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ABACABA pattern
Mathematical fractal pattern

The ABACABA pattern is a recursive fractal pattern that shows up in many places in the real world (such as in geometry, art, music, poetry, number systems, literature and higher dimensions). Patterns often show a DABACABA type subset. AA, ABBA, and ABAABA type forms are also considered.

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Generating the pattern

In order to generate the next sequence, first take the previous pattern, add the next letter from the alphabet, and then repeat the previous pattern. The first few steps are listed here.6

StepPatternLetters
1A21 − 1 = 1
2ABA3
3ABACABA7
4ABACABADABACABA15
5ABACABADABACABAEABACABADABACABA31
6ABACABADABACABAEABACABADABACABAFABACABADABACABAEABACABADABACABA63

ABACABA is a "quickly growing word", often described as chiastic or "symmetrically organized around a central axis" (see: Chiastic structure and Χ).7 The number of members in each iteration is a(n) = 2n − 1, the Mersenne numbers (OEIS: A000225).

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See also

Notes

References

  1. Naylor, Mike (February 2013). "ABACABA Amazing Pattern, Amazing Connections". Math Horizons. Retrieved June 13, 2019. http://digitaleditions.walsworthprintgroup.com/display_article.php?id=1300604 ↩

  2. SheriOZ (2016-04-21). "Exploring Fractals with ABACABA". Chicago Geek Guy. Archived from the original on 22 January 2021. Retrieved January 22, 2021. https://web.archive.org/web/20210122212731/http://chicagogeekguy.com/exploring-abacaba/ ↩

  3. Naylor, Mike (2011). "Abacaba! – Using a mathematical pattern to connect art, music, poetry and literature" (PDF). Bridges. Retrieved October 6, 2017. http://archive.bridgesmathart.org/2011/bridges2011-89.pdf ↩

  4. Conley, Craig (2008-10-01). Magic Words: A Dictionary. Weiser Books. p. 53. ISBN 9781609250508. 9781609250508 ↩

  5. Halter-Koch, Franz and Tichy, Robert F.; eds. (2000). Algebraic Number Theory and Diophantine Analysis, p.478. W. de Gruyter. ISBN 9783110163049. /wiki/ISBN_(identifier) ↩

  6. Conley, Craig (2008-10-01). Magic Words: A Dictionary. Weiser Books. p. 53. ISBN 9781609250508. 9781609250508 ↩

  7. Conley, Craig (2008-10-01). Magic Words: A Dictionary. Weiser Books. p. 53. ISBN 9781609250508. 9781609250508 ↩