Mathematics

=== Artistic expression ===
{{Main|Mathematics and art}}
Notes that sound well together to a Western ear are sounds whose fundamental [[frequencies]] of vibration are in simple ratios. For example, an octave doubles the frequency and a [[perfect fifth]] multiplies it by <math>\frac{3}{2}</math>.<ref>{{cite journal | last = Cazden | first = Norman | date = October 1959 | doi = 10.1177/002242945900700205 | issue = 2 | journal = Journal of Research in Music Education | jstor = 3344215 | pages = 197–220 | title = Musical intervals and simple number ratios | volume = 7| s2cid = 220636812 }}</ref><ref>{{cite journal | last = Budden | first = F. J. | date = October 1967 | doi = 10.2307/3613237 | issue = 377 | journal = The Mathematical Gazette | jstor = 3613237 | pages = 204–215 | publisher = Cambridge University Press ({CUP}) | title = Modern mathematics and music | volume = 51| s2cid = 126119711 }}</ref>

[[File:Julia set (highres 01).jpg|thumb|[[Fractal]] with a scaling symmetry and a central symmetry]]
Humans, as well as some other animals, find symmetric patterns to be more beautiful.<ref>{{Cite journal |last1=Enquist |first1=Magnus |last2=Arak |first2=Anthony |date=November 1994 |title=Symmetry, beauty and evolution |url=https://www.nature.com/articles/372169a0 |journal=Nature |language=en |volume=372 |issue=6502 |pages=169–172 |doi=10.1038/372169a0 |pmid=7969448 |bibcode=1994Natur.372..169E |s2cid=4310147 |issn=1476-4687 |access-date=December 29, 2022 |archive-date=December 28, 2022 |archive-url=https://web.archive.org/web/20221228052049/https://www.nature.com/articles/372169a0 |url-status=live }}</ref> Mathematically, the symmetries of an object form a group known as the [[symmetry group]].<ref>{{Cite web |last=Hestenes |first=David |date=1999 |title=Symmetry Groups |url=http://geocalc.clas.asu.edu/pdf-preAdobe8/SymmetryGroups.pdf |access-date=December 29, 2022 |website=geocalc.clas.asu.edu |archive-date=January 1, 2023 |archive-url=https://web.archive.org/web/20230101210124/http://geocalc.clas.asu.edu/pdf-preAdobe8/SymmetryGroups.pdf |url-status=live }}</ref>

For example, the group underlying mirror symmetry is the [[cyclic group]] of two elements, <math>\mathbb{Z}/2\mathbb{Z}</math>. A [[Rorschach test]] is a figure invariant by this symmetry,<ref>{{cite encyclopedia | last = Bender | first = Sara | editor1-last = Carducci | editor1-first = Bernardo J. | editor2-last = Nave | editor2-first = Christopher S. | editor3-last = Mio | editor3-first = Jeffrey S. | editor4-last = Riggio | editor4-first = Ronald E. | title = The Rorschach Test | date = September 2020 | doi = 10.1002/9781119547167.ch131 | pages = 367–376 | publisher = Wiley | encyclopedia = The Wiley Encyclopedia of Personality and Individual Differences: Measurement and Assessment| isbn = 978-1-119-05751-2 }}</ref> as are [[butterfly]] and animal bodies more generally (at least on the surface).<ref>{{cite book|title=Symmetry|volume=47|series=Princeton Science Library|first=Hermann|last=Weyl|author-link=Hermann Weyl|publisher=Princeton University Press|year=2015|isbn=978-1-4008-7434-7|page=[https://books.google.com/books?hl=en&lr=&id=GG1FCQAAQBAJ&pg=PA4 4]}}</ref> Waves on the sea surface possess translation symmetry: moving one's viewpoint by the distance between wave crests does not change one's view of the sea.{{Citation needed|date=December 2022}} [[Fractals]] possess [[self-similarity]].<ref>{{Cite web |last=Bradley |first=Larry |date=2010 |title=Fractals – Chaos & Fractals |url=https://www.stsci.edu/~lbradley/seminar/fractals.html |access-date=December 29, 2022 |website=www.stsci.edu |archive-date=March 7, 2023 |archive-url=https://web.archive.org/web/20230307054609/https://www.stsci.edu/~lbradley/seminar/fractals.html |url-status=live }}</ref><ref>{{Cite web |title=Self-similarity |url=https://math.bu.edu/DYSYS/chaos-game/node5.html |access-date=December 29, 2022 |website=math.bu.edu |archive-date=March 2, 2023 |archive-url=https://web.archive.org/web/20230302132911/http://math.bu.edu/DYSYS/chaos-game/node5.html |url-status=live }}</ref>