An ''n''-flake, polyflake, or Sierpinski ''n''-gon,
is a
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
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
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a
topological invariant, topologically invariant way.
Informal discussion
F ...
) 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
The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Cur ...
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
In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a '' scale factor'' that is the same in all directions. The result of uniform scaling is similar ...
''r'' for any ''n''-flake is:
:
where cosine is evaluated in radians and ''n'' is the number of sides of the ''n''-gon. The Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a ...
of a ''n''-flake is , 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
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a ...
is equal to ≈ 1.585. The is obtained because each iteration has 3 triangles that are scaled by 1/2.
File:Sierpinski triangle.svg, The sixth iteration of the Sierpinski triangle.
File:Sierpinski chaos animated.gif, The Sierpinski triangle created by the chaos game
In mathematics, the term chaos game originally referred to a method of creating a fractal, using a polygon and an initial point selected at random inside it. The fractal is created by iteratively creating a sequence of points, starting with the ...
.
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
In mathematics the Vicsek fractal, also known as Vicsek snowflake or box fractal, is a fractal arising from a construction similar to that of the Sierpinski carpet, proposed by Tamás Vicsek. It has applications including as compact antennas, pa ...
, 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 ≈ 1.4650. The 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
A pentaflake, or sierpinski pentagon, is formed by successive flakes of six regular pentagons.
Each flake is formed by placing a pentagon in each corner and one in the center. Its Hausdorff dimension is equal to ≈ 1.8617, where (golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
). The is obtained because each iteration has 6 pentagons that are scaled by . 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 ≈ 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.
Hexaflake
A hexaflake, is formed by successive flakes of seven regular hexagons.[.] 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 7''n''−1 hexagons in its ''n''th iteration, and its Hausdorff dimension is equal to ≈ 1.7712. The boundary of a hexaflake is the standard Koch curve of 60 degrees and infinitely many Koch snowflake
The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curv ...
s are contained within. Also, the projection of the cantor cube
In mathematics, a Cantor cube is a topological group of the form ''A'' for some index set ''A''. Its algebraic and topological structures are the group direct product and product topology over the cyclic group of order 2 (which is itself given th ...
onto the plane orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
to its main diagonal is a hexaflake.
The hexaflake has been applied in the design of antennas
In radio engineering, an antenna or aerial is the interface between radio waves propagating through space and electric currents moving in metal conductors, used with a transmitter or receiver. In transmission, a radio transmitter supplies a ...
and optical fiber
An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparent fiber made by drawing glass (silica) or plastic to a diameter slightly thicker than that of a human hair. Optical fibers are used most often as a means to ...
s.
Like the pentaflake, there is also a variation of the hexaflake, called the Sierpinski hexagon, that has no central hexagon. Its Hausdorff dimension equals ≈ 1.6309. This variation still contains infinitely many Koch curves of 60 degrees.
File:HexaFlake_5th_Iteration_Center.svg, Hexaflake
File:Hexaflake.gif, The first six iterations of the hexaflake.
File:HexaFlake_4th_Iteration_No_Center.svg, Fourth iteration of the Sierpinski hexagon.
File:Cantor cube as hexaflake.gif, Orthogonal projection of a cantor cube showing a hexaflake.
Polyflake
''n''-flakes of higher polygons also exist, though they are less common and don't usually have a central polygon. 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.
]
In three dimensions
''n''-flakes can generalized to higher dimensions, in particular to a topological dimension
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a
topological invariant, topologically invariant way.
Informal discussion
F ...
of three. Instead of polygons, regular polyhedra
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on t ...
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 called platonic solid fractals. In three dimensions, the fractals' volume is zero.
Sierpinski tetrahedron
A Sierpinski tetrahedron is formed by successive flakes of four regular tetrahedrons. Each flake is formed by placing a tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
scaled by 1/2 in each corner. Its Hausdorff dimension is equal to , which is exactly equal to 2. On every face there is a Sierpinski triangle and infinitely many are contained within.
File:Tetraedre Sierpinski.png, The third iteration of the Sierpinski tetrahedron.
Hexahedron flake
A hexahedron, or cube, flake defined in the same way as the Sierpinski tetrahedron is simply a cube and is not interesting as a fractal. However, there are two pleasing alternatives. One is the Menger Sponge
In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Si ...
, where every cube is replaced by a three dimensional ring of cubes. Its Hausdorff dimension is ≈ 2.7268.
Another hexahedron flake can be produced in a manner similar to the Vicsek fractal
In mathematics the Vicsek fractal, also known as Vicsek snowflake or box fractal, is a fractal arising from a construction similar to that of the Sierpinski carpet, proposed by Tamás Vicsek. It has applications including as compact antennas, pa ...
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
In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Si ...
where the cross is removed. However, it is not the Menger Sponge complement. Its Hausdorff dimension is ≈ 1.7712, because a cross of 7 cubes, each scaled by 1/3, replaces each cube.
File:Menger-Schwamm-farbig.png, The fourth iteration of the Menger Sponge.
File:3D Vicsek Fractal.gif, Third iteration of the 3D Vicsek fractal
3-D, 3D, or 3d may refer to:
Science, technology, and mathematics Relating to three-dimensionality
* Three-dimensional space
** 3D computer graphics, computer graphics that use a three-dimensional representation of geometric data
** 3D film, a ...
.
Octahedron flake
An octahedron flake, or sierpinski octahedron, is formed by successive flakes of six regular octahedra. Each flake is formed by placing an octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
scaled by 1/2 in each corner. Its Hausdorff dimension is equal to ≈ 2.5849. On every face there is a Sierpinski triangle and infinitely many are contained within.
File:Octaedron fractal.jpg, The third iteration of the octahedron flake.
Dodecahedron flake
A dodecahedron flake, or sierpinski dodecahedron, is formed by successive flakes of twenty regular dodecahedra. Each flake is formed by placing a dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
scaled by in each corner. Its Hausdorff dimension is equal to ≈ 2.3296.
File:Dodecaedron fractal.jpg, The second iteration of the dodecahedron fractal flake.
Icosahedron flake
An icosahedron flake, or sierpinski icosahedron, is formed by successive flakes of twelve regular icosahedra. Each flake is formed by placing an icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
scaled by in each corner. Its Hausdorff dimension is equal to ≈ 2.5819.
File:Icosaedron fractal.jpg, The third iteration of the icosahedron fractal flake.
See also
* List of fractals by Hausdorff dimension
According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension."
Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illus ...
References
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.
{{Fractals
Fractals
Fractal curves