Koch snowflake: Difference between revisions

Koch antisnowflake

First four iterations

Sixth iteration

The Koch snowflake (also known as the Koch curve, Koch star, or Koch island

The Koch snowflake can be built up iteratively, in a sequence of stages. The first stage is an equilateral triangle, and each successive stage is formed by adding outward bends to each side of the previous stage, making smaller equilateral triangles. The areas enclosed by the successive stages in the construction of the snowflake converge to 8/5 times the area of the original triangle, while the perimeters of the successive stages increase without bound. Consequently, the snowflake encloses a finite area, but has an infinite perimeter.

Construction

The Koch snowflake can be constructed by starting with an equilateral triangle, then recursively altering each line segment as follows:

1. divide the line segment into three segments of equal length.
2. draw an equilateral triangle that has the middle segment from step 1 as its base and points outward.
3. remove the line segment that is the base of the triangle from step 2.

The first iteration of this process produces the outline of a hexagram.

The Koch snowflake is the limit approached as the above steps are followed indefinitely. The Koch curve originally described by

A Koch curve–based representation of a nominally flat surface can similarly be created by repeatedly segmenting each line in a sawtooth pattern of segments with a given angle.^[4]

The Koch curve can also be generated starting with only a single black cell and applying a nested substitution, in a manner similar to a generalization of

an inverse power of three multiple of the original length. The perimeter of the snowflake after n iterations is:

${\displaystyle P_{n}=N_{n}\cdot S_{n}=3\cdot s\cdot {\left({\frac {4}{3}}\right)}^{n}\,.}$

The Koch curve has an

Limit of perimeter

As the number of iterations tends to infinity, the limit of the perimeter is:

${\displaystyle \lim _{n\rightarrow \infty }P_{n}=\lim _{n\rightarrow \infty }3\cdot s\cdot \left({\frac {4}{3}}\right)^{n}=\infty \,,}$

since |4/3| > 1.

An ln 4/ln 3-dimensional measure exists, but has not been calculated so far. Only upper and lower bounds have been invented.^[6]

Area of the Koch snowflake

In each iteration a new triangle is added on each side of the previous iteration, so the number of new triangles added in iteration n is:

${\displaystyle T_{n}=N_{n-1}=3\cdot 4^{n-1}={\frac {3}{4}}\cdot 4^{n}\,.}$

The area of each new triangle added in an iteration is 1/9 of the area of each triangle added in the previous iteration, so the area of each triangle added in iteration n is:

${\displaystyle a_{n}={\frac {a_{n-1}}{9}}={\frac {a_{0}}{9^{n}}}\,.}$

where a₀ is the area of the original triangle. The total new area added in iteration n is therefore:

${\displaystyle b_{n}=T_{n}\cdot a_{n}={\frac {3}{4}}\cdot {\left({\frac {4}{9}}\right)}^{n}\cdot a_{0}}$

The total area of the snowflake after n iterations is:

${\displaystyle A_{n}=a_{0}+\sum _{k=1}^{n}b_{k}=a_{0}\left(1+{\frac {3}{4}}\sum _{k=1}^{n}\left({\frac {4}{9}}\right)^{k}\right)=a_{0}\left(1+{\frac {1}{3}}\sum _{k=0}^{n-1}\left({\frac {4}{9}}\right)^{k}\right)\,.}$

Collapsing the geometric sum gives:

${\displaystyle A_{n}=a_{0}\left(1+{\frac {3}{5}}\left(1-\left({\frac {4}{9}}\right)^{n}\right)\right)={\frac {a_{0}}{5}}\left(8-3\left({\frac {4}{9}}\right)^{n}\right)\,.}$

Limits of area

The limit of the area is:

${\displaystyle \lim _{n\rightarrow \infty }A_{n}=\lim _{n\rightarrow \infty }{\frac {a_{0}}{5}}\cdot \left(8-3\left({\frac {4}{9}}\right)^{n}\right)={\frac {8}{5}}\cdot a_{0}\,,}$

since |4/9| < 1.

Thus, the area of the Koch snowflake is 8/5 of the area of the original triangle. Expressed in terms of the side length s of the original triangle, this is:^[7]

${\displaystyle {\frac {2s^{2}{\sqrt {3}}}{5}}.}$

Solid of revolution

The volume of the solid of revolution of the Koch snowflake about an axis of symmetry of the initiating equilateral triangle of unit side is ${\displaystyle {\frac {11{\sqrt {3}}}{135}}\pi .}$^[8]

Other properties

The Koch snowflake is self-replicating with six smaller copies surrounding one larger copy at the center. Hence, it is an irrep-7 irrep-tile (see Rep-tile for discussion).

The

The Koch curve is continuous everywhere, but differentiable nowhere.

