Substitutions

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Preview 2-component Rauzy Fractal
2-component Rauzy Fractal

A one dimensional substitution rule with a two component Rauzy Fractal. For a second example and more details see infinite component Rauzy fractal.

One Dimensional Euclidean Windowed Tiling Self Similar Substitution

Preview 2-component Rauzy Fractal (dual)
2-component Rauzy Fractal (dual)

In the cut and project scheme for the 2-component Rauzy Fractal, just interchange the roles of direct space and internal space: The Rauzy fractal and its decomposition define a plane substitution for fractal tiles. Here we replace the fractal tiles with appropriate parallelograms. This plane tiling now has one-dimensional internal space, and its window is just an interval.

Border Forcing

Preview A->AB, B->C, C->A
A->AB, B->C, C->A

A classic simple substitution rule with Rauzy Fractal:

One Dimensional Euclidean Windowed Tiling Self Similar Substitution Polytopal Tiles

Preview A->AB, B->C, C->A (dual)
A->AB, B->C, C->A (dual)

The dual tiling of the 1D tiling a->ab, b->c, c->a, resp. the version with polygonal tiles.

Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogramm Tiles

Preview Ammann A3
Ammann A3

In 1977 Robert Ammann discovered a number of sets of aperiodic prototiles, i.e., prototiles with matching rules forcing nonperiodic tilings. These were published as late as 1987 in [GS87] , where they were named Ammann A2 (our Ammann Chair), Ammann A3, Ammann A4 and Ammann A5 (better known as Ammann Beenker tiling). The substitution of this one uses the golden ratio as inflation factor. It is certainly true that this is a cut and project tiling, but to our knowledge, noone bothered to compute the window of it up to now.

Without Decoration Finite Rotations Euclidean Windowed Tiling Polytopal Tiles Self Similar Substitution

Preview Ammann A4
Ammann A4

One of the tilings discovered R. Ammann in 1977, when he found several sets of aperiodic prototiles, i.e., prototiles with matching rules forcing nonperiodic tilings. These were published much later, in 1987, in [GS87] , where they were named Ammann A2 (our Ammann Chair), Ammann A3, Ammann A4, and Ammann A5 (better known as Ammann Beenker). The A4 tilings are mld to the well-known Ammann Beenker tilings. Thus they share most properties with the latter.

With Decoration Finite Rotations Polytopal Windowed Tiling Canonical Substitution Tiling Parallelogram Tiles Self Similar Substitution Mld Class Ammann

Preview Ammann Chair
Ammann Chair

One of the tilings discovered by R. Ammann in 1977, published in [GS87] . The other ones (published there) are Ammann A3, Ammann A4, and Ammann A5 (better known as Ammann Beenker). The inflation factor of this substitution is quite small. It is the square root of the golden ratio, approx 1.272. These tilings are the dual tilings of the golden triangle tilings. The matching rules for the Ammann chair tilings can be expressed by using Ammann bars.

Without Decoration Finite Rotations Polytopal Windowed Tiling Polytopal Tiles Self Similar Substitution

Preview Ammann-Beenker
Ammann-Beenker

In 1977 R. Ammann found several sets of aperiodic tiles. This one (his set A5) is certainly the best-known of those. It allows tilings with perfect 8fold symmetry. The substitution factor is $1+\sqrt{2}$ - sometimes called the ‘silver mean’ - which was the first irrational inflation factor known which is not related to the golden mean. In 1982 F. Beenker described their algebraic properties, essentially how to obtain it by the projection method, following the lines of N.

With Decoration Finite Rotations Polytopal Windowed Tiling Canonical Substitution Tiling Rhomb Tiles Mld Class Ammann

Preview Ammann-Beenker rhomb triangle
Ammann-Beenker rhomb triangle

A self-similar version of the Ammann-Benker tiling. The colours of the triangles in the rule image indicate the orientation of the triangles: the orange triangle is just the ochre triangle reflected. Hence the rhomb supertile has two axes of mirror symmetry.

With Decoration Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogramm Tiles Self Similar Substitution

Preview Armchair
Armchair

A simple substitution rule with an L-shaped prototile. The tilings are mld. to the domino tilings.

Finite Rotations Polytopal Tiles Polyomio Tilings Rep Tiles Self Similar Substitution Mld Class Domino

Preview Bat in Cone
Bat in Cone

Denote the elements of the field F4 by {0, 1, w, w + 1}, where w satisfies the following equation with coefficients in F2: w2 + w + 1 = 0. Bat in Cone is a recurrent double sequence defined by a(i, 0) = a(0, j) = 1 and a(i, j) = f( a(i, j-1), a(i-1, j-1), a(i-1, j) ), where f(x, y, z ) = x + x2 + w y + z + z2.

Preview Binary
Binary

In [Lan88], energetic properties of certain decorations of Penrose Rhomb tilings were studied. A binary tiling was defined as a tiling by Penrose rhombs, where at each vertex all angles are either in {$\frac{\pi}{5}$, $3\frac{\pi}{5}$}, or in {$2\frac{\pi}{5}$, $4\frac{\pi}{5}$}. (‘Binary’ because the decorations were used to model binary alloys, i.e., alloys consisiting of two metallic elements). The authors did not mention the substitution rule explicitly, but it is obvious from the diagrams in this paper.

