View source for Koch snowflake

{{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://dspace.bcu-iasi.ro/handle/123456789/43818 |access-date=29 April 2026 |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 |date=24 November 1904 |pages=681–702 |language=fr |jfm=35.0387.02 |via=Central University Library of Iași}}</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"/>

== Origin and history ==
In his 1904 article, von Koch applies this recursive construction to a [[line segment]], obtaining the curve that forms <math>\tfrac{1}{3}</math> of the boundary of the Koch snowflake. However, the complete snowflake does not appear in the original article published in 1904,<ref name="Koch" /> nor in the extended 1906 memoir.<ref>{{Cite journal |last=von Koch |first=Helge |date=1906 |title=Une méthode géométrique élémentaire pour l'étude de certaines questions de la théorie des courbes planes |url=https://projecteuclid.org/journals/acta-mathematica/volume-30/issue-none/Une-m%c3%a9thode-g%c3%a9om%c3%a9trique-%c3%a9l%c3%a9mentaire-pour-l%c3%a9tude-de-certaines-questions-de/10.1007/BF02418570.full |journal=Acta Mathematica |language=French |volume=30 |issue= |pages=145–174 |doi=10.1007/BF02418570 |issn=0001-5962}}</ref> The Koch snowflake as a [[closed curve]] may instead be due to the American mathematician [[Edward Kasner]].<ref>{{Cite journal |last=Demichel |first=Yann |date=2024-09-13 |title=Who Invented von Koch's Snowflake Curve? |url=https://www.tandfonline.com/doi/full/10.1080/00029890.2024.2363737 |journal=The American Mathematical Monthly |language=en |volume=131 |issue=8 |pages=662–668 |arxiv=2308.15093 |doi=10.1080/00029890.2024.2363737 |issn=0002-9890 |via=}}</ref><ref>{{Cite book |last1=Kasner |first1=Edward |title=Mathematics and the imagination |last2=Newman |first2=James R. |date=2001 |publisher=Dover Publications |isbn=978-0-486-41703-5 |location=Mineola, N.Y}}</ref>

==Construction==
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.<ref>{{cite journal |last1=Alonso-Marroquin |first1=F. |last2=Huang |first2=P. |last3=Hanaor |first3=D. |last4=Flores-Johnson |first4=E. |last5=Proust |first5=G. |last6=Gan |first6=Y. |last7=Shen |first7=L. |title=Static friction between rigid fractal surfaces |journal=Physical Review E |year=2015 |volume=92 |issue=3 |article-number=032405 |doi=10.1103/PhysRevE.92.032405 |pmid=26465480 |bibcode=2015PhRvE..92c2405A |hdl=2123/13835 |url=https://ses.library.usyd.edu.au/bitstream/2123/13835/2/F%20Alonso-Marroquin%202015%20Static%20friction%20between%20rigid%20fractal%20surfaces.pdf |hdl-access=free }} — Study of fractal surfaces using Koch curves.</ref> 

==Properties==

===Perimeter of the Koch snowflake===

The [[arc length]] of the Koch snowflake is infinite. To show this, we note that each iteration of the construction is a polygonal approximation of the curve. Thus, it suffices to show that the perimeters of the iterates is unbounded.

The perimeter of the snowflake after <math>n</math> iterations, in terms of the side length <math>s</math> of the original triangle, is

<math display="block"> 3s \cdot {\left(\frac{4}{3}\right)}^n\, ,</math>

which diverges to infinity.

===Area of the Koch snowflake===

The total area of the snowflake after <math>n</math> iterations is, in terms of the original area <math>A</math> of the original triangle, is the geometric series

<math display="block">A\left(1 + \frac{3}{4} \sum_{k=1}^{n} \left(\frac{4}{9}\right)^{k} \right) = A \, \frac{1}{5} \left( 8 - 3 \left(\frac{4}{9}\right)^{n} \right)\, .</math>

Taking the limit as <math>n</math> approaches infinity, the area of the Koch snowflake is <math>\tfrac{8}{5}</math> of the area of the original triangle. Expressed in terms of the side length <math>s</math> of the original triangle, this is:<ref>{{cite web|url=http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/ksnow.htm|title=Koch Snowflake|website=ecademy.agnesscott.edu}}</ref> 
<math display=block>\frac{2s^2\sqrt{3}}{5}.</math>

