The **Koch snowflake** (also known as the **Koch curve**, **Koch star**, or **Koch island**^{ [1] }^{ [2] }) 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 Curve Without Tangents, Constructible from Elementary Geometry"^{ [3] } by the Swedish mathematician Helge von Koch.

- Construction
- Properties
- Perimeter of the Koch snowflake
- Area of the Koch snowflake
- Other properties
- Tessellation of the plane
- Thue–Morse sequence and turtle graphics
- Representation as Lindenmayer system
- Variants of the Koch curve
- See also
- References
- Further reading
- External links

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.

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

- divide the line segment into three segments of equal length.
- draw an equilateral triangle that has the middle segment from step 1 as its base and points outward.
- 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 Helge von Koch is constructed using only one of the three sides of the original triangle. In other words, three Koch curves make a Koch snowflake.

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] }

Each iteration multiplies the number of sides in the Koch snowflake by four, so the number of sides after n iterations is given by:

If the original equilateral triangle has sides of length s, the length of each side of the snowflake after n iterations is:

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

The Koch curve has an infinite length, because the total length of the curve increases by a factor of 4/3 with each iteration. Each iteration creates four times as many line segments as in the previous iteration, with the length of each one being 1/3 the length of the segments in the previous stage. Hence, the length of the curve after n iterations will be (4/3)^{n} times the original triangle perimeter and is unbounded, as n tends to infinity.

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

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.^{ [5] }

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:

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:

where *a*_{0} is the area of the original triangle. The total new area added in iteration n is therefore:

The total area of the snowflake after n iterations is:

Collapsing the geometric sum gives:

The limit of the area is:

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:^{ [6] }

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 ^{ [7] }

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 fractal dimension of the Koch curve is ln 4/ln 3 ≈ 1.26186. This is greater than that of a line (=1) but less than that of Peano's space-filling curve (=2).

The Koch curve is continuous everywhere, but differentiable nowhere.

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.^{ [8] } Koch snowflakes and Koch antisnowflakes of the same size may be used to tile the plane.

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.

The Koch curve can be expressed by the following rewrite system (Lindenmayer system):

**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.

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 ^{ [9] } | |

≤2D, 90° angle | Minkowski Island | |

≈1.37D, 90° angle | ||

≤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 Sierpiński pyramid and Menger sponge can be considered extensions of the Sierpinski triangle and 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.^{ [15] } 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:

while the total length of the perimeter is:

which approaches infinity as n increases.

- 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

In mathematics, a **geometric series** is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series

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 line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the *Hausdorff–Besicovitch dimension.*

A **perimeter** is either a path that encompasses/surrounds/outlines a shape or its length (one-dimensional). The perimeter of a circle or an ellipse is called its circumference.

The **Sierpiński triangle**, also called the **Sierpiński gasket** or **Sierpiński sieve**, is a fractal attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets—that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries before the work of Sierpiński.

In the context of complex dynamics, a topic of mathematics, the **Julia set** and the Fatou set are two complementary sets defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is "regular", while on the Julia set its behavior is "chaotic".

In mathematics, the **harmonic series** is the divergent infinite series

In mathematics, the **Menger sponge** is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension.

In mathematics, more specifically in fractal geometry, a **fractal dimension** is a ratio providing a statistical index of complexity comparing how detail in a pattern changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer.

**Gabriel's horn** is a particular geometric figure that has infinite surface area but finite volume. The name refers to the Christian tradition that identifies the archangel Gabriel as the angel who blows the horn to announce Judgment Day, associating the divine, or infinite, with the finite. The properties of this figure were first studied by Italian physicist and mathematician Evangelista Torricelli in the 17th century.

In mathematics, a **telescoping series** is a series whose general term can be written as , i.e. the difference of two consecutive terms of a sequence .

**Sierpiński curves** are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit completely fill the unit square: thus their limit curve, also called **the Sierpiński curve**, is an example of a space-filling curve.

In mathematics, the **T-square** is a two-dimensional fractal. It has a boundary of infinite length bounding a finite area. Its name comes from the drawing instrument known as a T-square.

In mathematics, a **de Rham curve** is a certain type of fractal curve named in honor of Georges de Rham.

