Hilbert Curves

Hilbert Curves

A Hilbert curve is a fractal space-filling curve that was presented first by German mathematician David Hilbert in 1891. The Hilbert curves, and its relatives, are widely used in computer science, partly because it preserves locality.

From its Wikipedia article:

If (x, y) are the coordinates of a point within the unit square, and d is the distance along the curve when it reaches that point, then points that have nearby d values will also have nearby (x, y) values.

The Hilbert curve can be expressed as a rewrite system (“Hilbert Curve” 2016):

Alphabet  : A, B
Constants : F + −
Axiom     : A

Production rules:
A -> − B F + A F A + F B −
B -> + A F − B F B − F A +
  • F: draw forward
  • : turn left 90°
  • +: turn right 90°

A and B are not drawn.

For an illustrative example, please visit the excellent Map of the internet on xkcd.

References

“Hilbert Curve.” 2016. Wikipedia.

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