Growth of a One Dimensional Quasiperiodic Covering with Locally Determined Decorations

TL;DR

This paper introduces a local growth rule for one-dimensional quasiperiodic structures using decorated tiles, demonstrating how perfect quasiperiodic order can emerge through local adjustments, offering insights into quasicrystal growth.

Contribution

It presents a novel local covering rule for growing perfect 1D quasiperiodic structures with decorated tiles, advancing understanding of quasicrystal formation.

Findings

Growth of perfect quasiperiodic structures is achievable with local rules.

Decorated tiles with movable string positions enable quasiperiodic order.

The method provides insights into the local mechanisms of quasicrystal growth.

Abstract

A growth mechanism for a perfect one-dimensional (1D) quasiperiodic structure is presented with a local covering rule. We use rectangular tiles with two different types of string decorations. The string position in a tile is allowed to move when the tile is attached to an existing patch. By adjusting the position properly with local information, we show that a growth of perfect quasiperiodic structure is possible. This observation may provide new insight into how quasicrystals grow with perfect quasiperiodic order.