n-flake
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.
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.
[1]: 2 Its boundary is the von Koch curve of varying types – depending on the n-gon – and infinitely many Koch curves are contained within.
The formula of the scale factor r for any n-flake is:[2] where cosine is evaluated in radians and n is the number of sides of the n-gon.
, where m is the number of polygons in each individual flake and r is the scale factor.
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.
The boundary of the Vicsek Fractal is a Type 1 quadratic Koch curve.
[3] Each flake is formed by placing a pentagon in each corner and one in the center.
The boundary of a pentaflake is the Koch curve of 72 degrees.
This variation still contains infinitely many Koch curves, but they are somewhat more visible.
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.
A hexaflake, is formed by successive flakes of seven regular hexagons.
[4] Each flake is formed by placing a scaled hexagon in each corner and one in the center.
Therefore the hexaflake has 7n−1 hexagons in its nth iteration, and its Hausdorff dimension is equal to
Also, the projection of the cantor cube onto the plane orthogonal to its main diagonal is a hexaflake.
The hexaflake has been applied in the design of antennas[4] and optical fibers.
This variation still contains infinitely many Koch curves of 60 degrees.
[If a central polygon is generated, the scale factor differs for odd and even
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.
Because of this, three-dimensional n-flakes are also called platonic solid fractals.
Each flake is formed by placing a tetrahedron scaled by 1/2 in each corner.
On every face there is a Sierpinski triangle and infinitely many are contained within.
Another hexahedron flake can be produced in a manner similar to the Vicsek fractal extended to three dimensions.
Every cube is divided into 27 smaller cubes and the center cross is retained, which is the opposite of the Menger sponge where the cross is removed.
Each flake is formed by placing an octahedron scaled by 1/2 in each corner.
On every face there is a Sierpinski triangle and infinitely many are contained within.
