Editing Mathematics (section)

== Cultural impact ==

=== 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 |year=1999 |title=Symmetry Groups |url=https://davidhestenes.net/geocalc/pdf/SymmetryGroups.pdf }}</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.<ref>{{Cite web|url=https://ocw.mit.edu/courses/8-03sc-physics-iii-vibrations-and-waves-fall-2016/pages/part-i-mechanical-vibrations-and-waves/lecture-8/|title=Lecture 8: Translation Symmetry &#124; Physics III: Vibrations and Waves &#124; Physics|website=MIT OpenCourseWare}}</ref> [[Fractals]] possess [[self-similarity]].<ref>{{Cite web |last=Bradley |first=Larry |year=2010 |title=Fractals – Chaos & Fractals |url=https://www.stsci.edu/~lbradley/seminar/fractals.html |access-date=December 29, 2022 |website=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>

=== Popularization ===
{{Main|Popular mathematics}}Popular mathematics is the act of presenting mathematics without technical terms.<ref>{{Cite conference |last=Kissane |first=Barry |date=July 2009 |title=Popular mathematics |url=https://researchrepository.murdoch.edu.au/id/eprint/6242/ |conference=22nd Biennial Conference of The Australian Association of Mathematics Teachers |location=Fremantle, Western Australia |publisher=Australian Association of Mathematics Teachers |pages=125–126 |access-date=December 29, 2022 |archive-date=March 7, 2023 |archive-url=https://web.archive.org/web/20230307054610/https://researchrepository.murdoch.edu.au/id/eprint/6242/ |url-status=live }}</ref> Presenting mathematics may be hard since the general public suffers from [[mathematical anxiety]] and mathematical objects are highly abstract.<ref>{{Cite book |last=Steen |first=L. A. |url={{GBurl|id=-d3TBwAAQBAJ|dq="popular mathematics" analogies|p=2}} |title=Mathematics Today Twelve Informal Essays |date=2012|publisher=Springer Science & Business Media |isbn=978-1-4613-9435-8 |page=2 |language=en |access-date=January 3, 2023 }}</ref> However, popular mathematics writing can overcome this by using applications or cultural links.<ref>{{Cite book |last=Pitici |first=Mircea |url={{GBurl|id=9nGQDQAAQBAJ|dq="popular mathematics" analogies|p=331}} |title=The Best Writing on Mathematics 2016 |date=2017|publisher=Princeton University Press |isbn=978-1-4008-8560-2 |language=en |access-date=January 3, 2023 }}</ref> Despite this, mathematics is rarely the topic of popularization in printed or televised media.

=== Awards and prize problems ===
{{Main category|Mathematics awards}}
[[File:FieldsMedalFront.jpg|thumb|The front side of the [[Fields Medal]] with an illustration of the Greek [[polymath]] [[Archimedes]]]]

The most prestigious award in mathematics is the [[Fields Medal]],{{sfn|Monastyrsky|2001|p=1|ps=: "The Fields Medal is now indisputably the best known and most influential award in mathematics."}}{{sfn|Riehm|2002|pp=778–782}} established in 1936 and awarded every four years (except around [[World War II in Yugoslavia|World War II]]) to up to four individuals.<ref>{{Cite web |title=Fields Medal {{!}} International Mathematical Union (IMU) |url=https://www.mathunion.org/imu-awards/fields-medal |access-date=February 21, 2022 |website=www.mathunion.org |archive-date=December 26, 2018 |archive-url=https://web.archive.org/web/20181226015744/https://www.mathunion.org/imu-awards/fields-medal |url-status=live }}</ref><ref name="StAndrews-Fields">{{Cite web |title=Fields Medal |url=https://mathshistory.st-andrews.ac.uk/Honours/FieldsMedal/ |access-date=February 21, 2022 |website=Maths History |language=en |archive-date=March 22, 2019 |archive-url=https://web.archive.org/web/20190322134417/http://www-history.mcs.st-andrews.ac.uk/Honours/FieldsMedal.html |url-status=live }}</ref> It is considered the mathematical equivalent of the [[Nobel Prize]].<ref name="StAndrews-Fields" />