Finite Rotations Polytopal Tiles Parallelogramm Tiles Rhomb Tiles

Preview Birds and Bees
Birds and Bees

A substitution tiling with three prototiles. The substitution rule is given for only two of the three tiles. The third tile (yellow) is substituted by nothing. The discoverer gives credits to Veit Elser for suggesting the shape of the tiles.

Preview Bowtie-Hexagon
Bowtie-Hexagon

Preview Bowtie-Hexagon-Decagon 1
Bowtie-Hexagon-Decagon 1

A tiling with fivefold symmetry using tiles inspired by Islamic Girih patterns.

Preview Bowtie-Hexagon-Decagon 2
Bowtie-Hexagon-Decagon 2

Preview Bowtie-Hexagon-Decagon 3
Bowtie-Hexagon-Decagon 3

Preview Bumerang
Bumerang

Denote the elements of the field $F_{4}$ by ${0, 1, w, w + 1}$, where w satisfies the following equation with coefficients in $F2: w^{2} + w + 1 = 0$. Bumerang is a recurrent double sequence defined by $a(i, 0) = a(0, j) = 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j) )$, where $f(x, y, z ) = w x^{2} + x + w y^{2} + w z^{2} + (w + 1) z$.

Preview Central Fibonacci
Central Fibonacci

The substitution rule a1->a1 b1, a2->b2 a2, b1->a2, b2->a1. The tilings generated become Fibonacci tilings under the projection a1,a2->a and b1,b2->b. Alternatively one can simply remove the colour labels on the tiles. The name comes from the projection structure of the tiling. The expansion predecessor of the tiling is itself a projection tiling with the window lying at the center of the window for the full tiling. For more information see [HL].

Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling One Dimensional Polytopal Tiles Parallelogram Tiles Self Similar Substitution Mld Class Fibonacci

Preview Cesi's Substitution
Cesi's Substitution

The substitution system uses 4 letters. With: $x = \frac{\pi}{7}$, $c = \cos(x)$ and $s = \sin(x)$ They are: two squares of side lengths $1$ and $2-c-s$; a rectangle with sides $c+s$ and $2-c-s$: and a right triangle with legs $c$ and $s$. The substitution is indicated in the figure. Up to our knowledge, this was the first example of a substitution where the tiles occur in infinitely many orientations. Obviously, the substitution is not primitive.

Preview Chaim's Cubic PV
Chaim's Cubic PV

Part of an infinite series, where most tilings in this series are not flc, this one is the exception. The reason is that the inflation factor is a - real - PV number. By an argument in [PR] this forces flc. Interestingly, the shape of the tiles can vary. That is, there is one free parameter $l$ , $0 < l < 1+s$, and the smallest prototile is the triangle with sides $1,s,l$ ($s$ the largest root of $x^{3}-x-1$).

Finite Rotations Euclidean Windowed Tiling Polytopal Tiles Self Similar Substitution

Preview Chaim's Square Tiling
Chaim's Square Tiling

A simple non-periodic substitution tiling with just one decorated prototile mentioned in the extended version of [Goo98] .

Preview Chair
Chair

The chair tiling, as most tilings presented here, is nonperiodic. But there is a strong resemblance to periodic tiling. For instance, the set of vertex points in the tiling obviously spans a square lattice. Moreover, it is possible to detect large subsets in the tiling which are fully periodic. For instance, consider the pattern of white crosses (consisting of four tiles each) in the tiling. In fact, the chair tiling is the union of a countable set of fully periodic tile sets $L_{1}, L_{2}, L_{3}$…, where each $L_{i}$ possesses period vectors of length $2 \times 2^{i}$.

Finite Rotations P Adic Windowed Tiling Polytopal Tiles Rep Tiles Self Similar Substitution Mld Class Chair

Preview Chair variant (9 tiles)
Chair variant (9 tiles)

A more or less obvious variant of the chair substitution.

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Chord Quadrangle 3-3
Chord Quadrangle 3-3

A member of an infinite family of substitution rules for similar quadrangles possessing two right interior angles at opposite vertices. A big copy of such a quadrangle can be divided into (smaller) similar quadrangles in several ways. Some of them are compatible with a substitution rule. This one is the smallest possibility, using three prototiles. Examples with more prototiles are Chord-Quadrangle-4-3, Chord-Quadrangle-5-3, Chord-Quadrangle-5-5, and so on. (Search for chord-quadrangle.)

Preview Chord Quadrangle 4-3
Chord Quadrangle 4-3

Preview Chord Quadrangle 5-3
Chord Quadrangle 5-3

Preview Chord Quadrangle 5-5
Chord Quadrangle 5-5

Preview Chord Quadrangle 6-5
Chord Quadrangle 6-5

Preview Chord Quadrangle 7-5
Chord Quadrangle 7-5

Preview Clamshell
Clamshell

A substitution tiling with three prototiles. The substitution rule is given for only two of the three tiles. The third tile (yellow) is substituted by nothing. The discoverer gives credits to Veit Elser for suggesting the shape of the tiles.

