Bird (mathematical artwork) explained

Bird, also known as A Bird in Flight refers to bird-like mathematical artworks that are introduced by mathematical equations.^{[1] [2] [3] [4] [5] [6] [7]} A group of these figures are created by combing through tens of thousands of computer-generated images. They are usually defined by trigonometric functions.^{[8] [9] [10] [11] [12]} An example of A Bird in Flight is made up of 500 segments defined in a Cartesian plane where for each

k =1 , 2 , 3 , \ldots , 500 the endpoints of the k -th line segment are:\left( (3)/(2) \sin ^{7 \left( (2 \pi k)/(500)} + (\pi)/(\right), 3) (1)/(4) \cos ^{2} \left( (6 \pi k)/(500) \right)\right)

and

\left( (1)/(\sin\left( 5) (6 \pi k)/(+ 500) (\pi)/(\right), 5) (-2)/(3) \sin ^{2 \left( (2 \pi k)/(500)} - (\pi)/(3) \right)\right) .

The 500 line segments defined above together form a shape in the Cartesian plane that resembles a bird with open wings. Looking at the line segments on the wings of the bird causes an optical illusion and may trick the viewer into thinking that the segments are curved lines. Therefore, the shape can also be considered as an optical artwork.^{[13] [14] [15] [16] [17]} Another version of A Bird in Flight was defined as the union of all of the circles with center

\left(A(k), B(k)\right) and radius R(k) , where k =-10000 , -9999 , \ldots , 9999 , 10000 , andA(k) = (3 k)/(+ \sin\left( 20000) (\pi)/(\left( 2) (k)/(10000) \right) ^{7\right)\cos} \right) + ^{6 \left( (41 \pi k)/(10000)} (1)/(4) \cos ^{16} \left( (41 \pi k)/(10000) \right)\cos ^{12} \left( (\pi k)/(\right)\sin\left( 20000) (6 \pi k)/(10000) \right), \begin{align} B(k) = & - \cos\left( (\pi)/(\left( 2) (k)/(10000) \right) ^{7 \right)\left( 1+ (3)/(2)} \cos ^{6 \left( (\pi k)/(20000)} \right)\cos ^{6 \left( (3 \pi k)/(20000)} \right)\right)\cos ^{6 \left( (41 \pi k)/(10000)} \right) \ & + (1)/(2) \cos ^{10} \left( (3 \pi k)/(100000) \right)\cos ^{10} \left( (9 \pi k)/(100000) \right)\cos ^{10} \left( (18 \pi k)/(100000) \right), \ \end{align} R(k) = (1)/(+ 50) (1)/(10) \sin ^{2 \left( (41 \pi k)/(10000)} \right)\sin ^{2 \left( (9 \pi k)/(100000)} \right) + (1)/(20) \cos ^{2 \left( (41 \pi k)/(10000)} \right)\cos ^{10} \left( (\pi k)/(20000) \right).

The set of the 20,001 circles defined above form a subset of the plane that resembles a flying bird. Although this version's equations are a lot more complicated than the version made of 500 segments, it has a better resemblance to a real flying bird.^{[18] [19]}

Notes and References

1. Von Formeln und Vögeln . . February 4, 2021 . 05/2021 . 47 . 9 April 2022 . German . 0170-2971 . 7 February 2021 . https://web.archive.org/web/20210207071721/https://www.spektrum.de/kolumne/freistetters-formelwelt-von-formeln-und-voegeln/1826563 . live .
2. Web site: November 2014 . Mathematical Concepts Illustrated by ... . April 3, 2022 . American Mathematical Society.
3. Web site: September 18, 2014 . Mathematical Works of Art . April 3, 2022 . Gustavus Adolphus College.
4. Web site: January 8, 2015 . This is not a bird (or a moustache) . April 3, 2022 . Plus Magazine.
5. Avian Arithmetic: The mathematics of bird flight . Peter . Cavanagh . National Museum of Mathematics' Events . MoMath Online, NY, United States . March 5, 2021 . April 3, 2022.
6. Web site: Gustlin . Deborah . 15.4: Digital Art . April 3, 2022 . LibreTexts. 17 November 2019 .
7. Web site: Mathematics Portal - IMKT . April 3, 2022 . International Mathematical Knowledge Trust.
8. News: Antonick . Gary . January 25, 2016 . Round Robin . . April 3, 2022.
9. News: Chung . Stephy . September 18, 2015 . Next da Vinci? Math genius using formulas to create fantastical works of art . .
10. Book: Baugher . Janée J. . The Ekphrastic Writer: Creating Art-Influenced Poetry, Fiction and Nonfiction . 2020 . . 9781476639611 . 56.
11. Web site: September 16, 2015 . "A Bird in Flight (2015)," . April 3, 2022 . . May 2, 2018 . https://web.archive.org/web/20180502211656/http://www.ams.org/mathimagery/displayimage.php?album=40&pid=616#top_display_media . dead .
12. News: Young . Lauren . January 19, 2016 . Math Is Beautiful . .
13. News: Mellow. Glendon. August 6, 2015. Mathematically Precise Crosshatching. Scientific American (blog). live. August 11, 2015. https://web.archive.org/web/20150925094554/http://blogs.scientificamerican.com/symbiartic/mathematically-precise-crosshatching/. September 25, 2015.
14. Web site: เมื่อคณิตศาสตร์ถูกสร้างเป็นภาพศิลปะ. When Mathematics is Made into Art. live. 2021-10-03. Faculty of Fine and Applied Arts. Chulalongkorn University. th. 2021-08-19. https://web.archive.org/web/20210819233835/https://www.faa.chula.ac.th/SelfLearningFaamai/detailform/119.
15. News: Mellow . Glendon . August 6, 2015 . Mathematically Precise Crosshatching . Scientific American (blog) .
16. Web site: March 23, 2016 . "A Bird in Flight (2016)," . April 3, 2022 . . March 29, 2017 . https://web.archive.org/web/20170329234641/http://www.ams.org/mathimagery/displayimage.php?album=40&pid=684#top_display_media . dead .
17. Web site: Passaro . Davide . Matematica e arti visive: percorsi interdisciplinari fra matematica, arte e coding . April 3, 2022 . Maddmaths! . SIMAI Società Italiana di Matematica Applicata e Industriale.
18. Web site: April 22, 2018 . "A Bird in Flight" . live . https://web.archive.org/web/20180423021644/https://www.futilitycloset.com/2018/04/22/a-bird-in-flight/ . April 23, 2018 . April 3, 2022 . Futility Closet.
19. News: 8 December 2020 . 수학적 아름다움, 프랙털 아트의 세계 . ko . Mathematical beauty, the world of fractal art . Sciencetimes . live . April 3, 2022 . https://web.archive.org/web/20201208103131/https://www.sciencetimes.co.kr/news/%EC%88%98%ED%95%99%EC%A0%81-%EC%95%84%EB%A6%84%EB%8B%A4%EC%9B%80-%ED%94%84%EB%9E%99%ED%84%B8-%EC%95%84%ED%8A%B8%EC%9D%98-%EC%84%B8%EA%B3%84/ . 8 December 2020.