Other prestigious mathematics awards include:<ref>{{cite web
 | title=Honours/Prizes Index
 | website=MacTutor History of Mathematics Archive
 | url=https://mathshistory.st-andrews.ac.uk/Honours/
 | access-date=February 20, 2023
| archive-date=December 17, 2021
| archive-url=https://web.archive.org/web/20211217235828/https://mathshistory.st-andrews.ac.uk/Honours/
 | url-status=live
 }}</ref>
* The [[Abel Prize]], instituted in 2002<ref>{{Cite web|title=About the Abel Prize|publisher=The Abel Prize|url=https://abelprize.no/page/about-abel-prize|access-date=January 23, 2022|archive-date=April 14, 2022|archive-url=https://web.archive.org/web/20220414060442/https://abelprize.no/page/about-abel-prize|url-status=live}}</ref> and first awarded in 2003<ref>{{Cite encyclopedia|title=Abel Prize {{!}} mathematics award|encyclopedia=Encyclopedia Britannica|url=https://www.britannica.com/science/Abel-Prize|access-date=January 23, 2022|language=en|archive-date=January 26, 2020|archive-url=https://web.archive.org/web/20200126120202/https://www.britannica.com/science/Abel-Prize|url-status=live}}</ref>
* The [[Chern Medal]] for lifetime achievement, introduced in 2009<ref>{{Cite web |date=June 1, 2009 |title=Chern Medal Award|url=https://www.mathunion.org/fileadmin/IMU/Prizes/Chern/Chern_MedalPress_Release_090601.pdf |url-status=live |archive-url=https://web.archive.org/web/20090617012953/https://www.mathunion.org/fileadmin/IMU/Prizes/Chern/Chern_MedalPress_Release_090601.pdf |archive-date=June 17, 2009 |access-date=February 21, 2022 |website=mathunion.org}}</ref> and first awarded in 2010<ref>{{Cite web |title=Chern Medal Award|publisher=International Mathematical Union (IMU)|url=https://www.mathunion.org/imu-awards/chern-medal-award |access-date=January 23, 2022|archive-date=August 25, 2010 |archive-url=https://web.archive.org/web/20100825071850/http://www.mathunion.org/general/prizes/chern/details |url-status=live }}</ref>
* The [[American Mathematical Society|AMS]] [[Leroy P. Steele Prize]], awarded since 1970<ref>{{cite web
 | title=The Leroy P Steele Prize of the AMS
 | publisher=School of Mathematics and Statistics, University of St Andrews, Scotland
 | url=https://mathshistory.st-andrews.ac.uk/Honours/AMSSteelePrize/
 | access-date=November 17, 2022
| archive-date=November 17, 2022
| archive-url=https://web.archive.org/web/20221117201134/https://mathshistory.st-andrews.ac.uk/Honours/AMSSteelePrize/
 | url-status=live
 }}</ref>
* The [[Wolf Prize in Mathematics]], also for lifetime achievement,<ref>{{Cite book |last1=Chern |first1=S. S. |last2=Hirzebruch |first2=F. |date=September 2000 |title=Wolf Prize in Mathematics |url=https://www.worldscientific.com/worldscibooks/10.1142/4149 |language=en |doi=10.1142/4149 |isbn=978-981-02-3945-9 |access-date=February 21, 2022 |archive-date=February 21, 2022 |archive-url=https://web.archive.org/web/20220221171351/https://www.worldscientific.com/worldscibooks/10.1142/4149 |url-status=live }}</ref> instituted in 1978<ref>{{Cite web|title=The Wolf Prize|url=https://wolffund.org.il/the-wolf-prize/|url-status=live|archive-url=https://web.archive.org/web/20200112205029/https://wolffund.org.il/the-wolf-prize/|archive-date=January 12, 2020|access-date=January 23, 2022|website=Wolf Foundation|language=en-US}}</ref>

A famous list of 23 [[open problem]]s, called "[[Hilbert's problems]]", was compiled in 1900 by German mathematician David Hilbert.<ref name=":0">{{Cite web|date=May 6, 2020|title=Hilbert's Problems: 23 and Math|url=https://www.simonsfoundation.org/2020/05/06/hilberts-problems-23-and-math/|access-date=January 23, 2022|website=Simons Foundation|language=en-US|archive-date=January 23, 2022|archive-url=https://web.archive.org/web/20220123011430/https://www.simonsfoundation.org/2020/05/06/hilberts-problems-23-and-math/|url-status=live}}</ref> This list has achieved great celebrity among mathematicians,<ref>{{cite book
 | chapter=Deciding the undecidable: Wrestling with Hilbert's problems
 | first=Solomon
 | last=Feferman
 | author-link=Solomon Feferman
 | title=In the Light of Logic
 | year=1998
 | publisher=Oxford University Press
 | isbn=978-0-19-508030-8
 | pages=3–27
 | series=Logic and Computation in Philosophy series
 | chapter-url=https://math.stanford.edu/~feferman/papers/deciding.pdf
 | url={{GBurl|id=1rjnCwAAQBAJ}}
 | access-date=November 29, 2022
}}</ref> and at least thirteen of the problems (depending how some are interpreted) have been solved.<ref name=":0" /><!-- Namely: problems 1, 3, 4; 5, 7, 10; 13, 14, 17; 18, 19, 20; 21 have been solved. (The semicolons are to make counting easier). ~Duckmather -->

A new list of seven important problems, titled the "[[Millennium Prize Problems]]", was published in 2000. Only one of them, the [[Riemann hypothesis]], duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward.<ref>{{Cite web|title=The Millennium Prize Problems|publisher=Clay Mathematics Institute|url=http://www.claymath.org/millennium-problems/millennium-prize-problems|access-date=January 23, 2022|archive-date=July 3, 2015|archive-url=https://web.archive.org/web/20150703184941/http://www.claymath.org/millennium-problems/millennium-prize-problems|url-status=live}}</ref> To date, only one of these problems, the [[Poincaré conjecture]], has been solved by the Russian mathematician [[Grigori Perelman]].<ref>{{Cite web|title=Millennium Problems|publisher=Clay Mathematics Institute|url=http://www.claymath.org/millennium-problems|access-date=January 23, 2022|archive-date=December 20, 2018|archive-url=https://web.archive.org/web/20181220122925/http://www.claymath.org/millennium-problems|url-status=live}}</ref><!-- NOTE that this website describes the answer to each problem as "unknown" EXCEPT for the Poincaré conjecture, where it mentions "Perelman's proof". ~Duckmather -->