Dirk Frettlöh

Discovered Tilings

Preview Tipi-3-1
Tipi-3-1

One example in a series of substitutions with inflation factor $\sqrt{s}$, where $s^m-s^k-1=0$. The parameters m and k are arbitrary integers with m>k, m>2, k>0. It seems that all these tilings show statistical circular symmetry. Click on ‘Infinite rotations’ above in order to see more examples of statistical circular symmetric tilings. The substitution is a slight variation of the substitution underlying Chaim’s cubic PV. The trick is that the free parameter in Chaim’s rule is choosen such that the prototiles become equilateral triangles.

Infinite Rotations Finite Local Complexity Polytopal Tiles Self Similar Substitution

Preview Squiral
Squiral

This substitution arises from a reptile with infinitely many straight edges, cf. [GS87]. It answers the question ‘Are there selfsimilar substitution tilings where the prototiles have infinitely many straight edges?’ positively. The colours of the tiles indicate their chirality. The substitution rule is shown for the right handed tile only, the substitution of the left-handed tile is the reflected image. One can easily define a substitution using only the right handed tile, but this generates periodic tilings only.

Finite Rotations Rep Tiles

Preview Semi-detached House Squared
Semi-detached House Squared

This one is mld to the semi-detached house tiling. A view at the latter (hopefully) explains the name. This version was realized in order to prove (or disprove) that the semi detached house tiling is a cut and project tiling with p-adic internal space. This is not the case, as was shown in [FS].

Finite Rotations Polytopal Tiles Parallelogram Tiles Rhomb Tiles Self Similar Substitution

Preview Rorschach
Rorschach

A substitution rule for a tiling with prototiles based on 12-fold dihedral symmetry. However, the tilings show only 4-fold dihedral symmetry. In contrast to the usual suspects related to 12-fold symmetry, like the shield tilings or the Socolar tilings, the inflation factor of this one is not an algebraic unit. It is still a PV number, which makes this tiling a candidate for a model set with mixed p-adic and Euclidean window.

Finite Rotations Polytopal Tiles

Preview Pythia-5-1
Pythia-5-1

Preview Pythia-4-1
Pythia-4-1

Preview Pythagoras-5-1
Pythagoras-5-1

Preview Pythagoras-4-1
Pythagoras-4-1

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 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 Golden Pinwheel
Golden Pinwheel

Using the prototiles of the golden triangle tiling, this substitution yields tilings where the tiles occur in infinitely many orientations. The inflation factor is $\tau + 1 = 2.618033988 \ldots $, the square of the golden mean. This is a PV number of algebraic degree 2. The expansion contains no rotational part. Nevertheless, the first substitution of the larger tile shows two small tiles, rotated against each other by an angle a incommensurate to pi (i.e., $\frac{a}{\pi}$ is irrational).

Finite Local Complexity Polytopal Tiles Self Similar Substitution

Preview Kite Domino
Kite Domino

This is a variation of the pinwheel substitution. The kite-domino tilings are mld to the pinwheel tilings. The two prototiles are made of two pinwheel triangles, glued together at their long edge. There are two ways to do so, one gives a kite (a quadrilateral with edge lengths 1,1,2,2) and a domino (a rectangle with edge lengths 1,2,1,2). Then the substitution rule is obtained by considering two steps of the pinwheel substitution as one step.

With Decoration Finite Local Complexity Polytopal Tiles Self Similar Substitution Mld Class Pinwheel

Preview Semi-detached House
Semi-detached House

A simple substitution rule with inflation factor 2, using two prototiles only. A glimpse on the image hopefully explains the name. The translation module is a square lattice, which is a hint that the semi-detached house tilings may be a model set with p-adic internal space. This question (model set or not) was raised in [Fre02] and was answered in [FS] .

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Pythia-3-1
Pythia-3-1

A simple example of an infinite series of substitutions with tilings of statistical circular symmetry. It is shown in [Frettloeh:STWCS not found], that all tilings in this series posses statistical circular symmetry. The substitution factors are $s2m$, where s is the largest root of $xm-xk-1$. Each pair of integers $(m,k)$, where $m>k, m>2, k>0$, encodes a such a Pythia substitution. The case $m=4, k=2$ yields the golden pinwheel substitution.

Finite Local Complexity Polytopal Tiles Self Similar Substitution