⛋𖢌⛋⯏⛋𖢌⛋⠀𖥚⚙𖥚᪣𖥚⚙𖥚𖥕𖥚⚙𖥚᪣𖥚⚙𖥚᯽𖥚⚙𖥚᪣𖥚⚙𖥚𖥕𖥚⚙𖥚᪣𖥚⚙𖥚⠀𖥚⚙𖥚᪣𖥚⚙𖥚𖥕𖥚⚙𖥚᪣𖥚⚙𖥚᯽𖥚⚙𖥚᪣𖥚⚙𖥚𖥕𖥚⚙𖥚᪣𖥚⚙𖥚⠀⛋𖢌⛋⯏⛋𖢌⛋

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ABACABA pattern

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From Wikipedia, the free encyclopedia This is an old revision of this page, as edited by 37.214.56.5 (talk) at 08:29, 15 June 2025 (◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision. (diff) ← Previous revision | Latest revision (diff) | Newer revision → (diff) DABACABA patterns in (3-bit) binary numbers

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).[1][2][3][4] Patterns often show a DABACABA type subset. AA, ABBA, and ABAABA type forms are also considered.[5] 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.[4] Step Pattern Letters 1 A 21 − 1 = 1 2 ABA 3 3 ABACABA 7 4 ABACABADABACABA 15 5 ABACABADABACABAEABACABADABACABA 31 6 ABACABADABACABAEABACABADABACABAFABACABADABACABAEABACABADABACABA 63

ABACABA is a "quickly growing word", often described as chiastic or "symmetrically organized around a central axis" (see: Chiastic structure and Χ).[4] The number of members in each iteration is a(n) = 2n − 1, the Mersenne numbers ((sequence A000225 in the OEIS)). Gallery

Sierpinski triangle:,
Sierpinski triangle[1][2]:
ABACABA
Ruler, excluding 1 and 2:, excluding 2:,
Ruler,[1][2] excluding 1 and 2:
ABACABADABACABA
excluding 2:
EABACABADABACABA
Cantor set:,
Cantor set:
ABACABADABACABA
Binary tree/upside down family tree:,
Binary tree[1][2]/upside down family tree:
ABACABADABACABA
Koch curve: is ABA, is ABACABA, and : ABACABADABACABA
Koch curve:[1] n = 1 {\displaystyle n=1} is ABA, n = 2 {\displaystyle n=2} is ABACABA, and n = 3 {\displaystyle n=3}: ABACABADABACABA
Metric hierarchy:,
Metric hierarchy:
ABACABADABACABA[a]
Metric levels:
Metric levels:[1]
EABACABADABACABA
When counting in binary (here 4-bit), the final 0s form an ABACABA pattern
When counting in binary (here 4-bit), the final 0s form an ABACABA pattern[1]
A staircase with each box double the size of the previous one:
A staircase with each box double the size of the previous one:
ABACABADABACABA[1]
A "circle fractal" superimposed with a 2 × 2 box fractal:
A "circle fractal"[1] superimposed with a 2 × 2 box fractal:
ABACABADABACABA
The Tower of Hanoi with four disks:
The Tower of Hanoi[1] with four disks:
ABACABADABACABA
Binary tree array:
Binary tree array:
to O
Binary-reflected Gray code (BRGC):
Binary-reflected Gray code (BRGC):
to G
Rotary encoder:
Rotary encoder:
to I
3-bit Gray code visualized as a traversal of vertices of a cube (0,1,3,2,6,7,5,4):
3-bit Gray code visualized as a traversal of vertices of a cube (0,1,3,2,6,7,5,4):[1]
ABACABA
Double harmonic scale () with steps of H-3H-H-W-H-3H-H:
Double harmonic scale (Playⓘ) with steps of H-3H-H-W-H-3H-H:
ABACABA
Château de Chambord:
Château de Chambord:
ABACABA[6]
Gray code along the number line ((sequence A003188 in the OEIS)):
Gray code along the number line[1] ((sequence A003188 in the OEIS)):
ABACABADABACABAEABACABADABACABA
Devil's needle:
Devil's needle:[1]
ABACABADABACABA
Size of hexagrams on a diagonal of a section of a Menger sponge model:
Size of hexagrams on a diagonal of a section of a Menger sponge model:
ABACABADABACABA

◦୦◦◯◦୦◦⠀       ⠀◦୦◦◯◦୦◦ See also

Arch form
Farey sequence
Rondo
Sesquipower

Notes

The strength, emphasis, or importance of the beginning of each duration 1 / 8 {\displaystyle 1/8} the length of a single measure in 4
4 (eighth-notes) is, divisively ( 2 / 2 1 = 1 {\displaystyle 2/2^{1}=1}, 4 / 2 2 = 1 {\displaystyle 4/2^{2}=1}, 8 / 2 3 = 1 {\displaystyle 8/2^{3}=1}), determined by each eighth-note's position in a DABACABA structure, while the eighth notes of two measures grouped, additively ( 8 × 2 = 16 {\displaystyle 8\times 2=16}), are determined by an EABACABADABACABA structure.[3]

References

Naylor, Mike (February 2013). "ABACABA Amazing Pattern, Amazing Connections". Math Horizons. Retrieved June 13, 2019. SheriOZ (2016-04-21). "Exploring Fractals with ABACABA". Chicago Geek Guy. Archived from the original on 22 January 2021. Retrieved January 22, 2021. Naylor, Mike (2011). "Abacaba! – Using a mathematical pattern to connect art, music, poetry and literature" (PDF). Bridges. Retrieved October 6, 2017. Conley, Craig (2008-10-01). Magic Words: A Dictionary. Weiser Books. p. 53. ISBN 9781609250508. Halter-Koch, Franz and Tichy, Robert F.; eds. (2000). Algebraic Number Theory and Diophantine Analysis, p.478. W. de Gruyter. ISBN 9783110163049.

Wright, Craig (2016). Listening to Western Music, p.48. Cengage Learning. ISBN 9781305887039.

External links

Naylor, Mike: abacaba.org

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This page was last edited on 15 June 2025, at 08:29 (UTC).
This version of the page has been revised. Besides normal editing, the reason for revision may have been that this version contains factual inaccuracies, vandalism, outdated information, or material not compatible with the Creative Commons Attribution-ShareAlike 4.0 License.

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