ABACABA pattern
Mathematical fractal pattern
The ABACABA pattern is a 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.
01Generating 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.
Sources and credits
This article is adapted from the Wikipedia article “ABACABA pattern”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.
Images, from Wikimedia Commons:
- Binary number dabacaba pattern.png by Hyacinth, CC BY-SA 4.0
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