ABACABA pattern

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 ( OEIS:  A000225 ). Replacing each letter with their respective numbers yields the ruler function.

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] ${\displaystyle n=1}$ is ABA, ${\displaystyle n=2}$ is ABACABA, and ${\displaystyle n=3}$: ABACABADABACABA
• 
  Metric hierarchy:
  

  ABACABADABACABA ^[a]
• 
  Metric levels: ^[1]

  EABACABADABACABA
• 
  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:

  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
• 
  Size of hexagrams on a diagonal of a section of a Menger sponge model:

  ABACABADABACABA

See also

• Arch form
• Farey sequence
• Rondo
• Sesquipower

Notes

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

References

1. Naylor, Mike (February 2013). "ABACABA Amazing Pattern, Amazing Connections". Math Horizons. Retrieved June 13, 2019.
2. SheriOZ (2016-04-21). "Exploring Fractals with ABACABA". Chicago Geek Guy. Archived from the original on 22 January 2021. Retrieved January 22, 2021.
3. Naylor, Mike (2011). "Abacaba! – Using a mathematical pattern to connect art, music, poetry and literature" (PDF). Bridges. Retrieved October 6, 2017.
4. Conley, Craig (2008-10-01). Magic Words: A Dictionary. Weiser Books. p. 53. ISBN   9781609250508.
5. Halter-Koch, Franz and Tichy, Robert F.; eds. (2000). Algebraic Number Theory and Diophantine Analysis, p.478. W. de Gruyter. ISBN   9783110163049.
6. Wright, Craig (2016). Listening to Western Music, p.48. Cengage Learning. ISBN   9781305887039.

External links

• Naylor, Mike: abacaba.org