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Koch Antisnowflake

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KochAntisnowflakeMotif

The fractal derived from the Koch snowflake when the triangular point of the motif faces inward when placed on the base curve as illustrated above may be termed the Koch antisnowflake.

KochAntisnowflake

The first few iterations of the Koch antisnowflake are illustrated above.

KochAntisnowflakeFilled

Let N_n be the number of segments, L_n be the length of a single segment, l_n be the length of the perimeter, and A_n the enclosed area after the nth iteration. Denote the area of the initial n=0 triangle Delta=A_0, and let the length of an initial n=0 side be 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[2+3(4/9)^n]Delta,
(7)

so as n->infty,

A_infty=2/5Delta.
(8)

See also

Exterior Snowflake, Gosper Island, Koch Snowflake, Pentaflake, Sierpiński Curve

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References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 66-67, 1989.Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 36-37, 1991.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 136, 1991.

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Koch Antisnowflake

Cite this as:

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

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