A Family of NxN Space-Filling Z-Functions, N>1

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In [1], Wunderlich explores variations on Peano's space-filling curve, and in Figure 5 gives a rule, which looks noticeably similar to Hilbert's rule. Up to one initial "+" or "−" sign, the Lindenmayer definition of Wunderlich's Figure 5 can be obtained from the Lindenmayer definition of Hilbert's specimen, simply by concatenating additional symbols to the image words of "L" and "R" axioms. The extra symbols draw a right angle around empty squares in the complements of the and grids.


In general, it is possible to simply draw an extra right angle around the empty squares in the complements of the and grids. Thus by induction of concatenation, a Lindenmayer rule is obtained for any grid, with integer . The Lindenmayer notation explicitly determines the space-filling -function, but more work is necessary to construct the densely-interpolated -function.


Contributed by: Brad Klee (October 2019)
Open content licensed under CC BY-NC-SA


For an explicit implementation of the concatenation procedure, refer to the source code above.


[1] W. Wunderlich, "Über Peano-Kurven," Elemente der Mathematik, 28(1), 1973 pp. 1–10. www.mathcurve.com/fractals/wunderlich/125.pdf.


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