Converting tilings with multiple types of rhombuses to pentagonal tilings

TL;DR

This paper presents a method to convert complex rotationally symmetric tilings with multiple rhombuses into tilings using various types of pentagons, expanding the understanding of tiling transformations.

Contribution

It introduces a novel conversion technique from rhombus-based tilings to pentagonal tilings, including convex, concave, and degenerated forms.

Findings

Conversion preserves rotational symmetry.

Convex pentagons belong to Type 2 family.

Applicable to tilings by Penrose, Ammann, Beenker, Socolar.

Abstract

The results involving rotationally symmetric tilings with multiple types of rhombuses, discovered by Penrose, Ammann, Beenker, or Socolar, are converted to tilings with multiple types of pentagons are presented. The pentagons can be convex or concave, and can be degenerated into a trapezoid or parallelogram. If the pentagons are convex, they belong to the Type 2 family.