3 (icosahedral), the regular grids have intersection points that are arbitrarily close to each other, but the Fibonacci spacing resolves this to provide finite minimal spacing, and therefore a discrete Fourier spectrum. In Part I of this series, Fang and Irwin have studied an H 3 version [21,22], referred to as the Fibonacci icosagrid (FIG), and have shown that it is closely related to sections of the Elser-Sloane quasicrystal (ESQC), a 4D quasicrystal derived by direct cut-and-project from the E 8 root lattice [23].…”