==== 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 <math>\frac{11\sqrt{3}}{135} \pi.</math><ref>{{Cite journal|last=McCartney|first=Mark|date=2020-04-16|title=The area, centroid and volume of revolution of the Koch curve|journal=International Journal of Mathematical Education in Science and Technology|volume=52|issue=5|pages=782–786|doi=10.1080/0020739X.2020.1747649|s2cid=218810213|issn=0020-739X|url=https://pure.ulster.ac.uk/en/publications/f9bb27ae-7638-406b-9eb3-98b0e29c8596}}</ref>

===Other properties===
The Koch snowflake is self-replicating <!-- (insert image here!) --> 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 [[Hausdorff dimension]] of the Koch curve is <math>d = \tfrac{\ln 4}{\ln 3} \approx 1.26186</math>. This is greater than that of a line (<math>=1</math>) but less than that of [[Peano]]'s [[space-filling curve]] (<math>=2</math>).

The [[Hausdorff measure]] of the Koch curve <math>S</math> satisfies <math> 0.032 < \mathcal{H}^d(S) < 0.6 </math>, but its exact value is unknown. It is conjectured that <math> 0.528 < \mathcal{H}^d(S) < 0.590 </math>.
<ref>{{cite journal
 | last = Jia
 | first = Baoguo
 | title = Bounds of the Hausdorff measure of the Koch curve
 | journal = Applied Mathematics and Computation
 | volume = 190
 | issue = 1
 | pages = 559–565
 | date = 1 June 2007
 | doi = 10.1016/j.amc.2007.01.046
 }}</ref>

It is impossible to draw a [[tangent line]] to any point of the curve.

==Representation as a de Rham curve==
The Koch curve arises as a special case of a [[de Rham curve]]. The de Rham curves are mappings of [[Cantor space]] into the plane, usually arranged so as to form a continuous curve. Every point on a continuous de Rham curve corresponds to a real number in the unit interval. For the Koch curve, the tips of the snowflake correspond to the [[dyadic rational]]s: each tip can be uniquely labeled with a distinct dyadic rational.

==Tessellation of the plane==
[[File:Koch similarity tiling.svg|thumb|[[Tessellation]] by two sizes of Koch snowflake]]
It is possible to [[tessellation|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.<ref>{{cite journal|title=Fractal tilings|journal=Mathematical Gazette|first=Aidan|last=Burns|volume=78|issue=482|year=1994|pages=193–6|doi=10.2307/3618577|jstor=3618577|s2cid=126185324 }}.</ref> Koch snowflakes and Koch antisnowflakes of the same size may be used to tile the plane.

== Thue–Morse sequence and turtle graphics ==
A [[turtle graphics|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 <math>t(n) = 0</math>, move ahead by one unit,
* If <math>t(n) = 1</math>, rotate counterclockwise by an angle of <math>\tfrac{\pi}{3}</math> radians or 60 degrees,

the resulting curve converges to the Koch snowflake.

==Representation as Lindenmayer system==
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.

== 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 |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"/>
{{-}}

