Heighway Dragon FASS-Curve Substitution Tiling

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Info

The original Heighway Dragon Curve as described in [gar1967] , can be derived by a substitution tiling with one substitution rule and appropriate decoration. However, it is not a FASS-curve because it is not self avoiding. With the results in [pau2021] it is possible to derive a substitution tiling which generates a Heighway Dragon FASS-Curve without disturbing self similarity. In detail the decoration on the proto tile is shifted away from the corners in different ways. As a consequence of the combinatorics the number of substitution rules increases to four. Furthermore the vertices of the FASS-curve are not preserved by the substitution rules. The inflation factor $q$ is $\sqrt{2}$.

Substitution Rule

Rule Heighway Dragon FASS-Curve Substitution Tiling

Patch

Patch Heighway Dragon FASS-Curve Substitution Tiling download vectorformat Heighway Dragon FASS-Curve Substitution Tiling

References

[gar1967]
Gardner, M
Mathematical Games
Scientific American 1967, Volume 216, Issue 3, 124-127, 10.1038/scientificamerican0367-124

[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