ABACABA 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). 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.

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 $a (n) = 2 n - 1$ , the Mersenne numbers (OEIS: A000225).

Gallery

Sierpinski triangle:
ABACABA
Ruler, excluding 1 and 2:
ABACABADABACABA
excluding 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
Metric levels:
EABACABADABACABA
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
$A "circle fractal" superimposed with a 2 × 2 box fractal:$
A "circle fractal" superimposed with a $2 \times 2$ box fractal:
ABACABADABACABA
The Tower of Hanoi 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):
ABACABA
Double harmonic scale (Play) with steps of H-3H-H-W-H-3H-H:
ABACABA
Château de Chambord:
ABACABA
Gray code along the number line (OEIS: A003188):
ABACABADABACABAEABACABADABACABA
Devil's needle:
ABACABADABACABA
Size of hexagrams on a diagonal of a section of a Menger sponge model:
ABACABADABACABA

Notes

References

External links

Naylor, Mike: abacaba.org