Ten colours in quasiperiodic and regular hyperbolic tilings

TL;DR

This paper presents explicit constructions of ten-colour symmetries in hyperbolic tilings, confirming theoretical predictions and contrasting new examples with previously known colourings.

Contribution

It provides the first explicit constructions of several ten-colour symmetries in hyperbolic tilings, expanding on prior group theoretical predictions.

Findings

Explicit constructions of ten-colour symmetries are achieved.

The new constructions differ from previously known ten-colourings.

The work confirms theoretical predictions about colour symmetries in hyperbolic tilings.

Abstract

Colour symmetries with ten colours are presented for different tilings. In many cases, the existence of these colourings were predicted by group theoretical methods. Only in a few cases explicit constructions were known, sometimes using combination of two-colour and five-colour symmetries. Here we present explicit constructions of several of the predicted colourings for the first time, and discuss them in contrast to already known colourings with ten colours.