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

Preview Conch
Conch

This tiling and Nautilus are dual tilings generated by non-PV morphisms. As such they are the first step in a generalisation of the work of G. Rauzy, P. Arnoux, S. Ito and others for PV substitution rules. The work that developed out of G. Rauzy’s seminal paper [Rau82] . The inflation factor for this substitution rule is either of the expanding roots of: $x^{4}-x+1 = 0$. Note that this it is related to R.

Finite Rotations Euclidean Windowed Tiling

Preview Conch (Volume Hierarchic)
Conch (Volume Hierarchic)

A volume hierarchic version of Conch.

Finite Rotations Euclidean Windowed Tiling Self Similar Substitution

Preview Crown
Crown

Preview Cubic Pinwheel
Cubic Pinwheel

A pinwheel substitution rule with cubic scaling. As the scaling and the rotations for the tiles are all given by algebraic units, every vertex of the tiling lies within a finitely generated Z-module.

Infinite Rotations Infinite Local Complexity Polytopal Tiles Self Similar Substitution

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 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 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 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 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 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 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 Diamond
Diamond

Denote the elements of the field $F_{4}$ by $\{0, 1, w, w + 1\}$, where w satisfies the following equation with coefficients in $F_{2}: w^{2} + w + 1 = 0$. Diamond is a recurrent double sequence defined by $a(i, 0) = a(0, j) = 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z) = w x^{2} + y + (w + 1) y^{2} + w z^{2}$.

Preview Diamond Triangle
Diamond Triangle

Preview Dite and Kart 2D
Dite and Kart 2D

Preview Domino
Domino

Also known as ‘table tiling’. In [Sol97] was shown that its dynamical spectrum has a continuous component. Thus it cannot be a cut and project tiling. The same was shown in [Rob99] , where a topological model of the dynamical system of the domino tilings is obtained.

Polyomio Tiling Finite Rotations Polyomio Tilings Rep Tiles Self Similar Substitution

Preview Domino variant
Domino variant

A simple variant of the domino tilings (aka table tilings). C. Goodman-Strauss pointed out in [Goo98] the following. B. Solomyak proved in Sol98, that for each nonperiodic substitution tiling the substitution rule is invertible: One can tell from $\sigma(T)$ its predecessor $T$ uniquely. But this is true only if the prototiles have the same symmetry group as the first order supertiles. By using decorated tiles this can always be achieved. (And now Chaims remark:) Here we see a case where such a decoration is necessary.

Finite Rotations P Adic Windowed Tiling Polytopal Tiles Parallelogram Tiles Polyomio Tiling Rep Tiles Self Similar Substitution

Preview Domino variant (9 tiles)
Domino variant (9 tiles)

An obvious generalization of the domino substitution. There are several possibilities to play with 1x2 rectangles (dominos) in order to generate non-periodic tilings.

Finite Rotations Polytopal Tiles Polyomio Tiling Rep Tiles Self Similar Substitution

Preview Double Halfhex
Double Halfhex

Preview Dragonul
Dragonul

Denote the elements of the field $F_{4}$ by $\{0, 1, w, w + 1\}$, where $w$ satisfies the following equation with coefficients in $F_{2}: w^{2} + w + 1 = 0$. Dragonul is a recurrent double sequence defined by $a(i, 0) = a(0, j) = 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z ) = x^{2} + w x + (w + 1) y^{2} + w y + w z$.

Preview Equithirds
Equithirds

A substitution tiling found by Bill Kalahurka, Texas, in 2009 (?). It is mld to T2000 by L. Danzer in 2000.

Finite Rotations P Adic Windowed Tiling Polytopal Tiles Rep Tiles Self Similar Substitution Two Dimensional

Preview Example of Canonical 1
Example of Canonical 1

In his PhD thesis, E. Harriss classified all substitution tilings which are canonical projection tilings. Here one example is shown, derived from the cut and project scheme of the Ammann-Beenker tilings.

Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogram Tiles

Preview Example of Canonical 2
Example of Canonical 2

In his PhD thesis, E. Harriss classified all substitution tilings which are canonical projection tilings. Here one example is shown, derived from the cut and project scheme of the Ammann-Beenker tilings.

Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogram Tiles

Preview Example of Canonical 3
Example of Canonical 3

In his PhD thesis, E. Harriss classified all substitution tilings which are canonical projection tilings. Here one example is shown, derived from the cut and project scheme of the Ammann-Beenker tilings.

Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogram Tiles Rhomb Tiles

Preview Example of Canonical 4
Example of Canonical 4

In his PhD thesis, E. Harriss classified all substitution tilings which are canonical projection tilings. Here one example is shown, derived from the cut and project scheme of the Ammann-Beenker tilings.

Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogram Tiles

Preview Fibonacci
Fibonacci

The classical example to explain the cut and project method (see figure, lower part): In the standard square lattice $\mathbb{Z}^2$, choose a stripe with slope $\frac{1}{\tau}$ (where tau is the golden ratio $\frac{1+\sqrt{5}}{2}$ ) of a certain width $\cos(\arctan(\frac{1}{\tau})) + \sin(\arctan(\frac{1}{\tau})) = \frac{1+\tau}{\sqrt{2+\tau}}$. Then take all lattice points within the strip and project them orthogonally to a line parallel to the strip. This yields a sequence of points. There are two values of distances between neighboured points, say, $S$ (short) and $L$ (long).

Finite Rotations Polytopal Windowed Tiling Canonical Substitution Tiling One Dimensional Parallelogram Tiles Self Similar Substitution Mld Class Fibonacci

Preview Fibonacci Times Fibonacci
Fibonacci Times Fibonacci

The 2dim analogue of the famous Fibonacci tiling in one dimension. It is just the Cartesian product of two Fibonacci tilings $F_{1}$, $ F_{2} : \{ T_{1} \times T_{2}\ |\ T_{i}\ in\ F_{i}\}$. Obviously, it can be generated by a substitution with three prototiles. It shares a lot of nice features with the 1dim Fibonacci tiling: It is a model set (better: it’s mld with one), so it has pure point spectrum.

Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Polytopal Tiles Parallelogram Tiles Rhomb Tiles Self Similar Substitution

Preview Fibonacci Times Fibonacci (variant)
Fibonacci Times Fibonacci (variant)

A simple variant of Fibonacci times Fibonacci, the latter arising from the one-dimensional Fibonacci tiling.

Finite Rotations Polytopal Tiles Parallelogram Tiles Rhomb Tiles Self Similar Substitution

Preview Fractal Single-Tile Pinwheel
Fractal Single-Tile Pinwheel

It is hard to find substitution tilings with dense tile orientations (like the pinwheel tiling) that uses a single prototile with fractal boundary. The fractal version of he pinwheel tiling by Frank and whittaker uses 13 different prototiles. An extensive computer search revealed that there are only two substitutions producing tilings with dense tile orientations using five copies of a single fractal prototile. Both substitutions use the same prototile. One substitution is shown here, the other one is obtained by dissecting the blue two tile patch in the supertile in exactly the opposite way into two of the tiles.

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 Golden Rhomboid Triangle
Golden Rhomboid Triangle

Preview Golden Triangle
Golden Triangle

The substitution can be expressed by using the real inflation factor $\sqrt{\tau} = 1.272\ldots$, where $\tau=\frac{\sqrt{5}+1}{2}$ is the golden mean. This factor is not a PV number. Nevertheless, the tiling is pure point diffractive, and it is a cut and project tiling, see [Gel97] , [Dv00] . Thus the right point of view is to consider it as a tiling with the inflation factor sqrt(-tau), which is a complex PV number.

With Decoration Finite Rotations Polytopal Windowed Tiling Polytopal Tiles Self Similar Substitution

Preview Goodman-Strauss 7-fold rhomb
Goodman-Strauss 7-fold rhomb

Whereas it is simple to generate rhomb tilings with n-fold symmetry by the cut and project method, it can be hard to find a substitution rule for such tilings. Here we see a rule for n=7. This one was later generalized by E. Harriss to arbitrary n.

Finite Rotations Canonical Substitution Tiling Polytopal Tiles Parallelogram Tiles Rhomb Tiles Harrisss Rhomb

Preview Half-Hex
Half-Hex

This one is easily seen to be limitperiodic: A large portion of the tiling is periodic. Thus it is a cut and project tiling. A detailed description of the corresponding cut and project scheme is contained in [Fre02]. The substitution occurs already in [GS87], see Exercise 10.1.3.

Finite Rotations P Adic Windowed Tiling Polytopal Tiles Rep Tiles Self Similar Substitution

Preview Half-Par 3-5
Half-Par 3-5

Preview Half-Par 5-5
Half-Par 5-5

Preview Harriss's 9-fold rhomb
Harriss's 9-fold rhomb

Finite Rotations Polytopal Tiles Parallelogram Tiles Rhomb Tiles Harrisss Rhomb

Preview Hexagonal Aperiodic Monotile
Hexagonal Aperiodic Monotile

In 2009 Joan Taylor (Burnie, Tasmania) found a decoration of the hexagon, which - together with few local matching rules - allows only aperiodic tilings of the plane. This is certainly the best example of an aperiodic monotile we have today. This decorated hexagonal tile, together with the local matching rule, is shown in Fig 1 of a joint paper by J. Socolar and J. Taylor: http://arxiv.org/pdf/1003.4279v2. It contains also a non-decorated version of the prototile, but then the prototile is not longer a connected set.