Tessellation of the plane

It is possible to tessellate the plane by copies of Koch snowflakes in two different sizes. However, such a tessellation is not possible using only snowflakes of one size. Since each Koch snowflake in the tessellation can be subdivided into seven smaller snowflakes of two different sizes, it is also possible to find tessellations that use more than two sizes at once.^[9] Koch snowflakes and Koch antisnowflakes of the same size may be used to tile the plane.

Thue–Morse sequence and turtle graphics

A turtle graphic is the curve that is generated if an automaton is programmed with a sequence. If the Thue–Morse sequence members are used in order to select program states:

• If t(n) = 0, move ahead by one unit,
• If t(n) = 1, rotate counterclockwise by an angle of π/3,

the resulting curve converges to the Koch snowflake.

Representation as Lindenmayer system

The Koch curve can be expressed by the following

Alphabet : F

Constants : +, −

Axiom : F

Production rules:

F → F+F--F+F

Here, F means "draw forward", - means "turn right 60°", and + means "turn left 60°".

To create the Koch snowflake, one would use F--F--F (an equilateral triangle) as the axiom.

Variants of the Koch curve

Following von Koch's concept, several variants of the Koch curve were designed, considering right angles (quadratic), other angles (Cesàro), circles and polyhedra and their extensions to higher dimensions (Sphereflake and Kochcube, respectively)

Variant (dimension, angle)	Illustration	Construction
≤1D, 60-90° angle		The Cesàro fractal is a variant of the Koch curve with an angle between 60° and 90°.[citation needed]
≈1.46D, 90° angle
1.5D, 90° angle		Minkowski Sausage[10]
≤2D, 90° angle		Minkowski Island
≈1.37D, 90° angle		Minkowski Sausage",[11][12][13] its fractal dimension equals ln 3/ln √5 = 1.36521.[14]
≤2D, 90° angle		Anticross-stitch curve, the quadratic flake type 1, with the curves facing inwards instead of outwards (Vicsek fractal)
≈1.49D, 90° angle		Another variation. Its fractal dimension equals ln 3.33/ln √5 = 1.49.
≤2D, 90° angle
≤2D, 60° angle
≤2D, 90° angle		Extension of the quadratic type 1 curve. The illustration at left shows the fractal after the second iteration .
≤3D, any		A three-dimensional fractal constructed from Koch curves. The shape can be considered a three-dimensional extension of the curve in the same sense that the Sierpinski carpet . The version of the curve used for this shape uses 85° angles.

Squares can be used to generate similar fractal curves. Starting with a unit square and adding to each side at each iteration a square with dimension one third of the squares in the previous iteration, it can be shown that both the length of the perimeter and the total area are determined by geometric progressions. The progression for the area converges to 2 while the progression for the perimeter diverges to infinity, so as in the case of the Koch snowflake, we have a finite area bounded by an infinite fractal curve.^[16] The resulting area fills a square with the same center as the original, but twice the area, and rotated by π/4 radians, the perimeter touching but never overlapping itself.

The total area covered at the nth iteration is:

${\displaystyle A_{n}={\frac {1}{5}}+{\frac {4}{5}}\sum _{k=0}^{n}\left({\frac {5}{9}}\right)^{k}\quad {\mbox{giving}}\quad \lim _{n\rightarrow \infty }A_{n}=2\,,}$

while the total length of the perimeter is:

${\displaystyle P_{n}=4\left({\frac {5}{3}}\right)^{n}a\,,}$

which approaches infinity as n increases.

See also

• List of fractals by Hausdorff dimension
• Gabriel's Horn
  (infinite surface area but encloses a finite volume)
• Gosper curve (also known as the Peano–Gosper curve or flowsnake)
• Osgood curve
• Self-similarity
• Teragon
• Weierstrass function
• Coastline paradox

References

Further reading

• Kasner, Edward; Newman, James (2001) [1940]. "IX Change and Changeability § The snowflake".
  ISBN 0-486-41703-4
  .

External links

External videos
Koch Snowflake Fractal – Khan Academy

Wikimedia Commons has media related to Koch curve.

Wikimedia Commons has media related to Koch snowflake.

• (2000) "von Koch Curve", efg's Computer Lab at the Wayback Machine (archived 20 July 2017)
• The Koch Curve poem by Bernt Wahl, Wahl.org. Retrieved 23 September 2019.
• Weisstein, Eric W. "Koch Snowflake". MathWorld. Retrieved 23 September 2019.
  • "7 iterations of the Koch curve".
    Wolfram Alpha
    Site. Retrieved 23 September 2019.
  • "Square Koch Fractal Curves". Wolfram Demonstrations Project. Retrieved 23 September 2019.
  • "Square Koch Fractal Surface". Wolfram Demonstrations Project. Retrieved 23 September 2019.
• Application of the Koch curve to an antenna
• A WebGL animation showing the construction of the Koch surface, tchaumeny.github.io. Retrieved 23 September 2019.
• "A mathematical analysis of the Koch curve and quadratic Koch curve" (PDF). Archived from the original (pdf) on 26 April 2012. Retrieved 22 November 2011.