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# Koch Snowflake

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 . The zeroth through third iterations of the construction are shown above.

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

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

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

Let be the number of sides, be the length of a single side, be the length of the perimeter, and the snowflake's area after the th iteration. Further, denote the area of the initial triangle , and the length of an initial side 1. Then

 (1) (2) (3) (4) (5) (6)

Solving the recurrence equation with gives

 (7)

so as ,

 (8)

The capacity dimension is then

 (9) (10) (11) (12)

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

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

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

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.

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.

Koch Snowflake

## Cite this as:

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