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ABACABA pattern
open-in-new
<h2 id="generating-the-pattern">Generating the pattern</h2> <p>In order to <a href="/facts/Iterated_function_system/j56g6yEM">generate</a> 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.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> <table><tbody><tr><th>Step</th><th>Pattern</th><th>Letters</th></tr><tr><td>1</td><td>A</td><td>21 − 1 = 1</td></tr><tr><td>2</td><td>ABA</td><td>3</td></tr><tr><td>3</td><td>ABACABA</td><td>7</td></tr><tr><td>4</td><td>ABACABADABACABA</td><td>15</td></tr><tr><td>5</td><td>ABACABADABACABAEABACABADABACABA</td><td>31</td></tr><tr><td>6</td><td>ABACABADABACABAEABACABADABACABAFABACABADABACABAEABACABADABACABA</td><td>63</td></tr></tbody></table> <p>ABACABA is a "quickly <a href="/facts/Exponential_growth/m4yEqKJ7">growing</a> word", often described as <i><a href="/facts/Chiastic/oXfb8P29">chiastic</a></i> or "symmetrically organized around a central axis" (see: <a href="/facts/Chiastic_structure/BpAxPtCU">Chiastic structure</a> and <a href="/facts/Chi_(letter)/pj4UT8Pf">Χ</a>).<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> The number of members in each iteration is <i>a</i>(<i>n</i>) = 2<i>n</i> − 1, the <a href="/facts/Mersenne_prime/IyF3sBNf">Mersenne numbers</a> (<a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A000225). </p> <h2 id="gallery">Gallery</h2> <table><tbody><tr><th><a href="HTTP://ARCHIVE.TⓄDAY/OOOOOOOOOOOOOOOOOOOOOOOOOOO.CARRD.CO">◦୦◦◯◦୦◦⠀ ⠀◦୦◦◯◦୦◦</a></th></tr></tbody></table> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Arch_form/jRNHI7nU">Arch form</a></li> <li><a href="/facts/Farey_sequence/P6n10vaU">Farey sequence</a></li> <li><a href="/facts/Rondo/TQWwaAop">Rondo</a></li> <li><a href="/facts/Sesquipower/GPvxjMR1">Sesquipower</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="external-links">External links</h2> <ul><li>Naylor, Mike: <a href="http://abacaba.org/">abacaba.org</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Naylor, Mike (February 2013). "ABACABA Amazing Pattern, Amazing Connections". Math Horizons. Retrieved June 13, 2019. <a href="http://digitaleditions.walsworthprintgroup.com/display_article.php?id=1300604" target="_blank">http://digitaleditions.walsworthprintgroup.com/display_article.php?id=1300604</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>SheriOZ (2016-04-21). "Exploring Fractals with ABACABA". Chicago Geek Guy. Archived from the original on 22 January 2021. Retrieved January 22, 2021. <a href="https://web.archive.org/web/20210122212731/http://chicagogeekguy.com/exploring-abacaba/" target="_blank">https://web.archive.org/web/20210122212731/http://chicagogeekguy.com/exploring-abacaba/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Naylor, Mike (2011). "Abacaba! – Using a mathematical pattern to connect art, music, poetry and literature" (PDF). Bridges. Retrieved October 6, 2017. <a href="http://archive.bridgesmathart.org/2011/bridges2011-89.pdf" target="_blank">http://archive.bridgesmathart.org/2011/bridges2011-89.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Conley, Craig (2008-10-01). Magic Words: A Dictionary. Weiser Books. p. 53. ISBN 9781609250508. <a href="9781609250508" target="_blank">9781609250508</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Halter-Koch, Franz and Tichy, Robert F.; eds. (2000). Algebraic Number Theory and Diophantine Analysis, p.478. W. de Gruyter. ISBN 9783110163049. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Conley, Craig (2008-10-01). Magic Words: A Dictionary. Weiser Books. p. 53. ISBN 9781609250508. <a href="9781609250508" target="_blank">9781609250508</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Conley, Craig (2008-10-01). Magic Words: A Dictionary. Weiser Books. p. 53. ISBN 9781609250508. <a href="9781609250508" target="_blank">9781609250508</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> </ol>