Preview Imbalanced orientations
Imbalanced orientations

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Infinite component Rauzy Fractal
Infinite component Rauzy Fractal

An invertible substitution rule with a disconnected Rauzy Fractal. For two letter substitution rules the Rauzy fractal is connected if and only if the substitution is invertible. In fact as the window is one dimensional for these tilings it is an interval. It was hoped that the connectedness property extended to the higher dimensional case. Unfortunately, as this example shows, this is not the case. A second example, with just two components is 2-component Rauzy fractal.

One Dimensional Euclidean Windowed Tiling Self Similar Substitution Polytopal Tiles

Preview Infinite component Rauzy Fractal (dual)
Infinite component Rauzy Fractal (dual)

Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogram Tiles

Preview Infinity
Infinity

Denote the elements of the field $F_{4}$ by $\{0, 1, w, w + 1\}$, where $w$ satisfies the following equation with coefficients in $F_{2}: w^{2} + w + 1 = 0$. Infinity is a recurrent double sequence defined by $a(i, 0) = a(0, j) = 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z) = x^{2} + (w + 1) y^{2} + z$. This recurrent double sequence can be also obtained using a system of substitutions of type 2 -> 4 with 5 rules.

Preview Kenyon (1,2,1)
Kenyon (1,2,1)

As well as showing that there are substitution rules with any Perron inflation factor, in [Ken96] , R. Kenyon gives an explicit construction for the Perron numbers that satsify: $xn - a xn-1 + b x + c$, where $a, b$, and $c$ are natural numbers. This is an example of that method given in that paper. A locally isomorphic version with polygonal tiles is Kenyon (1,2,1) Polygon.

Finite Rotations Self Similar Substitution Kenyons Construction

Preview Kenyon (1,2,1) Polygon
Kenyon (1,2,1) Polygon

A polygonal version of Kenyon (1,2,1). The boundary is generated by the morphism $a \to b, b \to c, c \to c a' b' b'$ (where $x'$ is the inverse of $x$).

Finite Rotations Polytopal Windowed Tiling Canonical Substitution Tiling Parallelogram Tiles Kenyons Construction

Preview Kenyon 2
Kenyon 2

A substitution rule shown on R. Kenyon’s homepage: http://www.math.brown.edu/~rkenyon/gallery/gallery.html with inflation factor that satisfies: $x^4+x+1 = 0$.

Finite Rotations Self Similar Substitution

Preview Kenyon 2 Polygonal
Kenyon 2 Polygonal

A polygonal version of Kenyon 2. The edges are generated by the morphism: a->b, b->c, c->d, d-> b’a’ (where x’ is the inverse of x).

Finite Rotations Euclidean Windowed Tiling Polytopal Tiles Parallelogram Tiles Kenyon'S Construction

Preview Kenyon's non FLC
Kenyon's non FLC

A simple substitution, yielding a tiling which is not of finite local complexity (flc). The substitution maps the single prototile, a unit square, to three columns of three squares each, where the third column is shifted by an irrational amount t. In higher iterates of the substitution, there are neighboured squares shifted against each other by t, 3t+t, 9t+3t+t,… mod 1. Since t is irrational, these sequence contains infinitely many values (mod 1), thus there are infinitely many pairwise incongruent pairs of tiles.

Polytopal Tiles Parallelogram Tiles Rhomb Tiles

Preview Kenyon's non FLC (volume hierarchic)
Kenyon's non FLC (volume hierarchic)

A simple substitution rule, generating tilings which don’t possess flc. This fractally shaped tiles make it volume hierarchic. Despite the fractal apperance, the dimension of the boundary of the prototile is one almost everywhere. For more details, see Kenyon non-FLC.

Rep Tiles Self Similar Substitution

Preview Kidney and its dual
Kidney and its dual

The substitution $a \rightarrow ab, b \rightarrow cb, c \rightarrow a$ is the composition of the one with the smallest PV scaling factor, $a \rightarrow bc, b \rightarrow a, c \rightarrow b$, and its mirror image, $a \rightarrow cb, b \rightarrow a, c \rightarrow b$. As such, it is MLD to its own mirror image, $a \rightarrow ba, b \rightarrow bc, c \rightarrow a$. The scaling factor $\lambda \approx$ 1.7549 is the largest root of $x^3-2x^2+x-1=0$.

Euclidean Windowed Tiling One Dimensional 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 Kolakoski-(3,1) symmmetric variant, dual
Kolakoski-(3,1) symmmetric variant, dual

The substitution $a \rightarrow aca, b \rightarrow a, c \rightarrow b$ has palindromic and thus mirror symmetric variant of the Kolakoski-(3,1) substitution, which is in the same MLD class, along with the further variants A (mirror symmetric) and B (with its mirror image). The scaling factor $\lambda \approx $ 2.20557 is the largest root of $x^3-2x^2-1=0$. This substitution has a simple dual, with three mildly fractal tiles, which are all similar to each other.

