N-flake

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.

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. 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 \textstyle \frac{\log m}{\log r} , where m is the number of polygons in each individual flake and r is the scale factor. ===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 \textstyle{\frac {\log(5)} {\log(3)}} ≈ 1.4650. The \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. Each flake is formed by placing a pentagon in each corner and one in the center. Its Hausdorff dimension is equal to \textstyle{\frac {\log(6)} {\log(1+\varphi)}} ≈ 1.8617, where \textstyle{\varphi = \frac{1+\sqrt 5}{2}} (golden ratio). The \textstyle{\frac {\log(6)} {\log(1+\varphi)}} is obtained because each iteration has 6 pentagons that are scaled by \textstyle{\frac {1} {1+\varphi}} . The boundary of a pentaflake is the Koch curve of 72 degrees. There is also a variation of the pentaflake that has no central pentagon. Its Hausdorff dimension equals \textstyle{\frac {\log(5)} {\log(1+\varphi)}} ≈ 1.6723. This variation still contains infinitely many Koch curves, but they are somewhat more visible. File:Pentaflake-C 2nd Iteration Blue.svg File:Pentaflake-C 3rd Iteration Blue.svg|3rd iteration, with center pentagons File:Pentaflake-C 4th Iteration Blue.svg|4th iteration, with center pentagons File:Pentaflake-C 5th Iteration Blue.svg|5th iteration, with center pentagons File:Pentaflake-NC 2nd Iteration Blue.svg|2nd iteration, without center pentagons File:Pentaflake-NC 3rd Iteration Blue.svg|3rd iteration, without center pentagons File:Pentaflake-NC 4th Iteration Blue.svg|4th iteration, without center pentagons File:Pentaflake-NC 5th Iteration Blue.svg|5th iteration, without center pentagons Concentric patterns of pentaflake boundary shaped tiles can cover the plane, with the central point being covered by a third shape formed of segments of 72-degree Koch curve, also with 5-fold rotational and reflective symmetry. File:PentaflakeTiling1.jpg|Pentaflake tiling. Center point not covered. File:PentaflakeTiling2.jpg|Pentaflake tiling. Center point covered. n-flakes of higher polygons also exist, though they are less common and usually do not have a central polygon. [If a central polygon is generated, the scale factor differs for odd and even n : R = 1-2r for even n and \textstyle{R=\frac{1-r}{cos(\pi/n)}-r} for odd n .] Some examples are shown below; the 7-flake through 12-flake. While it may not be obvious, these higher polyflakes still contain infinitely many Koch curves, but the angle of the Koch curves decreases as n increases. Their Hausdorff dimensions are slightly more difficult to calculate than lower n-flakes because their scale factor is less obvious. However, the Hausdorff dimension is always less than two but no less than one. An interesting n-flake is the ∞-flake, because as the value of n increases, an n-flake's Hausdorff dimension approaches 1, File:Heptaflake-NC Iterations 01-04.svg|The first four iterations of the heptaflake or 7-flake. File:Octoflake-NC Iterations 01-04.svg|The first four iterations of the octoflake or 8-flake. File:Enneaflake-NC Iterations 01-04.svg|The first four iterations of the enneaflake or 9-flake. File:Decaflake-NC Iterations 01-04.svg|The first four iterations of the decaflake or 10-flake. File:Hendecaflake-NC Iterations 01-04.svg|The first four iterations of the hendecaflake or 11-flake. File:Dodecaflake-NC Iterations 01-04.svg|The first four iterations of the dodecaflake or 12-flake. ==External links== Quadraflakes, Pentaflakes, Hexaflakes and more – includes Mathematica code to generate these fractals Javascript for covering the plane with 5-fold symmetric Pentaflake tiles.

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