Ann-flake,polyflake, orSierpinskin-gon,^[1]: 1 is afractal constructed starting from ann-gon. Thisn-gon is replaced by a flake of smallern-gons, such that the scaled polygons are placed at thevertices, and sometimes in the center. This process is repeated recursively to result in the fractal. Typically, there is also the restriction that then-gons must touch yet not overlap.

The most common variety ofn-flake is two-dimensional (in terms of itstopological 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 thevon Koch curve of varying types – depending on then-gon – and infinitely many Koch curves are contained within. The fractals occupy zero area yet have an infinite perimeter.

The formula of thescale factorr for anyn-flake is:^[2]

${\displaystyle r=\frac{1}{2\left(1+{\displaystyle \sum _{k=1}^{\lfloor n/4\rfloor }\cos \frac{2\pi k}{n}}\right)}}$

where cosine is evaluated in radians andn is the number of sides of then-gon. TheHausdorff dimension of an-flake is${\displaystyle \frac{\log m}{\log r}}$, wherem is the number of polygons in each individual flake andr is the scale factor.

TheSierpinski triangle is ann-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. ItsHausdorff dimension is equal to${\displaystyle \frac{\log (3)}{\log (2)}}$ ≈ 1.585. The${\displaystyle \frac{\log (3)}{\log (2)}}$ is obtained because each iteration has 3 triangles that are scaled by 1/2.

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  The sixth iteration of the Sierpinski triangle.
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  The Sierpinski triangle created by thechaos game.

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, theVicsek 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${\displaystyle \frac{\log (5)}{\log (3)}}$ ≈ 1.4650. The${\displaystyle \frac{\log (5)}{\log (3)}}$ is obtained because each iteration has 5 squares that are scaled by 1/3. The boundary of the Vicsek Fractal is aType 1 quadratic Koch curve.

A pentaflake, or sierpinski pentagon, is formed by successive flakes of six regular pentagons.^[3]Each flake is formed by placing a pentagon in each corner and one in the center. Its Hausdorff dimension is equal to${\displaystyle \frac{\log (6)}{\log (1+\phi )}}$ ≈ 1.8617, where${\displaystyle \phi =\frac{1+\sqrt{5}}{2}}$ (golden ratio). The${\displaystyle \frac{\log (6)}{\log (1+\phi )}}$ is obtained because each iteration has 6 pentagons that are scaled by${\displaystyle \frac{1}{1+\phi }}$. 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${\displaystyle \frac{\log (5)}{\log (1+\phi )}}$ ≈ 1.6723. This variation still contains infinitely many Koch curves, but they are somewhat more visible.

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  3rd iteration, with center pentagons
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  4th iteration, with center pentagons
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  5th iteration, with center pentagons

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  2nd iteration, without center pentagons
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  3rd iteration, without center pentagons
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  4th iteration, without center pentagons
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  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.

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  Pentaflake tiling. Center point not covered.
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  Pentaflake tiling. Center point covered.

Ahexaflake, 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. Each iteration has 7 hexagons that are scaled by 1/3. Therefore the hexaflake has 7ⁿ⁻¹ hexagons in itsnth iteration, and its Hausdorff dimension is equal to${\displaystyle \frac{\log (7)}{\log (3)}}$ ≈ 1.7712. The boundary of a hexaflake is the standard Koch curve of 60 degrees and infinitely manyKoch snowflakes are contained within. Also, the projection of thecantor cube onto the planeorthogonal to its main diagonal is a hexaflake.The hexaflake has been applied in the design ofantennas^[4] andoptical fibers.^[5]

Like the pentaflake, there is also a variation of the hexaflake, called the Sierpinski hexagon, that has no central hexagon.^[6] Its Hausdorff dimension equals${\displaystyle \frac{\log (6)}{\log (3)}}$ ≈ 1.6309. This variation still contains infinitely many Koch curves of 60 degrees.

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  Hexaflake
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  The first six iterations of the hexaflake.
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  Fourth iteration of the Sierpinski hexagon.
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  Orthogonal projection of a cantor cube showing a hexaflake.