Euclidean Windowed Tiling One Dimensional Self Similar Substitution

Preview Kolakoski-(3,1) variant A, with dual
Kolakoski-(3,1) variant A, with dual

The substitution $a \rightarrow bcc, b \rightarrow ba, c \rightarrow bc$ is a member of the MLD class of the Kolakoski-(3,1) sequence. As the reversed substitution generates the same hull, it is mirror symmetric. The scaling factor $\lambda \approx $ 2.20557 is the largest root of $x^3-2x^2-1=0$. This substitution has a simple dual, with three mildly fractal tiles, which are all similar to each other. The dual substitution scales by about 1.485, and rotates clockwise by about 81.22°.

Euclidean Windowed Tiling One Dimensional Self Similar Substitution

Preview Kolakoski-(3,1) variant B, with dual
Kolakoski-(3,1) variant B, with dual

The substitution $a \rightarrow abcc, b \rightarrow a, c \rightarrow bc$ is a member of the MLD class of the Kolakoski-(3,1) sequence. The scaling factor $\lambda \approx $ 2.20557 is the largest root of $x^3-2x^2-1=0$. This substitution has a simple dual, with three mildly fractal tiles, which are all similar to each other. The dual substitution scales by about 1.485, and rotates clockwise by about 81.22°.

Euclidean Windowed Tiling One Dimensional Self Similar Substitution

Preview Kolakoski-(3,1), with dual
Kolakoski-(3,1), with dual

The substitution $a \rightarrow abc, b \rightarrow ab, c \rightarrow b$ is closely related to the Kolakoski-(3,1) sequence, and is one of the examples whose windows (dual tiles, Rauzy fractals) have been analysed in detail [BaS04] . It is MLD to the mirror symmetric variant given by the palindromic substitution $a \rightarrow aca, b \rightarrow a, c \rightarrow b$. As a consequence, the Kolakoski-(3,1) substitution is MLD to its mirror image, even though it is not mirror symmetric itself.

Euclidean Windowed Tiling One Dimensional Self Similar Substitution

Preview Labyrinth
Labyrinth

Preview Limhex
Limhex

A substitution yielding tilings with statistical 6-fold symmetry, with inflation factor 2. It is not known whether this one is a cut and project tiling or not. If it is, it has necessarily a p-adic internal space.

Finite Rotations Polytopal Tiles

Preview Lord
Lord

A substitution tiling with inflation factor sqrt(3), using a single prototile, namely a 60º rhomb. The substitution sends one rhomb to seven rhombs (instead of three, as one would expect from the inflation factor), thus the tiles in higher iterations do overlap. But the substitution is chosen in a way such that tiles do either overlap completely, or not at all. So overlapping tiles can be identified, and the substitution yields a proper tiling.

Finite Rotations P Adic Windowed Tiling Polytopal Tiles Parallelogram Tiles Rhomb Tiles

Preview Maloney's 7-fold
Maloney's 7-fold

A substitution for three triangular prototiles, based on 7-fold symmetry. The lengths of the edges of the tiles are $\sin(\frac{\pi}{7})$, $\sin(\frac{2\pi}{7})$, $\sin(\frac{3\pi}{7})$. These tilings are essentially different from Danzer’s 7-fold examples, see for instance Danzer’s 7-fold.

Preview Minitangram
Minitangram

A simple substitution rule, using three Tangram pieces as prototiles.

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Mothman
Mothman

Denote the elements of the field $F_{4}$ by $\{0, 1, w, w + 1\}$, where $w$ satisfies the following equation with coefficients in $F_{2}: w^{2} + w + 1 = 0$. Mothman is a recurrent double sequence defined by $a(i, 0) = a(0, j) = 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z) = x^{2} + (w + 1) y^{2} + z^{2}$. This recurrent double sequence can be also obtained using a system of substitutions of type 2 -> 4 with 15 rules, as it follows.

Preview Nautilus
Nautilus

This is the dual partner of Conch, which has more details. The scaaling factor of this rule is either of the (complex conjugate) expanding roots of $x^4 - x^3 + 1 = 0$.

Finite Rotations Euclidean Windowed Tiling Polytopal Tiles

Preview Nautilus (Volume Hierarchic)
Nautilus (Volume Hierarchic)

A volume hierarchic version of Nautilus

Finite Rotations Euclidean Windowed Tiling Self Similar Substitution

Preview Non Unique Decomposition
Non Unique Decomposition

Preview Non-Pinwheel
Non-Pinwheel

A variant of the Pinwheel tiling without infinitely many rotations.

Preview Non-invertible connected Rauzy Fractal
Non-invertible connected Rauzy Fractal

A companion to infinite component Rauzy fractal. As mentioned for that rule, it was hoped that the result for two symbol substitution rules that the window is connected if and only if the rule is invertible. This substitution rules is not invertible and yet the Rauzy fractal is connected:

Euclidean Windowed Tiling One Dimensional Polytopal Tiles Self Simmilar Substitution

Preview Non-reducible 4-letter
Non-reducible 4-letter

One Dimensional Self Similar Substitution Polytopal Tiles

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 Octiamond Reptile
Octiamond Reptile

Preview Open Peano
Open Peano

Denote the elements of the field $F_4$ by $\{0, 1, w, w + 1\}$, where $w$ satisfies the following equation with coefficients in $F_2: w2 + w + 1 = 0$. Open Peano is a recurrent double sequence defined by $a(i, 0) = a(0, j) = w + 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z) = w x^2 + w y + w z^2$.

