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The Snowflake Curve

Figure 2. Drawn this way shows the reason for the name snowflake curve. It actually consists of three copies (placed along the sides of an equilateral triangle) of the original (in blue) von Koch curve. Also available is a larger, higher resolution version (6kb). Source code is available.

We can use iterated function systems to give a quick complete definition of the snowflake curve. It's the attractor determined by following four similarities in the complex plane:

f₀: Shrinks by a factor of 1/3 toward 0.

f₁: f₀ followed by a rotation of 60° about 0 and a shift of 1/3.

f₂: f₃ followed by a rotation of -60° about 1 and a shift of -1/3.

f₃: Shrinks by a factor of 1/3 toward 1.

Since 0 is the fixed point of f₀ and 1 is the fixed point of f₃ and for k=0,1,2 we have f_k(1)=f_k+1(0), then the attractor of the IFS {f₀, f₁, f₂, f₃} can naturally be described as a continuous curve. (See [Hu, p. 730] for the general theorem.) Note that to have f₁(1)=f₂(0) we must have that Re(w)=1/(2s)-1.

Now, if we let w vary in equation (1), with w=e^it for t in [0, p/2], and also let s vary so that f₁(1)=f₂(0), namely s=1/(2(1+Re(w))), we get a family of IFS's that gives us a continuous curve at each stage. (Note that we always have f₀(1)=f₁(0) and f₂(1)=f₃(0).) As t (the angle of rotation for f₁) increases from 0 to p/2, s will increase from 1/4 to 1/2. For t=0 the attractor is just the unit interval, for t in (0,1) we get a snowflake like curve with dimension increasing from 1 to 2 as t increases. For t=p/3 we get the classical snowflake curve, and for t=1 we get what's known as the Sierpinski-Knopp space-filling curve with image a filled in right isosceles triangle.

Four examples are shown in Figure 3 with s equal to about .252, .288, .336, and .444 respectively.

Definition of the Hilbert Curve Table of contents Mathematical Background