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Koch Snowflake


KochSnowflake

The Koch snowflake is a fractal curve, also known as the Koch island, which was first described by Helge von Koch in 1904. It is built by starting with an equilateral triangle, removing the inner third of each side, building another equilateral triangle at the location where the side was removed, and then repeating the process indefinitely. The Koch snowflake can be simply encoded as a Lindenmayer system with initial string "F--F--F", string rewriting rule "F" -> "F+F--F+F", and angle 60 degrees. The zeroth through third iterations of the construction are shown above.

Each fractalized side of the triangle is sometimes known as a Koch curve.

KochSnowflakeMotif

The fractal can also be constructed using a base curve and motif, illustrated above.

The nth iterations of the Koch snowflake is implemented in the Wolfram Language as KochCurve[n].

Let N_n be the number of sides, L_n be the length of a single side, l_n be the length of the perimeter, and A_n the snowflake's area after the nth iteration. Further, denote the area of the initial n=0 triangle Delta, and the length of an initial n=0 side 1. Then

N_n=3·4^n
(1)
L_n=(1/3)^n
(2)
l_n=N_nL_n
(3)
=3(4/3)^n
(4)
A_n=A_(n-1)+1/4N_nL_n^2Delta
(5)
=A_(n-1)+1/3(4/9)^(n-1)Delta.
(6)

Solving the recurrence equation with A_0=Delta gives

 A_n=1/5[8-3(4/9)^n]Delta,
(7)

so as n->infty,

 A_infty=8/5Delta.
(8)

The capacity dimension is then

d_(cap)=-lim_(n->infty)(lnN_n)/(lnL_n)
(9)
=log_34
(10)
=(2ln2)/(ln3)
(11)
=1.261859507...
(12)

(OEIS A100831; Mandelbrot 1983, p. 43).

KochSnowflakeTilings

Some beautiful tilings, a few examples of which are illustrated above, can be made with iterations toward Koch snowflakes.

KochSnowflakeTiling

In addition, two sizes of Koch snowflakes in area ratio 1:3 tile the plane, as shown above.

KochFrillFlake3

Another beautiful modification of the Koch snowflake involves inscribing the constituent triangles with filled-in triangles, possibly rotated at some angle. Some sample results are illustrated above for 3 and 4 iterations.


See also

Cesàro Fractal, Exterior Snowflake, Gosper Island, Koch Antisnowflake, Peano-Gosper Curve, Pentaflake, Sierpiński Sieve

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References

Bulaevsky, J. "The Koch Curve Fractal." http://ejad.best.vwh.net/java/fractals/koch.shtml.Cesàro, E. "Remarques sur la courbe de von Koch." Atti della R. Accad. della Scienze fisiche e matem. Napoli 12, No. 15, 1-12, 1905. Reprinted as §228 in Opere scelte, a cura dell'Unione matematica italiana e col contributo del Consiglio nazionale delle ricerche, Vol. 2: Geometria, analisi, fisica matematica. Rome: Edizioni Cremonese, pp. 464-479, 1964.Charpentier, M. "L-Systems in PostScript." http://www.cs.unh.edu/~charpov/Programming/L-systems/.Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 65-66, 1989.Dickau, R. M. "Two-Dimensional L-Systems." http://mathforum.org/advanced/robertd/lsys2d.html.Dixon, R. Mathographics. New York: Dover, pp. 175-177 and 179, 1991.Flake, G. W. The Computational Beauty of Nature. Cambridge, MA: MIT Press, p. 89, 1998.Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 227, 1984.Gleick, J. Chaos: Making a New Science. New York: Penguin Books, p. 99 and center plate (following p. 114), 1988.Harris, J. W. and Stocker, H. "Koch's Curve" and "Koch's Snowflake." §4.11.5-4.11.6 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 114-115, 1998.King, B. W. "Snowflake Curves." Math. Teacher 57, 219-222, 1964.Knopp, K. "Einheitliche Erzeugung und Darstellung der Kurven von Peano, Osgood und v. Koch." Arch. f. Math. u. Phys. 26, 103-115, 1917.Koch, H. von. "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire." Archiv för Matemat., Astron. och Fys. 1, 681-702, 1904.Koch, H. von. "Une méthode géométrique élémentaire pour l'étude de certaines questions de la théorie des courbes planes." Acta Math. 30, 145-174, 1906.Kosmulski, M. "Modulus Origami--Fractals, IFS." http://hektor.umcs.lublin.pl/~mikosmul/origami/fractals.html.Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 28-29 and 32-36, 1991.Mandelbrot, B. B. The Fractal Geometry of Nature. New York: W. H. Freeman, pp. 42-45, 1983.Pappas, T. "The Snowflake Curve." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 78 and 160-161, 1989.Peitgen, H.-O.; Jürgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, 1992.Peitgen, H.-O. and Saupe, D. (Eds.). "The von Koch Snowflake Curve Revisited." §C.2 in The Science of Fractal Images. New York: Springer-Verlag, pp. 275-279, 1988.Schneider, J. E. "A Generalization of the Von Koch Curves." Math. Mag. 38, 144-147, 1965.Sloane, N. J. A. Sequence A100831 in "The On-Line Encyclopedia of Integer Sequences."Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 185-195, 1991.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 135-136, 1991.

Referenced on Wolfram|Alpha

Koch Snowflake

Cite this as:

Weisstein, Eric W. "Koch Snowflake." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KochSnowflake.html

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