Nischke-Danzer-Deltoid
K. P. Nischke and L. Danzer derived an algorithm in [ND96] to derive substitution tilings with n-fold symmetry for the cases ‘$odd\,n\geqq5,3\nmid n$ ‘ based of the tangents of a deltoid: “A unit-circle S is rolled around inside a circle of radius 3, a point fixed on S will move along a hypocycloid” with three peaks, hence a deltoid. Other cases of ‘$n$’ as the case ‘$n=6$’ may be derived by a similar approach.
All interior angles of all prototiles are integer multiples of $\frac{\pi}{n}$ and all edges have an orientation.
Within this website we use the following notation for the different Nischke-Danzer-Deltoid substitution tilings:
“Nischke-Danzer-Deltoid n-fold-k-‘$\varepsilon$‘” - ‘$n$’ stands for the symmetry. - ‘$k$’ describes the inflation multiplier ‘$\mu_{n,k}=\frac{\sin\left(\frac{k\pi}{n}\right)}{\sin\left(\frac{\pi}{n}\right)}$’ with ‘$\lfloor\frac{n}{2}\rfloor\geqq k\geqq2$‘. - ‘$\varepsilon$‘=1 means that a substitution preserves the orientation of all edges. - ‘$\varepsilon$‘=2 means that a substitution alternates the orientation of all edges.
For the case $n=5$ the Robinson Triangle can be derived.
References
[ND96]
Nischke, K-P and Danzer, L
A construction of inflation rules based on $n$-fold symmetry
Discrete and Computational Geometry
1996,
15,2,
pp. 221-236,
96j:52035