Ludwig Danzer
Discovered Tilings
The Nischke-Danzer-Deltoid 7-fold-2-2 was discussed and derived in [ND96] but not shown. A figure with the tiling can be found in [Pau2017] .
Polytopal Tiles Self Similar Substitution Finite Local Complexity Finite Rotations Nischke Danzer Deltoid
The Nischke-Danzer-Deltoid 6-fold-2-2 was discussed and derived in [ND96] .
Polytopal Tiles Self Similar Substitution Finite Local Complexity Finite Rotations Nischke Danzer Deltoid Limitperiodic
A substitution tiling with statistical eight-fold symmetry. This example answers a question of L. Danzer, whether there is a substitution tiling with substitution matrix with entries 1,2,2,5.
Part of an infinite series of triangle susbstitutions described by L.Danzer. Most of them are not flc, this one being one of the simplest examples in this series. The substitution factor is of algebraic degree 5. The positions where one can ‘see’ the non-flc property are fault-lines throughout the …
Finite Rotations Polytopal Tiles Self Similar Substitution
Substitution tiling with isosceles triangles as prototiles allow several variations: For each tile in the first order supertiles, one can choose whether it is a left-handed or a right-handed version. By playing around with these possibilities, one obtains this variant from Danzer’s 7-fold.
Finite Rotations Polytopal Tiles Self Similar Substitution Generalized Robinson Triangles
A tiling based on 7-fold (resp. 14-fold) symmetry [ND96].
The inflation factor is $1+{\sin(\frac{2\pi}{7})}/{\sin(\frac{\pi}{7})}$.
The three different edge lengths are proportional to
$\sin(\frac{\pi}{7})$, $\sin(\frac{2\pi}{7})$,
$\sin(\frac{3\pi}{7})$.
On a first glance, there seems to exist a …
Without Decoration Finite Rotations Polytopal Tiles Self Similar Substitution
A substitution tiling with three triangles as prototiles,
based on 7-fold symmetry.
The four different edge lengths occurring are $\sin(\frac{\pi}{7})$, $\sin(\frac{2\pi}{7})$,
$\sin(\frac{3\pi}{7})$, $\sin(\frac{2\pi}{7}) + \sin(\frac{3\pi}{7})$,
The inflation factor is …
Without Decoration Finite Rotations Polytopal Tiles Self Similar Substitution Matching Rules
In order to generalize Danzer’s 7-fold tiling to n-fold symmetry,
where n>5 is odd, L. Danzer and D. Frettlöh introduced trapezoidal tiles,
each one the union of two triangles with edge lengths of the form $\sin(k \frac{\pi}{n})$.
It needs some further effort, including the introduction of three …
In order to generalize Danzer’s 7-fold tiling to n-fold symmetry,
where n>5 is odd, L. Danzer and D. Frettlöh introduced trapezoidal tiles,
each one the union of two triangles with edge lengths of the form $\sin(k \frac{\pi}{n})$.
It needs some further effort, including the introduction of three …
In order to generalize Danzer’s 7-fold tiling to n-fold symmetry,
where n>5 is odd, L. Danzer and D. Frettlöh introduced trapezoidal tiles,
each one the union of two triangles with edge lengths of the form $\sin(k \frac{\pi}{n})$.
It needs some further effort, including the introduction of three …
In order to generate the golden triangle tilings by matching rules, L. Danzer and G. van Ophuysen found this substitution for coloured prototiles. The list of its vertex stars serves as matching rules. For more details, see golden triangle and the references there.
Without Decoration Finite Rotations Euclidean Windowed Tiling Polytopal Tiles Self Similar Substitution