Hilbert Curve Substitution Tiling

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The Hilbert Curve is one of the earliest FASS-curves. The original algorithm in [hil1891] bases on one substitution rule and an additional rule which describes how the substitutes have to be connected. As briefly mentioned in [pau2021] it is also possible to create the Hilbert Curve by a substitution tiling with two substitution rules and appropriate decorations. The inflation factor $q$ is 2 and the lines are shifted slightly away from the center of the sides to illustrate the matching rules.

Substitution Rule

Rule Hilbert Curve Substitution Tiling

Patch

Patch Hilbert Curve Substitution Tiling download vectorformat Hilbert Curve Substitution Tiling

References

[hil1891]
Hilbert, D
Ueber die stetige Abbildung einer Line auf ein Flaechenstueck
Mathematische Annalen 1891, 38, 459-460, doi.org/10.1007/BF01199431

[pau2021]
Pautze, S
Space-Filling, Self-Similar Curves of Regular Pentagons, Heptagons and Other n-Gons
Proceedings of Bridges 2021: Mathematics, Art, Music, Architecture, Culture 2021, 38, 157-164, bridges2021-157.pdf