It is simple to define a mapping from the unit interval into the unit square
. Georg Cantor found a one-to-one map from
onto
, showing that the one-dimensional interval and the two-dimensional square have the same cardinality. Cantor’s map was not continuous, but Giuseppe Peano found a continuous surjection from
onto
, that is, a curve that fills the entire unit square. Shortly afterwards, David Hilbert found an even simpler space-filling curve, which we discussed in Part I of this post.
Continue reading ‘Space-Filling Curves, Part II: Computing the Limit Function’