ABACABA pattern

"ABACABA" redirects here. For other uses, see Sonata rondo form.

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.

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. A generator can be found here

Step	Pattern
1	A
2	ABA
3	ABACABA
4	ABACABADABACABA
5	ABACABADABACABAEABACABADABACABA
6	ABACABADABACABAEABACABADABACABAFABACABADABACABAEABACABADABACABA

Gallery

• 
  Sierpinski triangle:
  ABACABA
• 
  Ruler, excluding 1 and 2:
  ABACABADABACABAexcluding 2:
  EABACABADABACABA
• 
  Cantor set:
  ABACABADABACABA
• 
  Binary tree/upside down family tree:
  ABACABADABACABA
• 
  Koch curve: $n=1$ is ABA, $n=2$ is ABACABA, and $n=3$: ABACABADABACABA
• 
  Metric hierarchy:
  ABACABADABACABA
• 
  When counting in binary (here 4-bit), the final 0s form an ABACABA pattern
• 
  A staircase built with the largest possible squares/cubes while allowing equally sized steps: ABACABADABACABA
• 
  A "circle fractal" superimposed with a 2 × 2 box fractal: ABACABADABACABA

See also

• Chiastic structure (X)
• Farey sequence
• Rondo
• Sesquipower
• Tower of Hanoi

External links

• Naylor, Mike: abacaba.org