Mathematical fractal pattern
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 ).[ 1] [ 2] [ 3] [ 4] D ABACABA type subset. AA ABB A , and ABAA BA type forms are also considered.[ 5]
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.[ 4]
Step
Pattern
Letters
1
A 21 − 1 = 1
2
AB A 3
3
ABAC ABA 7
4
ABACABAD ABACABA 15
5
ABACABADABACABAE ABACABADABACABA 31
6
ABACABADABACABAEABACABADABACABAF ABACABADABACABAEABACABADABACABA 63
ABACABA is a "quickly growing word", often described as chiastic Chiastic structure and Χ ).[ 4] a (n ) = 2n Mersenne numbers (OEIS : A000225
Gallery
Ruler ,
[ 1] [ 2] excluding 1 and 2:
ABACABAD ABACABA
excluding 2:
E ABACABADABACABA
Cantor set :
ABACABAD ABACABA
Binary tree [ 1] [ 2] /upside down
family tree :
ABACABAD ABACABA
Koch curve :
[ 1]
n
=
1
{\displaystyle n=1}
is A
B A,
n
=
2
{\displaystyle n=2}
is ABA
C ABA, and
n
=
3
{\displaystyle n=3}
: ABACABA
D ABACABA
Metric levels:
[ 1] E ABACABADABACABA
When counting in
binary (here 4-
bit ), the final 0s form an ABA
C ABA pattern
[ 1]
A staircase built with the largest possible squares/cubes while allowing equally sized steps:
A "circle fractal"
[ 1] superimposed with a
2 × 2 box fractal :
ABACABAD ABACABA
The
Tower of Hanoi [ 1] with four disks:
ABACABAD ABACABA
Binary-reflected
Gray code (BRGC):
to G
3-bit Gray code visualized as a traversal of vertices of a cube (0,1,3,2,6,7,5,4):
[ 1] ABAC ABA
Double harmonic scale (
Play ⓘ ) with steps of
H-3H-H-W-H-3H-H :
ABAC ABA
Gray code along the number line
[ 1] (
OEIS : A003188 ):
ABACABADABACABAE ABACABADABACABA
Devil's needle :
[ 1] ABACABAD ABACABA
Size of
hexagrams on a diagonal of a section of a
Menger sponge model:
ABACABAD ABACABA
See also
Notes
^ The strength, emphasis, or importance of the beginning of each duration
1
/
8
{\displaystyle 1/8}
4 4 divisively (
2
/
2
1
=
1
{\displaystyle 2/2^{1}=1}
4
/
2
2
=
1
{\displaystyle 4/2^{2}=1}
8
/
2
3
=
1
{\displaystyle 8/2^{3}=1}
D ABACABA structure, while the eighth notes of two measures grouped, additively (
8
×
2
=
16
{\displaystyle 8\times 2=16}
E ABACABADABACABA structure.[ 3]
References
^ a b c d e f g h i j k l m Naylor, Mike (February 2013). "ABACABA Amazing Pattern, Amazing Connections" . Math Horizons . Retrieved June 13, 2019 . ^ a b c d SheriOZ (2016-04-21). "Exploring Fractals with ABACABA" . Chicago Geek Guy . Archived from the original on 22 January 2021. Retrieved January 22, 2021 . ^ a b Naylor, Mike (2011). "Abacaba! – Using a mathematical pattern to connect art, music, poetry and literature" (PDF) . Bridges . Retrieved October 6, 2017 . ^ a b c Conley, Craig (2008-10-01). Magic Words: A Dictionary ISBN 9781609250508 ^ Halter-Koch, Franz and Tichy, Robert F.; eds. (2000). Algebraic Number Theory and Diophantine Analysis , p.478. W. de Gruyter. ISBN 9783110163049 .
^ Wright, Craig (2016). Listening to Western Music , p.48. Cengage Learning. ISBN 9781305887039 .
External links
![Sierpinski triangle[1][2]: ABACABA](File:ABACABA_Sierpinski_triangle.svg)
![Ruler,[1][2] excluding 1 and 2: ABACABADABACABA excluding 2: EABACABADABACABA](File:Measuring_-_Fractions_of_an_inch.svg)
![Binary tree[1][2]/upside down family tree: ABACABADABACABA](File:Ex_001.png)
![Koch curve:[1] '"`UNIQ--postMath-00000001-QINU`"' is ABA, '"`UNIQ--postMath-00000002-QINU`"' is ABACABA, and '"`UNIQ--postMath-00000003-QINU`"': ABACABADABACABA](File:KochCurve1_abacaba_pattern.png)
![Metric hierarchy: ABACABADABACABA[a]](File:Solfege_subdivision_de_la_ronde_a_la_croche.svg)
![Metric levels:[1] EABACABADABACABA](File:Metric_levels.svg)
![When counting in binary (here 4-bit), the final 0s form an ABACABA pattern[1]](File:Abacaba_binary.svg){kind=small}
![A staircase built with the largest possible squares/cubes while allowing equally sized steps: ABACABADABACABA[1]](File:Staircase_of_largest_squares_abacaba_pattern.svg)
![A "circle fractal"[1] superimposed with a 2 × 2 box fractal: ABACABADABACABA](File:Circle-box_fractal_abacaba_pattern.svg)
![The Tower of Hanoi[1] with four disks: ABACABADABACABA](File:Tower_of_Hanoi_recursion_SMIL.svg)
![3-bit Gray code visualized as a traversal of vertices of a cube (0,1,3,2,6,7,5,4):[1] ABACABA](File:Blue_cube_with_Gray_code_path.png)
![Château de Chambord: ABACABA[6]](File:Chambord_Castle_Northwest_facade.jpg)
![Gray code along the number line[1] (OEIS: A003188): ABACABADABACABAEABACABADABACABA](File:Gray_code_number_line_arcs.svg)
![Devil's needle:[1] ABACABADABACABA](File:Baguenaudier.svg)
