# About quasiperiodic tilings, possessing five-fold symmetry

## TL;DR

This paper presents a group-theoretical method for constructing five-fold symmetric quasiperiodic tilings of the plane, identifying a universal tile set capable of generating diverse tilings with complex rotational symmetries.

## Contribution

It introduces a universal set of five tiles for creating various five-fold symmetric quasiperiodic tilings using a group-theoretical approach.

## Key findings

- Identified geometric characteristics of the tiles.
- Demonstrated tilings with fifth and tenth order rotational symmetries.
- Showed the versatility of a single tile set for different tilings.

## Abstract

A group-theoretical approach to the construction of quasiperiodic tilings of a Euclidean plane, possessing five-fold symmetry, is applied. Of the infinitely many of variants of quasiperiodic partitions of the plane, possessing the dihedral group of symmetry D5, special attention is paid to those that, can be obtained by one universal set, with consists five different tiles. Geometric characteristics of tiles from this set are determined. It is shown, by this set of tiles it is possible to carry out topologically different tilings of the plane, possessing rotating symmetries of both the fifth and tenth orders.

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Source: https://tomesphere.com/paper/1908.02550