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The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a
fractal curve A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectif ...
and one of the earliest
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 ill ...
s 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" by the Swedish mathematician
Helge von Koch Niels Fabian Helge von Koch (25 January 1870 – 11 March 1924) was a Swedish mathematician who gave his name to the famous fractal known as the Koch snowflake, one of the earliest fractal curves to be described. He was born to Swedish nobility ...
. 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 \tfrac 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 In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
, 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 Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
of this process produces the outline of a
hexagram , can be seen as a compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green). A hexagram ( Greek language, Greek) or sexagram ( Latin) is a six-pointe ...
. The Koch snowflake is the limit approached as the above steps are followed indefinitely. The Koch curve originally described by
Helge von Koch Niels Fabian Helge von Koch (25 January 1870 – 11 March 1924) was a Swedish mathematician who gave his name to the famous fractal known as the Koch snowflake, one of the earliest fractal curves to be described. He was born to Swedish nobility ...
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.


Properties


Perimeter of the Koch snowflake

Each iteration multiplies the number of sides in the Koch snowflake by four, so the number of sides after n iterations is given by: N_ = N_ \cdot 4 = 3 \cdot 4^\, . If the original equilateral triangle has sides of length s, the length of each side of the snowflake after n iterations is: S_ = \frac = \frac\, , an inverse
power of three In mathematics, a power of three is a number of the form where is an integer – that is, the result of exponentiation with number three as the base and integer  as the exponent. Applications The powers of three give the place values in ...
multiple of the original length. The perimeter of the snowflake after n iterations is: P_ = N_ \cdot S_ = 3 \cdot s \cdot ^n\, . The Koch curve has an infinite length, because the total length of the curve increases by a factor of \tfrac with each iteration. Each iteration creates four times as many line segments as in the previous iteration, with the length of each one being \tfrac the length of the segments in the previous stage. Hence, the length of the curve after n iterations will be (\tfrac)^ times the original triangle perimeter and is unbounded, as n tends to infinity.


Limit of perimeter

As the number of iterations tends to infinity, the limit of the perimeter is: \lim_ P_n = \lim_ 3 \cdot s \cdot \left(\frac \right)^n = \infty\, , since \tfrac > 1. An \tfrac-dimensional measure exists, but has not been calculated so far. Only upper and lower bounds have been invented.


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: T_ = N_ = 3 \cdot 4^ = \frac \cdot 4^n\, . The area of each new triangle added in an iteration is \tfrac of the area of each triangle added in the previous iteration, so the area of each triangle added in iteration n is: a_ = \frac = \frac\, . where a_ is the area of the original triangle. The total new area added in iteration n is therefore: b_ = T_ \cdot a_ = \frac \cdot ^ \cdot a_ The total area of the snowflake after n iterations is: A_ = a_0 + \sum_^ b_k = a_0\left(1 + \frac \sum_^ \left(\frac\right)^ \right)= a_0\left(1 + \frac \sum_^ \left(\frac\right)^ \right)\, . Collapsing the geometric sum gives: A_ = a_0 \left( 1 + \frac \left( 1 - \left(\frac\right)^ \right) \right) = \frac \left( 8 - 3 \left(\frac\right)^ \right)\, .


Limits of area

The limit of the area is: \lim_ A_n = \lim_ \frac \cdot \left(8 - 3 \left(\frac \right)^n \right) = \frac \cdot a_\, , since \tfrac < 1. Thus, the area of the Koch snowflake is \tfrac of the area of the original triangle. Expressed in terms of the side length s of the original triangle, this is: \frac.


Solid of revolution

The volume of the
solid of revolution In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the '' axis of revolution'') that lies on the same plane. The surface created by this revolution and which bounds the solid is ...
of the Koch snowflake about an axis of symmetry of the initiating equilateral triangle of unit side is \frac \pi.


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 In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by ...
for discussion). The
fractal 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 (strictly speaking, a fractal pattern) changes with the scale at which it is me ...
of the Koch curve is \tfrac \approx 1.26186. This is greater than that of a line (=1) but less than that of
Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...
's
space-filling curve In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, ...
(=2). The Koch curve is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
everywhere, but differentiable nowhere.


Tessellation of the plane

It is possible to
tessellate A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
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. 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 In mathematics, the Thue–Morse sequence, or Prouhet–Thue–Morse sequence, is the binary sequence (an infinite sequence of 0s and 1s) obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus ...
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 \tfrac, the resulting curve converges to the Koch snowflake.


Representation as Lindenmayer system

The Koch curve can be expressed by the following
rewrite system In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, or reduc ...
(
Lindenmayer system An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into so ...
): :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 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 ...
and their extensions to higher dimensions (Sphereflake and Kochcube, respectively) 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.Demonstrated by James McDonald in a public lecture at KAUST University on January 27, 2013. retrieved 29 January 2013. The resulting area fills a square with the same center as the original, but twice the area, and rotated by \tfrac radians, the perimeter touching but never overlapping itself. The total area covered at the nth iteration is: A_ = \frac + \frac \sum_^n \left(\frac\right)^k \quad \mbox \quad \lim_ A_n = 2\, , while the total length of the perimeter is: P_ = 4 \left(\frac\right)^na\, , which approaches infinity as n increases.


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 ...
*
Gabriel's Horn Gabriel's horn (also called Torricelli's trumpet) is a particular geometry, geometric figure that has infinite surface area but finite volume. The name refers to the Christian tradition where the archangel Gabriel blows the horn to announce Last ...
(infinite surface area but encloses a finite volume) * Gosper curve (also known as the Peano–Gosper curve or ''flowsnake'') * Osgood curve *
Self-similarity __NOTOC__ In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
*
Teragon A teragon is a polygon with an infinite number of sides, the most famous example being the Koch snowflake ("triadic Koch teragon"). The term was coined by Benoît Mandelbrot from the words Classical Greek (''teras'', monster) + (''gōnía'' ...
*
Weierstrass function In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass. The Weierstr ...
* Coastline paradox


References


Further reading

*


External links

*
The Koch Curve poem by Bernt Wahl
''Wahl.org''. Retrieved 23 September 2019. * ** ** **
A WebGL animation showing the construction of the Koch surface
''tchaumeny.github.io''. Retrieved 23 September 2019. * {{Authority control De Rham curves L-systems