Lab 1: The Space-Filling Curves of Hilbert and Peano
Fractals have captured the public imagination for a relatively short time, beginning with the work of Benoit Mandelbrot (1979), but their history and roots go back more than a century.
In the late 1800's, mathematicians were struggling with the notion of "dimension" and some of the rather unusual aspects of limits in calculus. Even today, "dimension" is not an obvious concept. We take it for granted that we can tell the difference between one-dimensional, two-dimensional and three-demensional, but exactly what is it that makes something two-dimensional?
Our science-fiction writers and physicists create extra dimensions at will (four, five, six, ... twenty-three!), but what about a fractional dimension? Can you draw a figure that has dimension 1 1/2?
One of the first hints that something was "wrong" with the "obvious" notion of dimension came in 1872, when Georg Cantor announced that he had discovered a set of "dimension 0" that had just as many points as the real number line(!)
Then in swift succession Giuseppe Peano (1890) and David Hilbert (1891) announced that they had constructed curves (one-dimensional objects) that could completely fill a (two-dimensional) space.
Try to carry out Peano's or Hilbert's construction at stage 3.
