ABACABA pattern

"ABACABA" redirects here. For the YouTube channel, see Abacaba.

File:Binary number dabacaba pattern.png

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). Patterns often show a DABACABA type subset. AA, ABBA, and ABAABA type forms are also considered.

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.

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). The number of members in each iteration is a(n) = 2ⁿ − 1, the Mersenne numbers ((sequence A000225 in the OEIS)). Replacing each letter with their respective numbers yields the ruler function.

Gallery

• Sierpinski triangle:,
  Sierpinski triangle:
  

  ABACABA
• Ruler, excluding 1 and 2:, excluding 2:,
  Ruler, excluding 1 and 2:
  

  ABACABADABACABA

  excluding 2:
  

  EABACABADABACABA
• Cantor set:,
  Cantor set:
  

  ABACABADABACABA
• Binary tree/upside down family tree:,
  Binary tree/upside down family tree:
  

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

  ABACABADABACABA
• Metric levels:
  Metric levels:

  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
• 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
• A "circle fractal" superimposed with a 2 × 2 box fractal:
  A "circle fractal" superimposed with a 2 × 2 box fractal:

  ABACABADABACABA
• The Tower of Hanoi with four disks:
  The Tower of Hanoi 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):

  ABACABA
• Double harmonic scale (Audio file "Double harmonic scale.mid" not found) with steps of H-3H-H-W-H-3H-H:
  Double harmonic scale () with steps of H-3H-H-W-H-3H-H:

  ABACABA
• Château de Chambord:
  Château de Chambord:

  ABACABA
• Gray code along the number line ((sequence A003188 in the OEIS)):
  Gray code along the number line ((sequence A003188 in the OEIS)):

  ABACABADABACABAEABACABADABACABA
• Devil's needle:
  Devil's needle:

  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

See also

• Arch form
• Farey sequence
• Rondo
• Sesquipower

External links

• Naylor, Mike: abacaba.org