Manuel Joseph C. Loquias

Colourings of lattices and coincidence site lattices

The relationship between the coincidence indices of a lattice Γ1 and a sublattice Γ2 of Γ1 is examined via the colouring of Γ1 that is obtained by assigning a unique colour to each coset of Γ2. In addition, the idea of colour symmetry, originally defined for symmetries of lattices, is extended to coincidence isometries of lattices. An example involving the Ammann–Beenker tiling is provided to illustrate the results in the quasicrystal setting.

Coincidence isometries of a shifted square lattice

We consider the coincidence problem for the square lattice that is translated by an arbitrary vector. General results are obtained about the set of coincidence isometries and the coincidence site lattices of a shifted square lattice by identifying the square lattice with the ring of Gaussian integers. To illustrate them, we calculate the set of coincidence isometries, as well as generating functions for the number of coincidence site lattices and coincidence isometries, for specific examples.

The coincidence problem for shifted lattices and crystallographic point packings

A coincidence site lattice is a sublattice formed by the intersection of a lattice

Rational digit systems over finite fields and Christol's Theorem

\Gamma

Coincidence indices of sublattices and coincidences of colorings

in

Dually weighted Stirling-type sequences

\mathbb{R}^d

Color groups of colorings ofN-planar modules

with the image of

Symmetries and color symmetries of a family of tilings with a singular point.

\Gamma

Tilings with Singularity and Spiral Tilings Obtained from Archimedean Tilings

under a linear isometry. Such a linear isometry is referred to as a linear coincidence isometry of

Perfect colorings of hyperbolic buckyball tilings

\Gamma