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${\displaystyle n}$:${\displaystyle R=1-2r}$ for even${\displaystyle n}$ and${\displaystyle R=\frac{1-r}{cos(\pi /n)}-r}$ for odd${\displaystyle n}$.] 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 asn increases. Their Hausdorff dimensions are slightly more difficult to calculate than lowern-flakes because their scale factor is less obvious. However, the Hausdorff dimension is always less than two but no less than one. An interestingn-flake is the ∞-flake, because as the value ofn increases, ann-flake's Hausdorff dimension approaches 1,^[1]: 7

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  The first four iterations of the heptaflake or 7-flake.
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  The first four iterations of the octoflake or 8-flake.
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  The first four iterations of the enneaflake or 9-flake.
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  The first four iterations of the decaflake or 10-flake.
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  The first four iterations of the hendecaflake or 11-flake.
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  The first four iterations of the dodecaflake or 12-flake.

n-flakes can generalized to higher dimensions, in particular to atopological dimension of three.^[8] Instead of polygons, regularpolyhedra are iteratively replaced. However, while there are an infinite number of regular polygons, there are only five regular, convex polyhedra. Because of this, three-dimensional n-flakes are also calledplatonic solid fractals.^[9] In three dimensions, the fractals' volume is zero.

ASierpinski tetrahedron is formed by successive flakes of four regular tetrahedrons. Each flake is formed by placing atetrahedron scaled by 1/2 in each corner. Its Hausdorff dimension is equal to${\displaystyle \frac{\log (4)}{\log (2)}}$, which is exactly equal to 2. On every face there is a Sierpinski triangle and infinitely many are contained within.

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  The third iteration of the Sierpinski tetrahedron.

A hexahedron, or cube, flake defined in the same way as the Sierpinski tetrahedron is simply a cube^[10] and is not interesting as a fractal. However, there are two pleasing alternatives. One is theMenger Sponge, where every cube is replaced by a three dimensional ring of cubes. Its Hausdorff dimension is${\displaystyle \frac{\log (20)}{\log (3)}}$ ≈ 2.7268.

Another hexahedron flake can be produced in a manner similar to theVicsek fractal extended to three dimensions. Every cube is divided into 27 smaller cubes and the center cross is retained, which is the opposite of theMenger sponge where the cross is removed. However, it is not the Menger Sponge complement. Its Hausdorff dimension is${\displaystyle \frac{\log (7)}{\log (3)}}$ ≈ 1.7712, because a cross of 7 cubes, each scaled by 1/3, replaces each cube.

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  The fourth iteration of the Menger Sponge.
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  Third iteration of the3D Vicsek fractal.

An octahedron flake, or sierpinski octahedron, is formed by successive flakes of six regular octahedra. Each flake is formed by placing anoctahedron scaled by 1/2 in each corner. Its Hausdorff dimension is equal to${\displaystyle \frac{\log (6)}{\log (2)}}$ ≈ 2.5849. On every face there is a Sierpinski triangle and infinitely many are contained within.

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  The third iteration of the octahedron flake.

A dodecahedron flake, or sierpinski dodecahedron, is formed by successive flakes of twenty regular dodecahedra. Each flake is formed by placing adodecahedron scaled by${\displaystyle \frac{1}{2+\phi }}$ in each corner. Its Hausdorff dimension is equal to${\displaystyle \frac{\log (20)}{\log (2+\phi )}}$ ≈ 2.3296.

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  The second iteration of the dodecahedron fractal flake.

An icosahedron flake, or sierpinski icosahedron, is formed by successive flakes of twelve regular icosahedra. Each flake is formed by placing anicosahedron scaled by${\displaystyle \frac{1}{1+\phi }}$ in each corner. Its Hausdorff dimension is equal to${\displaystyle \frac{\log (12)}{\log (1+\phi )}}$ ≈ 2.5819.

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  The third iteration of the icosahedron fractal flake.

• List of fractals by Hausdorff dimension

• Quadraflakes, Pentaflakes, Hexaflakes and more – includes Mathematica code to generate these fractals
• Javascript for covering the plane with 5-fold symmetric Pentaflake tiles.

• Fractals
• Fractal curves