ABACABA_pattern

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).^[1]^[2]^[3]^[4] Patterns often show a DABACABA type subset. AA, ABBA, and ABAABA type forms are also considered.^[5]

DABACABA patterns in (3-bit) binary numbers

Gallery

Sierpinski triangle^[1]^[2]:

ABACABA
Ruler,^[1]^[2] excluding 1 and 2:

ABACABADABACABA
excluding 2:

EABACABADABACABA
Cantor set:

ABACABADABACABA
Binary tree^[1]^[2]/upside down family tree:

ABACABADABACABA
Koch curve:^[1] $n=1$ is ABA, $n=2$ is ABACABA, and $n=3$ : ABACABADABACABA
Metric hierarchy:

ABACABADABACABA^{[lower-alpha 1]}
Metric levels:^[1]
EABACABADABACABA
When counting in binary (here 4-bit), the final 0s form an ABACABA pattern^[1]
A staircase built with the largest possible squares/cubes while allowing equally sized steps:
ABACABADABACABA^[1]
A "circle fractal"^[1] superimposed with a 2 × 2 box fractal:
ABACABADABACABA
The Tower of Hanoi^[1] with four disks:
ABACABADABACABA
Binary tree array:
to O
Binary-reflected Gray code (BRGC):
to G
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):^[1]
ABACABA
Double harmonic scale (Play) with steps of H-3H-H-W-H-3H-H:
ABACABA
Château de Chambord:
ABACABA^[6]
Gray code along the number line^[1] (OEIS: A003188):
ABACABADABACABAEABACABADABACABA
Devil's needle:^[1]
ABACABADABACABA

Notes

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

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This article uses material from the Wikipedia article ABACABA_pattern, and is written by contributors. Text is available under a CC BY-SA 4.0 International License; additional terms may apply. Images, videos and audio are available under their respective licenses.

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

[Naylor-1] [1]
Naylor, Mike (February 2013). "ABACABA Amazing Pattern, Amazing Connections". Math Horizons. Retrieved June 13, 2019.

[Chicago-2] [2]
SheriOZ (2016-04-21). "Exploring Fractals with ABACABA". Chicago Geek Guy. Archived from the original on 22 January 2021. Retrieved January 22, 2021.

[Bridges-3] [3]
Naylor, Mike (2011). "Abacaba! – Using a mathematical pattern to connect art, music, poetry and literature" (PDF). Bridges. Retrieved October 6, 2017.

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

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

[7] [6]
Wright, Craig (2016). Listening to Western Music, p.48. Cengage Learning. ISBN 9781305887039.

[1]

[2]

[3]

[4]

[5]

[lower-alpha 1]

[6]

ABACABA_pattern

ABACABA pattern

Generating the pattern

Gallery

See also

Notes

References

External links

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Step	Pattern	Letters
1	A	2¹ − 1 = 1
2	ABA	3
3	ABACABA	7
4	ABACABADABACABA	15
5	ABACABADABACABAEABACABADABACABA	31
6	ABACABADABACABAEABACABADABACABAFABACABADABACABAEABACABADABACABA	63