An n-flake, polyflake, or Sierpinski n-gon, is a fractal constructed starting from an n-gon. This n-gon is replaced by a flake of smaller n-gons, such that the scaled polygons are placed at the vertices, and sometimes in the center. This process is repeated recursively to result in the fractal. Typically, there is also the restriction that the n-gons must touch yet not overlap.
In two dimensions The most common variety of n-flake is two-dimensional (in terms of its topological dimension) and is formed of polygons. The four most common special cases are formed with triangles, squares, pentagons, and hexagons, but it can be extended to any polygon. Its boundary is the von Koch curve of varying types – depending on the n-gon – and infinitely many Koch curves are contained within. The fractals occupy zero area yet have an infinite perimeter. The formula of the scale factor r for any n-flake is:
r = 1 2 ( 1 + ∑ k = 1 ⌊ n / 4 ⌋ cos 2 π k n ) {\displaystyle r={\frac {1}{2\left(1+\displaystyle \sum _{k=1}^{\lfloor n/4\rfloor }{\cos {\frac {2\pi k}{n}}}\right)}}}
where cosine is evaluated in radians and n is the number of sides of the n-gon. The Hausdorff dimension of a n-flake is log m log r {\displaystyle \textstyle {\frac {\log m}{\log r}}} , where m is the number of polygons in each individual flake and r is the scale factor.
Sierpinski triangle The Sierpinski triangle is an n-flake formed by successive flakes of three triangles. Each flake is formed by placing triangles scaled by 1/2 in each corner of the triangle they replace. Its Hausdorff dimension is equal to log ( 3 ) log ( 2 ) {\displaystyle \textstyle {\frac {\log(3)}{\log(2)}}} ≈ 1.585. The log ( 3 ) log ( 2 ) {\displaystyle \textstyle {\frac {\log(3)}{\log(2)}}} is obtained because each iteration has 3 triangles that are scaled by 1/2.
Vicsek fractal
If a sierpinski 4-gon were constructed from the given definition, the scale factor would be 1/2 and the fractal would simply be a square. A more interesting alternative, the Vicsek fractal, rarely called a quadraflake, is formed by successive flakes of five squares scaled by 1/3. Each flake is formed either by placing a scaled square in each corner and one in the center or one on each side of the square and one in the center. Its Hausdorff dimension is equal to log ( 5 ) log ( 3 ) {\displaystyle \textstyle {\frac {\log(5)}{\log(3)}}} ≈ 1.4650. The log ( 5 ) log ( 3 ) {\displaystyle \textstyle {\frac {\log(5)}{\log(3)}}} is obtained because each iteration has 5 squares that are scaled by 1/3. The boundary of the Vicsek Fractal is a Type 1 quadratic Koch curve.
Pentaflake
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