Ludwig Danzer

Discovered Tilings

Preview Triangle Duo
Triangle Duo

Infinite Rotations Finite Local Complexity Polytopal Tiles Self Similar Substitution

Preview Octagonal 1225
Octagonal 1225

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.

Preview T2000
T2000

A substitution with factor $\sqrt{3}$. The tilings are limitperiodic.

Preview Danzer's non-FLC 5
Danzer's non-FLC 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 tiling where the tiles don’t meet vertex-to-vertex. One of these fault lines is visible in the picture, it is located near the diagonal of the image.

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Danzer's 7-fold variant
Danzer's 7-fold variant

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

Preview Danzer's 7-fold original
Danzer's 7-fold original

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 centre of perfect 14-fold symmetry: a 14-tipped star in the upper right corner. But in fact it is only 2-fold symmetric. The symmetry is broken by the right- and left-handedness of the tiles. On rings around the 14-tipped star, this manifests in tiles pointing clockwise or counterclockwise, thus breaking the symmetry.

Without Decoration Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Danzer's 7-fold
Danzer's 7-fold

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 $1+{\sin(\frac{2\pi}{7})}/{\sin(\frac{\pi}{7})}$ , which is not a PV number. There are simple matching rules for the tiling. In fact, the list of all vertex stars occurring in the substitution tiling serves as one. This is stated in [ND96], but never really published, up to my knowledge.

Without Decoration Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Cyclotomic Trapezoids 9-fold
Cyclotomic Trapezoids 9-fold

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 additional prototiles (two pentagons, one non-trapezoidal quadrangle), but one obtains an infinite series of substitution rules based on n-fold symmetry (n odd). Unfortunately, none of these tilings show perfect n-fold symmetry, as Danzer’s 7-fold does, thus loosing aesthetic appeal.

Preview Cyclotomic Trapezoids 7-fold
Cyclotomic Trapezoids 7-fold

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 additional prototiles (two pentagons, one non-trapezoidal quadrangle), but one obtains an infinite series of substitution rules based on n-fold symmetry (n odd). Unfortunately, none of these tilings show perfect n-fold symmetry, as Danzer’s 7-fold does, thus loosing aesthetic appeal.

Preview Cyclotomic Trapezoids 11-fold
Cyclotomic Trapezoids 11-fold

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 additional prototiles (two pentagons, one non-trapezoidal quadrangle), but one obtains an infinite series of substitution rules based on n-fold symmetry (n odd). Unfortunately, none of these tilings show perfect n-fold symmetry, as Danzer’s 7-fold does, thus loosing aesthetic appeal.

Preview Coloured Golden Triangle
Coloured Golden Triangle

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