Fang Fang, Klee Irwin (2016)The construction of an icosahedral quasicrystal and a quasicrystalline spin network are obtained by spacing the parallel planes in an icosagrid with the Fibonacci sequence. This quasicrystal can also be thought of as a golden composition of five sets of Fibonacci tetragrids. We found that this quasicrystal embeds the quasicrystals that are golden compositions of the three-dimensional tetrahedral cross-sections of the Elser-Sloane quasicrystal, which is a four-dimensional cut-and-project of the E8 lattice. These compound quasicrystals are subsets of the quasicrystalline spin network, and the former can be enriched to form the latter. This creates a mapping between the quasicrystalline spin network and the E8 lattice.
An Icosahedral Quasicrystal and E8 derived quasicrystals
F. Fang∗ and K. Irwin
Quantum Gravity Research, Los Angeles, CA, U.S.A.
W
e present an icosahedral quasicrystal, a Fibonacci icosagrid, obtained by spacing
the parallel planes in an icosagrid with the Fibonacci sequence. This quasicrystal
can also be thought as a golden composition of five sets of Fibonacci tetragrids.
We found that this quasicrystal embeds the quasicrystals that are golden compositions of
the three-dimensional tetrahedral cross-sections of the Elser-Sloane quasicrystal, which
is a four-dimensional cut-and-project of the E8 lattice. These compound quasicrystals
are subsets of the Fibonacci icosagrid, and they can be enriched to form the Fibonacci
icosagrid. This creates a mapping between the Fibonacci icosagrid and the E−8 lattice.
It is known that the combined structure and dynamics of all gravitational and Standard
Model particle fields, including fermions, are part of the E8 Lie algebra. Because of this,
the Fibonacci icosagrid is a good candidates, for representing states and interactions
between particles and fields in quantum mechanics. We coin the name Quasicrystalline
Spin-Network (QSN) for this quasicrystalline structure.
I. Introduction
Until Shechtman et al.1 discovered them in nature,
quasicrystals were a pure mathematical curiosity, for-
bidden to exist physically by the established rules of
crystallography. This discovery intrigued scientists from
verious disciplines2–5. First there was a surge in the
interest in studying the mathematical aspects of qua-
sicrystals6–11. Then in recent years, the focus of the
majority of research in the field has shifted toward the
physical aspects of quasicrystals, i.e. their electronic
and optical properties12,13 and quasicrystal growth14.
As a field, quasicrystallography is still very new. The
interesting mathematics that accompanies the field is
relatively unexplored and provides opportunity for dis-
coveries that could have far reaching consequences in
physics and other disciplines.
In thi