## Abstract

The notion of the informational measure of symmetry is introduced according to: (Formula Presented), where P(G_{i} ) is the probability of appearance of the symmetry operation G_{i} within the given 2D pattern. H_{sym} (G) is interpreted as an averaged uncertainty in the presence of symmetry elements from the group G in the given pattern. The informational measure of symmetry of the “ideal” pattern built of identical equilateral triangles is established as H_{sym} (D_{3} ) = 1.792. The informational measure of symmetry of the random, completely disordered pattern is zero, H_{sym} = 0. The informational measure of symmetry is calculated for the patterns generated by the P3 Penrose tessellation. The informational measure of symmetry does not correlate with either the Voronoi entropy of the studied patterns nor with the continuous measure of symmetry of the patterns. Quantification of the “ordering” in 2D patterns performed solely with the Voronoi entropy is misleading and erroneous.

Original language | English |
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Article number | 2146 |

Journal | Symmetry |

Volume | 13 |

Issue number | 11 |

DOIs | |

State | Published - Nov 2021 |

## Keywords

- Continuous symmetry measure
- Informational measure
- Ordering
- Penrose tiling
- Symmetry
- Voronoi entropy