Quasilattices of the Spectre monotile

TL;DR

This paper explores how point decorations of the Spectre monotile can generate a wide variety of non-periodic quasilattices, offering a new approach to designing physical potential landscapes with controllable properties.

Contribution

It introduces a lattice generating function that maps point decorations to quasilattices, revealing new non-periodic structures beyond traditional Bravais lattices.

Findings

Point decorations support diverse non-periodic quasilattices.

Some lattices derive from nearest-neighbor distances.

Near-periodicity observed in certain projections.

Abstract

The Spectre is a family of recently discovered aperiodic monotiles that tile the plane only in non-periodic ways, and novel physical phenomena have been predicted for planar systems made of aperiodic monotiles. It is shown that point decorations of Tile(1,1), the base tile for all Spectres, supports the generation of a large variety of non-periodic quasilattices, in contrast to Bravais-lattices in which all point decorations would be periodic. A lattice generating function is introduced as a mapping from point decorations to quasilattice space, and investigated systematically. It is found that some lattices result from the properties of nearest-neighbor distances of point decorations, and that other lattices show near-periodicity in projections along one of the symmetry axes of the tiling. It is concluded that the lattice generating function can serve as a template for the design of physical potential landscapes that can be controlled by the point decoration as a parameter.