In mathematics, the infinite series **1/4 + 1/16 + 1/64 + 1/256 + ⋯** is an example of one of the first infinite series to be summed in the history of mathematics; it was used by Archimedes circa 250–200 BC. As it is a geometric series with first term 1/4 and common ratio 1/4, its sum is

In geometry, the **spiral of Theodorus** is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene.

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, particularly in cellular phones.

An ** n-flake**,

The **Fibonacci word fractal** is a fractal curve defined on the plane from the Fibonacci word.

The **Ulam–Warburton cellular automaton** (UWCA) is a 2-dimensional fractal pattern that grows on a regular grid of cells consisting of squares. Starting with one square initially ON and all others OFF, successive iterations are generated by turning ON all squares that share precisely one edge with an ON square. This is the von Neumann neighborhood. The automaton is named after the Polish-American mathematician and scientist Stanislaw Ulam and the Scottish engineer, inventor and amateur mathematician Mike Warburton.

- ↑ Addison, Paul S. (1997).
*Fractals and Chaos: An Illustrated Course*. Institute of Physics. p. 19. ISBN 0-7503-0400-6. - ↑ Lauwerier, Hans (1991).
*Fractals: Endlessly Repeated Geometrical Figures*. Translated by Gill-Hoffstädt, Sophia. Princeton University Press. p. 36. ISBN 0-691-02445-6.Mandelbrot called this a Koch island.

- ↑ von Koch, Helge (1904). "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire".
*Arkiv för matematik, astronomi och fysik*(in French).**1**: 681–704. JFM 35.0387.02. - ↑ Alonso-Marroquin, F.; Huang, P.; Hanaor, D.; Flores-Johnson, E.; Proust, G.; Gan, Y.; Shen, L. (2015). "Static friction between rigid fractal surfaces" (PDF).
*Physical Review E*.**92**(3): 032405. doi:10.1103/PhysRevE.92.032405. hdl: 2123/13835 . PMID 26465480. — Study of fractal surfaces using Koch curves. - ↑ Zhu, Zhi Wei; Zhou, Zuo Ling; Jia, Bao Guo (October 2003). "On the Lower Bound of the Hausdorff Measure of the Koch Curve".
*Acta Mathematica Sinica*.**19**(4): 715–728. doi:10.1007/s10114-003-0310-2. S2CID 122517792. - ↑ "Koch Snowflake".
*ecademy.agnesscott.edu*. - ↑ McCartney, Mark (2020-04-16). "The area, centroid and volume of revolution of the Koch curve".
*International Journal of Mathematical Education in Science and Technology*.**0**: 1–5. doi:10.1080/0020739X.2020.1747649. ISSN 0020-739X. - ↑ Burns, Aidan (1994). "Fractal tilings".
*Mathematical Gazette*.**78**(482): 193–6. doi:10.2307/3618577. JSTOR 3618577.. - ↑ Paul S. Addison,
*Fractals and Chaos: An illustrated course*, p. 19, CRC Press, 1997 ISBN 0849384435. - ↑ Weisstein, Eric W. (1999). "Minkowski Sausage",
*archive.lib.msu.edu*. Accessed: 21 September 2019. - ↑ Pamfilos, Paris. "Minkowski Sausage",
*user.math.uoc.gr/~pamfilos/*. Accessed: 21 September 2019. - ↑ Weisstein, Eric W. "Minkowski Sausage".
*MathWorld*. Retrieved 22 September 2019. - ↑ Mandelbrot, B. B. (1983).
*The Fractal Geometry of Nature*, p.48. New York: W. H. Freeman. ISBN 9780716711865. Cited in Weisstein, Eric W. "Minkowski Sausage".*MathWorld*. Retrieved 22 September 2019.. - ↑ Appignanesi, Richard; ed. (2006).
*Introducing Fractal Geometry*. Icon. ISBN 978-1840467-13-0. - ↑ Demonstrated by James McDonald in a public lecture at KAUST University on January 27, 2013. "Archived copy". Archived from the original on 2013-01-12. Retrieved 2013-01-29.CS1 maint: archived copy as title (link) retrieved 29 January 2013.

- Kasner, Edward; Newman, James (2001) [1940]. "IX Change and Changeability § The snowflake".
*Mathematics and the Imagination*. Dover Press. pp. 344–351. ISBN 0-486-41703-4.

External video | |
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Koch Snowflake Fractal |

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.

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