Generalization of Non-periodic Rhomb Substitution Tilings

TL;DR

This paper develops general substitution rules for non-periodic rhomb tilings, allowing for complex constructions including negative area prototiles, expanding the theoretical framework of aperiodic tilings.

Contribution

It introduces a generalized method for deriving substitution rules for non-periodic rhomb tilings, including the use of negative area prototiles.

Findings

Derived substitution rules for non-periodic rhomb tilings.

Established conditions for prototile composition and edge fitting.

Extended the framework to include subtraction tiles with negative area.

Abstract

General substitution rules for non-periodic rhomb tilings are derived. From the requirement that all substitution tiles consist of a discrete number of prototiles, it follows that a substitution tile with angle s*pi/n must be built out of pairs of prototiles with angles (s+t)*pi/n and (s-t)*pi/n, except if t=0. In addition, we require that a discrete number of prototile edges must fit between the beginning and endpoint of the substitution tile edge. By comparing the total area of the discrete number of prototiles constituting the substitution tile and the total area derived from the assumed edge shape, a set of substitution rules is derived. The generalization involves the introduction of prototiles with a negative area or subtraction tiles.