Editing Koch snowflake (section)

{{Short description|Fractal curve}}
[[File:KochFlake.svg|thumb|upright=1.35|The first four [[iteration]]s of the Koch snowflake]]
[[File:Von Koch curve.gif|thumb|upright=1.35|The first seven iterations in animation]]
[[File:Kochsim.gif|thumb|Zooming into a vertex of the Koch curve]]
[[File:Zooming in a point of Koch curve that is not a vertex.gif|thumb|Zooming into a point that is not a vertex may cause the curve to rotate]]
{{multiple image
| align    =right
| direction=vertical
| width    =200
| header   =Koch antisnowflake 
| image1   =Koch antisnowflake 1 through 4.svg
| caption1 =First four iterations
| image2   =KochAntiSnowflake.svg
| caption2 =Sixth iteration
}}

The '''Koch''' '''snowflake''' (also known as the '''Koch curve''', '''Koch star''', or '''Koch island'''<ref>{{cite book|last=Addison |first=Paul S. |title=Fractals and Chaos: An Illustrated Course |publisher=Institute of Physics |date=1997 |isbn=0-7503-0400-6 |page=19}}</ref><ref>{{cite book|last=Lauwerier |first=Hans |title=Fractals: Endlessly Repeated Geometrical Figures |publisher=Princeton University Press |date=1991 |isbn=0-691-02445-6 |page=36 |quote=Mandelbrot called this a Koch island. |translator-last1=Gill-Hoffstädt |translator-first1=Sophia }}<!--Original 1987 in Dutch.--></ref>) is a [[fractal curve]] and one of the earliest [[fractal]]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"<ref name="Koch">{{cite journal|url=https://babel.hathitrust.org/cgi/pt?id=inu.30000100114564;view=1up;seq=691|last=von Koch |first=Helge|title=Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire|journal=[[Arkiv för matematik, astronomi och fysik]]|volume=1|year=1904|pages=681–704 |language=fr |jfm=35.0387.02}}</ref> by the Swedish mathematician [[Helge von Koch]].

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 <math>\tfrac{8}{5}</math> 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 [[Arc length#Curves with infinite length|infinite perimeter]].

The Koch snowflake has been constructed as an example of a continuous curve where drawing a [[tangent line]] to any point is impossible. Unlike the earlier [[Weierstrass function]] where the proof was purely analytical, the Koch snowflake was created to be possible to geometrically represent at the time, so that this property could also be seen through "naive intuition".<ref name="Koch"/>