Methods for Calculating Empires in Quasicrystals

Abstract

This paper reviews the empire problem for quasiperiodic tilings and the existing methods for generating the empires of the vertex configurations in quasicrystals, while introducing a new and more efficient method based on the cut-and-project technique. Using Penrose tiling as an example, this method finds the forced tiles with the restrictions in the high dimensional lattice (the mother lattice) that can be cut-and-projected into the lower dimensional quasicrystal. We compare our method to the two existing methods, namely one method that uses the algorithm of the Fibonacci chain to force the Ammann bars in order to find the forced tiles of an empire and the method that follows the work of N.G. de Bruijn on constructing a Penrose tiling as the dual to a pentagrid. This new method is not only conceptually simple and clear, but it also allows us to calculate the empires of the vertex configurations in a defected quasicrystal by reversing the configuration of the quasicrystal to its higher dimensional lattice, where we then apply the restrictions. These advantages may provide a key guiding principle for phason dynamics and an important tool for self error-correction in quasicrystal growth.

Non-Local Game of Life in 2D Quasicrystals

“…The cut-and-project method for generating quasiperiodic tilings from a higher dimensional lattice-the mother lattice-expanded to generating empires in quasicrystals is a well-developed method [17]. We will review here some notions related to the non-local properties of quasicrystals, namely the acceptance domain/QC window and the empire and possibility space windows using the Z 2 lattice and its 1D quasicrystal, the Fibonacci chain, as an example.…”

“…The distribution and frequency of the vertex configurations or local patches are strictly governed by the higher dimensional 'mother lattice' and the manner in which the quasicrystal is generated from it. Each vertex configuration or local patch has an empire [14][15][16][17], a feature unique to quasicrystals representing the totality of all the tiles whose existence and positions are forced by the local patch. Therefore, the local patches propagate under the influence of the empire fields.…”

On the Poincaré Group at the Fifth Root of Unity

“…It has a finite number of prototiles or "letters" as its finite set of symbolic objects and it has a discrete diffraction pattern indicating order but not periodicity. An example of spatiotemporal codes naturally occurring in nature are quasicrystals such as DNA, which Schrödinger called aperiodic crystals [46], and various metallic quasicrystals [47][48][49][50]. Quasicrystalline codes are dynamic geometrical spatiotemporal codes based on the first principles of Euclidean projective geometry.…”

“…All of these quasicrystals can be understood as projections of higher dimensional lattices such as the pure mathematical four dimensional Elser-Sloane quasicrystal [53,54], which is a cut-and-projection of the E 8 lattice. The simplest quasicrystals possible are the 1D class with only two letters or lengths, such as the two length Fibonacci chain [47,48]. The Penrose tiling, a 2D quasicrystal, is a network of 1D quasicrystals.…”

Universal Quantum Computing and Three-Manifolds

“…For doing this, one may use the picture of the Riemann sphere C ∪ ∞ in parallel to that of the Bloch sphere and follow F. Klein lectures on the icosahedron to perceive the platonic solids within the landscape [18]. This picture fits well the Hopf fibrations [19], their entanglements described in [20,21] and quasicrystals [22,23]. However, we can be more ambitious and dress S 3 in an alternative way that reproduces the historic thread of the proof of Poincaré conjecture.…”