Preview Overlapping Robinson Triangle I
Overlapping Robinson Triangle I

A substitution rule where the tiles are allowed to overlap. The image left indicates, that the yellow and the green tiles do overlap. It is unknown whether these tilings are mld to the Penrose Rhomb tilings.

Finite Rotations Euclidean Windowed Tiling Polytopal Tiles

Preview Overlapping Robinson Triangle II
Overlapping Robinson Triangle II

As Overlapping Robinson Triangles I, this is a variant of the Penrose Rhomb tiling, using only one prototile, and the tiles are allowed to overlap. Here, the overlap happens after applying the substitution rule twice on one tile.

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Pairs of Squares
Pairs of Squares

Denote the elements of the field $F_4$ by $\{0, 1, w, w + 1\}$, where $w$ satisfies the following equation with coefficients in $F_2: w^2 + w + 1 = 0$. Pairs of Squares is a recurrent double sequence defined by $a(i, 0) = a(0, j) = 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z) = (w + 1) x + (w + 1) y^2 + (w + 1) z$.

Preview Penrose Kite Dart
Penrose Kite Dart

A classic, using a kite (blue) and a dart (orange) as prototiles. See Penrose Rhomb for more details.

Without Decoration Finite Rotations Polytopal Windowed Tiling Polytopal Tiles Mld Class Penrose

Preview Penrose Pentagon Boat Star
Penrose Pentagon Boat Star

One manifestation of the famous Penrose tilings. In fact, this is the first manifestation found by Penrose, the Penrose rhomb, the Penrose kite-dart and the Robinson triangle tilings are refinements of this one. (You may also click ‘Penrose’ below ‘MLD-class’ above to see the others.) Their properties are discussed on the page Penrose rhomb. For a more detailed discussion see [GS87] .

Finite Rotations Polytopal Tiles Self Similar Substitution Mld Class Penrose

Preview Penrose Rhomb
Penrose Rhomb

Certainly the most popular substitution tilings. Discovered in 1973 and 1974 by R. Penrose in - at least - three versions (Rhomb, Penrose kite-dart and Penrose Pentagon boat star), all of them forcing nonperiodic tilings by matching rules. It turns out that the three versions are strongly related: All three generate the same mld-class. These tiles, their matching rules and the corresponding substitution was studied thoroughly in [GS87] . A lot of information can be found there.

Without Decoration Finite Rotations Polytopal Windowed Tiling Canonical Substitution Tiling Rhomb Tiles Mld Class Penrose

Preview Penrose triangle (without rotations)
Penrose triangle (without rotations)

A simple variant of the Robinson triangle substitution. This substitution uses no reflections. The resulting tilings are not longer vertex-to-vertex, but still flc.

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Pentiamond AC factor 2
Pentiamond AC factor 2

Preview Pentiamond AC factor 3
Pentiamond AC factor 3

Preview Pentiamond AD factor 2
Pentiamond AD factor 2

Preview Pentomino
Pentomino

A substitution arising from a polyomio rep-tile. This one is made of five unit squares, thus the name. The tiles are coloured blue or ochre, according to their chirality (left-handed vs right-handed).

Finite Rotations Polytopal Tiles Polyomio Tilings Rep Tiles Self Similar Substitution

Preview Period Doubling
Period Doubling

In some sense, the simplest cut and project tiling. It arises from the symbolic substitution a -> ab, b -> aa. Its internal space are the 2-adic integers.

P Adic Windowed Tiling One Dimensional Rep Tiles Self Similar Substitution

Preview Pinwheel
Pinwheel

This substitution tiling is the example of substitution tilings with infinite rotations. Its statistical and dynamical properties were studied in several papers by C. Radin, see for instance [Rad92] , [Rad97] . In particular, it was shown that the orientations of triangles in the pinwheel tiling are equally distributed in the circle. Despite the occurrance of irrational edge lengths and incommensurate angles, all vertices of the pinwheel tiling have rational coordinates.

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

Preview Pinwheel variant (10 tiles)
Pinwheel variant (10 tiles)

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Pinwheel variant (13 tiles)
Pinwheel variant (13 tiles)

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Pinwheel variant (65 tiles I)
Pinwheel variant (65 tiles I)

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Pinwheel variant (65 tiles II)
Pinwheel variant (65 tiles II)

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Pinwheel-1-2
Pinwheel-1-2

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Pinwheel-1-3
Pinwheel-1-3

Preview Pinwheel-1/2
Pinwheel-1/2

One member of an infinite series of tilings generated by a more general construction than a tile-substitution, [Sad98]. In particular, Sadun’s construction yields tilings with infinitely many prototiles, as well as with finitely many prototiles. Each tiling in this series is described by two positive integer parameters. This one, with parameter 1⁄2, is one of the ‘simplest’, in the sense that there are only two prototiles. THE simplest in the series is the one with parameter 1, which is the well known pinwheel tiling.

