The Koch snowflake
The curious property of the Koch curve is that it has a finite area, because it
always stays within a circle, but at each stage of its generation its length
increases. It is a curve which encloses a finite area but has an ‘infinite’
circumference!
Another famous fractal is named after the Polish mathematician Wacław
Sierpinski. It is found by subtracting triangles from an equilateral triangle; and by
continuing this process, we get the Sierpinski gasket (generated by a different
process on page 54).
The Sierpinski gasketFractional dimension
The way Felix Hausdorff looked at dimension was innovative. It has to do with
scale. If a line is scaled up by a factor of 3 it is 3 times longer than it was
previously. Since this 3 = 3^1 a line is said to have dimension 1. If a solid square