Archive for August, 2022

Space-Filling Curves, Part II: Computing the Limit Function

The Approximating Functions

It is simple to define a mapping from the unit interval {I := [0,1]} into the unit square {Q:=[0,1]\times[0,1]}. Georg Cantor found a one-to-one map from {I} onto {Q}, 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 {I} onto {Q}, 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.

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Space-Filling Curves, Part I: “I see it, but I don’t believe it”

We are all familiar with the concept of dimension: a point is zero-dimensional, a line is one-dimensional, a plane is two-dimensional and the space around us is three-dimensional. A position on a line can be specified by a single number, such as the distance from a fixed origin. In the plane, a point can be located by giving its Cartesian coordinates {(x,y)}, or its polar coordinates {(\rho,\theta)}. In space, we may specify the location by giving three numbers {(x,y,z)}.

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