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The ABACABA pattern is a
here
ABACABA is a "quickly growing word", often described as ''
abacaba.org
Fractals
The ABACABA pattern is a recursive
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
pattern
A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated l ...
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.Halter-Koch, Franz and Tichy, Robert F.; eds. (2000). ''Algebraic Number Theory and Diophantine Analysis'', p.478. W. de Gruyter. .
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 founhere
ABACABA is a "quickly growing word", often described as ''
chiastic
In rhetoric, chiasmus ( ) or, less commonly, chiasm (Latin term from Greek , "crossing", from the Greek , , "to shape like the letter Χ"), is a "reversal of grammatical structures in successive phrases or clauses – but no repetition of wo ...
'' or "symmetrically organized around a central axis" (see: Chiastic structure
Chiastic structure, or chiastic pattern, is a literary technique in narrative motifs and other textual passages. An example of chiastic structure would be two ideas, A and B, together with variants A' and B', being presented as A,B,B',A'. Chi ...
and Χ). The number of members in each iteration is , the Mersenne numbers
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17 ...
().
Gallery
See also
*Arch form In music, arch form is a sectional structure for a piece of music based on repetition, in reverse order, of all or most musical sections such that the overall form is symmetric, most often around a central movement. The sections need not be repeat ...
* Farey sequence
In mathematics, the Farey sequence of order ''n'' is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less than or equal to ''n'', arranged in ord ...
* Rondo
The rondo is an instrumental musical form introduced in the Classical period.
Etymology
The English word ''rondo'' comes from the Italian form of the French ''rondeau'', which means "a little round".
Despite the common etymological root, rondo ...
* Sesquipower In mathematics, a sesquipower or Zimin word is a string over an alphabet with identical prefix and suffix. Sesquipowers are unavoidable patterns, in the sense that all sufficiently long strings contain one.
Formal definition
Formally, let ''A'' ...
Notes
References
{{ReflistExternal links
*Naylor, Mikeabacaba.org
Fractals