Editing Koch snowflake (section)

== Variants of the Koch curve ==
{{anchor|Variants}}

Following von Koch's concept, several variants of the Koch curve were designed, considering right angles ([[Quadratic function|quadratic]]), other angles ([[De Rham curve#Cesàro curves|Cesàro]]), circles and [[polyhedron|polyhedra]] and their extensions to higher dimensions (Sphereflake and Kochcube, respectively)

{| class="wikitable"
! Variant ([[fractal dimension|dimension]], [[angle]]) !! Illustration !! Construction
|-
| ≤1D, 60-90° angle || [[Image:Koch Curve 85degrees.png|150px]] <br> Cesàro fractal (85°)|| The Cesàro fractal is a variant of the Koch curve with an angle between 60° and 90°.{{citation needed|date=September 2019|reason=Claim connecting a specific range of angles and a specific name.}} <br> [[File:Cesàro fractal outlines 1-4.svg|450px]] <br> First four iterations of a Cesàro antisnowflake (four 60° curves arranged in a 90° square)
|-
| ≈1.46D, 90° angle || [[Image:Quadratic Koch 2.svg|150px]] <br> Quadratic type 1 curve|| align="left"| [[File:Quadratic Koch curve type1 iterations.png|450px]] <br> First two iterations
|-
| 1.5D, 90° angle || [[Image:Quadratic Koch.svg|150px]] <br> Quadratic type 2 curve|| align="left"| [[Minkowski Sausage]]<ref>Paul S. Addison, ''Fractals and Chaos: An illustrated course'', p. 19, CRC Press, 1997 {{ISBN|0849384435}}.</ref> <br> [[Image:Quadratic Koch curve type2 iterations.png|450px]] <br> First two iterations. Its fractal dimension equals <math>\tfrac{3}{2}</math> and is exactly half-way between dimension 1 and 2. It is therefore often chosen when studying the physical properties of non-integer fractal objects.
|-
| ≤2D, 90° angle || [[File:Minkowski island 3.svg|150px]] <br> Third iteration|| align="left"| [[Minkowski Island]] <br> [[File:Minkowski island 1-3.svg|450px]] <br> Four quadratic type 2 curves arranged in a square
|-
| ≈1.37D, 90° angle || [[Image:Karperienflake.gif|150px]] <br> Quadratic flake|| align="left"| [[Image:Karperienflakeani2.gif|450px]] <br> 4 quadratic type 1 curves arranged in a polygon: First two iterations. Known as the "[[Minkowski Sausage]]",<ref>Weisstein, Eric W. (1999). "[https://archive.lib.msu.edu/crcmath/math/math/m/m263.htm Minkowski Sausage]", ''archive.lib.msu.edu''. Accessed: 21 September 2019.</ref><ref>Pamfilos, Paris. "[http://users.math.uoc.gr/~pamfilos/eGallery/problems/Minkowski.html Minkowski Sausage]", ''user.math.uoc.gr/~pamfilos/''. Accessed: 21 September 2019.</ref><ref>{{MathWorld |id=MinkowskiSausage  |title=Minkowski Sausage |access-date=22 September 2019}}</ref> its fractal dimension equals <math>\tfrac{\ln 3}{\ln \sqrt{5}} = 1.36521</math>.<ref>Mandelbrot, B. B. (1983). ''The Fractal Geometry of Nature'', p.48. New York: W. H. Freeman. {{ISBN|9780716711865}}. Cited in {{MathWorld |id=MinkowskiSausage  |title=Minkowski Sausage |access-date=22 September 2019}}.</ref>
|-
| ≤2D, 90° angle || [[File:Anticross-stitch curve 0-4.svg|150px]] <br> Quadratic antiflake ||align="left"| Anti'''cross-stitch curve''', the quadratic flake type 1, with the curves facing inwards instead of outwards ([[Vicsek fractal]])
|-
| ≈1.49D, 90° angle || [[Image:quadriccross.gif|150px]] <br> Quadratic Cross || align="left"| Another variation. Its fractal dimension equals <math>\frac{\ln 3.33}{\ln \sqrt{5}} = 1.49</math>.
|-
| ≤2D, 90° angle || [[File:Koch quadratic island L7 3.svg|150px]] <br> Quadratic island<ref>Appignanesi, Richard; ed. (2006). ''Introducing Fractal Geometry''. Icon. {{ISBN|978-1840467-13-0}}.</ref>|| align="left"| [[File:Koch quadratic L7 curves 0-2.svg|450px]] <br> Quadratic curve, iterations 0, 1, and 2; dimension of <math>\tfrac{\ln 18}{\ln 6} \approx  1.61</math><!--1.6131471928-->
|-
| ≤2D, 60° angle || [[Image:Koch surface 3.png|150px]] <br> von Koch surface|| [[Image:Koch surface 0 through 3.png|450px]] <br> First three iterations of a natural extension of the Koch curve in two dimensions.
|-
| ≤2D, 90° angle || [[Image:koch_quadratic_3d_fractal.svg|150px]] <br> First (blue block), second (plus green blocks), third (plus yellow blocks) and fourth (plus transparent blocks) iterations of the type 1 3D Koch quadratic fractal|| Extension of the quadratic type 1 curve. The illustration at left shows the fractal after the second iteration <br>  [[Image:KochCube Animation Gray.gif|300px]] <br> Animation quadratic surface
|-
| ≤3D, any || [[image:Koch_Curve_in_Three_Dimensions_("Delta"_fractal).jpg|150px]] <br> Koch curve in 3D || 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 <math>2</math> 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.<ref>Demonstrated by  [[James McDonald (writer)|James McDonald]]  in a public lecture at KAUST University on January 27, 2013. {{cite web |url=http://www.kaust.edu.sa/academics/wep/ |title=KAUST &#124; Academics &#124; Winter Enrichment Program |access-date=2013-01-29 |url-status=dead |archive-url=https://web.archive.org/web/20130112023924/http://www.kaust.edu.sa/academics/wep/ |archive-date=2013-01-12 }} retrieved 29 January 2013.</ref> The resulting area fills a square with the same center as the original, but twice the area, and rotated by <math>\tfrac{\pi}{4}</math> radians, the perimeter touching but never overlapping itself.

The total area covered at the <math>n</math>th iteration is:
<math display=block>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\, ,</math>

while the total length of the perimeter is:
<math display=block>P_{n} =  4  \left(\frac{5}{3}\right)^na\, ,</math>
which approaches infinity as <math>n</math> increases.

=== Functionalisation ===
[[File:Koch function graph.svg|thumb|520px|Graph of the Koch's function]]
In addition to the curve, the paper by Helge von Koch that has established the Koch curve shows a variation of the curve as an example of a [[continuous function|continuous]] everywhere yet [[nowhere differentiable]] function that was possible to represent geometrically at the time. From the base straight line, represented as AB, the graph can be drawn by recursively applying the following on each line segment:
* Divide the line segment (''XY'') into three parts of equal length, divided by dots ''C'' and ''E''.
* Draw a line ''DM'', where ''M'' is the middle point of ''CE'', and ''DM'' is perpendicular to the initial base of ''AB'', having the length of <math>\frac{CE\sqrt{3}}{2}</math>.
* Draw the lines ''CD'' and ''DE'' and erase the lines ''CE'' and ''DM''.
Each point of ''AB'' can be shown to converge to a single height. If <math>y = \phi(x)</math> is defined as the distance of that point to the initial base, then <math>\phi(x)</math> as a function is continuous everywhere and differentiable nowhere.<ref name="Koch"/>
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