== Applications ==
Because the Koch snowflake has a finite area but an infinitely long boundary, it serves as a model for designs that require maximized perimeter or surface length within limited space. In [[Antenna (radio)|antenna]] engineering, incorporating a Koch-type fractal design increases the perimeter of the material that transmits or receives [[electromagnetic radiation]], allowing the construction of compact antennas suited to confined or complex [[Electric Circuit|circuit]] layouts.<ref>{{Cite journal |last=Choukiker |first=Yogesh Kumar |last2=Behera |first2=Santanu Kumar |date=2017 |title=Wideband frequency reconfigurable Koch snowflake fractal antenna |url=https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/iet-map.2016.0238 |journal=IET Microwaves, Antennas & Propagation |language=en |volume=11 |issue=2 |pages=203–208 |doi=10.1049/iet-map.2016.0238 |issn=1751-8725|url-access=subscription }}</ref><ref>{{cite conference |last1=Saharsh |first1=S. B. |last2=Viswasom |first2=Sanoj |last3=Kumar |first3=Santhosh K. |year=2020 |title=Design and Analysis of Koch Snowflake Fractal Antenna Array |url=https://ieeexplore.ieee.org/abstract/document/9243518/ |conference=Fourth International Conference on I-SMAC (IoT in Social, Mobile, Analytics and Cloud) (I-SMAC) |pages=194-197 |doi=10.1109/I-SMAC49090.2020.9243518|url-access=subscription }}</ref><ref>{{cite conference |last1=Palacios |first1=Orlando F. G. |last2=Diaz Vargas |first2=Ricardo E. |last3=Perez |first3=Jorge A. H. |last4=Erazo |first4=Segundo B. C. |date=2016 |year= |title=S-band Koch snowflake fractal antenna for CubeSats |url=https://ieeexplore.ieee.org/document/7836227/ |conference=2016 IEEE ANDESCON |location=Arequipa, Peru |pages=1-4 |doi=10.1109/ANDESCON.2016.7836227|url-access=subscription }}</ref> In [[Acoustical engineering|acoustic engineering]], a Koch snowflake-inspired acoustic metasurface has been tested for broadband sound diffusion in automotive cabins.<ref>{{Cite journal |last=Li |first=Zhi-Han |last2=Xie |first2=Long-Xiang |last3=Gao |first3=Xinhua |last4=Huang |first4=Weichun |last5=Xu |first5=Youzhong |last6=Yang |first6=Jin |last7=Lu |first7=Ming-Hui |last8=Zhong |first8=Xu |date=2025-11-14 |title=Koch snowflake-inspired acoustic metasurface for broadband sound diffusion in automotive loudspeaker systems |url=https://pubs.aip.org/jap/article/138/18/183102/3371768/Koch-snowflake-inspired-acoustic-metasurface-for |journal=Journal of Applied Physics |language=en |volume=138 |issue=18 |doi=10.1063/5.0277003 |issn=0021-8979|doi-access=free }}</ref> The Koch snowflake geometry has also been applied to enhance [[heat transfer]] performance in double-pipe [[Heat exchanger|heat exchangers]].<ref>{{cite thesis |last=Gomez |first=Anton |title=Thermal Performance of a Double-Pipe Heat Exchanger with a Koch Snowflake Fractal Design |accessdate=18 November 2025 |type=Master’s thesis |url=https://digitalcommons.georgiasouthern.edu/etd/1697/ |year=2017 |institution=Georgia Southern University |location=Statesboro, GA, United States}}</ref>

== 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==
{{Reflist}}

== External links ==
{{Commons|Koch curve}}
{{Commons}}
{{External media
| width  = 200px
| float  = right
| video1 = [https://www.khanacademy.org/math/geometry/basic-geometry/koch_snowflake/v/koch-snowflake-fractal Koch Snowflake Fractal]

:– [[Khan Academy]]
}}
*{{webarchive |url=https://web.archive.org/web/20170720203543/http://www.efg2.com/Lab/FractalsAndChaos/vonKochCurve.htm |date=20 July 2017 |title=(2000) "von Koch Curve", ''efg's Computer Lab''}}
*[https://github.com/Alexey-T/Boleeman_Curves/tree/master/Koch%20Snowflake Koch Snowflake by Edward Bole (Lazarus src)]
*[http://www.wahl.org/fe/HTML_version/link/FE3W/c3.htm#koch The Koch Curve poem by Bernt Wahl], ''Wahl.org''. Retrieved 23 September 2019.
*{{MathWorld |title=Koch Snowflake |id=KochSnowflake |access-date=23 September 2019}}
**{{WolframAlpha |title=7 iterations of the Koch curve |id=7+iterations+of+the+koch+curve |access-date=23 September 2019}}
**{{WolframDemonstrations |title=Square Koch Fractal Curves |urlname=SquareKochFractalCurves |access-date=23 September 2019}}
**{{WolframDemonstrations |title=Square Koch Fractal Surface |urlname=SquareKochFractalSurface |access-date=23 September 2019}}
*[https://tchaumeny.github.io/KochGL/ A WebGL animation showing the construction of the Koch surface] {{Webarchive|url=https://web.archive.org/web/20200916103214/http://tchaumeny.github.io/KochGL/ |date=2020-09-16 }}, ''tchaumeny.github.io''. Retrieved 23 September 2019.
*{{cite web|url=https://ujdigispace.uj.ac.za/bitstream/handle/10210/1941/2Mathematical.pdf?sequence=2 |format=pdf |title=A mathematical analysis of the Koch curve and quadratic Koch curve |access-date=22 November 2011 |archive-url=https://web.archive.org/web/20120426000500/https://ujdigispace.uj.ac.za/bitstream/handle/10210/1941/2Mathematical.pdf?sequence=2 |archive-date=26 April 2012 }}

{{Fractals}}
{{Authority control}}

[[Category:De Rham curves]]
[[Category:L-systems]]
[[Category:Fractal curves]]