Peano Curve Substitution Tiling

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The Peano Curve is one the earliest known FASS-curves. The original algorithm in [pea1890] 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 Peano Curve by a substitution tiling with two substitution rules and appropriate decorations. The inflation factor $q$ is 3 and the lines are shifted slightly away from the center of the sides to illustrate the matching rules.

Substitution Rule

Rule Peano Curve Substitution Tiling

Patch

Patch Peano Curve Substitution Tiling download vectorformat Peano Curve Substitution Tiling

References

[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

[pea1890]
Peano, G
Sur une courbe, qui remplit toute une aire plane
Mathematische Annalen 1890, 36, 157-160, doi.org/10.1007/BF01199438