VSFCF  CVM 1.1 
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:
Since 0 is the fixed point of f_{0} and 1 is the fixed point of f_{3} and for k=0,1,2 we have f_{k}(1)=f_{k+1}(0), then the attractor of the IFS {f_{0}, f_{1}, f_{2}, f_{3}} can naturally be described as a continuous curve. (See [Hu, p. 730] for the general theorem.) Note that to have f_{1}(1)=f_{2}(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 f_{1}(1)=f_{2}(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 f_{0}(1)=f_{1}(0) and f_{2}(1)=f_{3}(0).) As t (the angle of rotation for f_{1}) 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 SierpinskiKnopp spacefilling 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.
Figure 3. Frames from the increasing dimension snowflake curve. Also available is an animated view (126 kb) and a speed controlled animation. Use the browser back button to return. Source code is available.A similar idea to these variations of the von Koch curve is attributed to J. Lighthill by Mandelbrot ([Ma1, pp. 3839]). The snowflake curve can also be produced using just two IFS functions, although then reflections are needed in addition to shrinking, rotating, and shifting. This is shown by example in source code.