Infinite Rotations Infinite Local Complexity Saduns Generalised Pinwheels Polytopal Tiles Self Similar Substitution

Preview Pinwheel-2-2
Pinwheel-2-2

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Plate Tiling
Plate Tiling

Preview Pregnant Chairs
Pregnant Chairs

Preview Pregnant Chairs (Variant)
Pregnant Chairs (Variant)

Preview Priebe Frank non PV
Priebe Frank non PV

A plane substitution tiling which does not possess flc. It arises from the 1-dimensional substitution a -> abbb, b -> a, which inflation factor is not a PV-number. In the last sentence, ‘arises’ is to be understood as follows: Whenever one has a 1-dimensional substitution, it defines a d-dimensional substitution just by taking the Cartesian product. For an example, see Fibonacci times Fibonacci. Then, a 1-dimensional cut through each such d-dimensional tiling along a direction of some edge is the 1-dimensional tiling itself.

Polytopal Tiles Parallelogram Tiles Self Similar Substitution

Preview Psychedelic Penrose variant I
Psychedelic Penrose variant I

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Psychedelic Penrose variant II
Psychedelic Penrose variant II

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Psychedelic Penrose variant III
Psychedelic Penrose variant III

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Psychedelic Penrose variant IV
Psychedelic Penrose variant IV

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Psychedelic Penrose variant V
Psychedelic Penrose variant V

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Pythagoras-3-1
Pythagoras-3-1

Preview Pythagoras-3-2
Pythagoras-3-2

Preview Pythagoras-4-1
Pythagoras-4-1

Preview Pythagoras-5-1
Pythagoras-5-1

Preview Pythagoras-5-2
Pythagoras-5-2

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

Preview Pythia-4-1
Pythia-4-1

Preview Pythia-5-1
Pythia-5-1

Preview Quarterhex
Quarterhex

A variant of semi-detached house. In contrast to the latter, this one is a model set with p-adic internal space.

Preview Quartic pinwheel
Quartic pinwheel

One of the rare examples of a tiling where the prototiles occur in infinitely many orientations. Apart from the pinwheel tiling and its generalizations [Sad98] there are only a few examples known which show infinite rotations. The inflation factor of this one is a complex algebraic PV number of degree four. As the scaling and the rotations for the tiles are all given by algebraic units, every vertex of the tiling lies within a finitely generated Z-module.

Infinite Rotations Infinite Local Complexity Polytopal Tiles Self Similar Substitution

Preview Rectangulo dorado
Rectangulo dorado

Imagine a rectangle which can be decomposed into three proportional rectangles. The three rectangles have different sizes. The smallest one has the same orientation as the primitive rectangle. The other two rectangles have a 90 degrees with respect to the large rectangle. Starting with the smallest rectangle, the small side is 1 and the long side is x [where x is the square root of the golden number $\frac{\sqrt{5}+1}{2}$]. The next rectangle has x as a lower side.

Preview Rhomb square oktagon
Rhomb square oktagon

Finite Rotations Polytopal Tiles

Preview Rhomboid Triangle
Rhomboid Triangle

Preview Robinson Triangle
Robinson Triangle

A variation of the Penrose rhomb tilings, suggested by R. M. Robinson. The rhombs are cut into triangles, thus making the substitution volume hierarchic. Thus, this one is obviously mld with the other Penrose tilings. For more details, see Penrose rhomb tilings. Each triangle comes either left- or right-handed, which is indicated by the different colours. This distinction is important since the triangles itself are mirror symmetric, but their first substitutions are not.

Without Decoration Finite Rotations Polytopal Windowed Tiling Polytopal Tiles Self Similar Substitution Mld Class Penrose

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 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 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 Shield
Shield

In connection with physical quasicrystals, the most interesting 2dim tilings are based on 5-, 8-, 10- and 12-fold rotational symmetry. This 12-fold tiling was studied by F. Gähler, in particular its cut and project scheme, the local matching rules and diffraction properties [Gah88]. The window of the vertex set of the shield It is mld to the Socolar tiling, thus they share many interesting properties. One is that they possess a local matching rules.

With Decoration Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Polytopal Tiles Mld Class Shield And Socolar

Preview Single Bat
Single Bat

Denote the elements of the field $F_4$ by $\{0, 1, w, w + 1\}$, where $w$ satisfies the following equation with coefficients in $F_2: w^2 + w + 1 = 0$. Single Bat is a recurrent double sequence defined by $a(i, 0) = a(0, j) = 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z) = x + (w + 1) x^2 + w y^2 + z + (w + 1) z^2$.

Preview Smallest PV
Smallest PV

The three letter substitution rule whose scaling is the smallest PV number, a root of the polynomial $x^3 - x - 1 = 0$. Though it might not look it at first glance, the Rauzy fractal is connected. This can be shown using the method of A. Siegel described in [Sie04]. The Rauzy fractal:

One Dimensional Euclidean Windowed Tiling Self Similar Substitution Polytopal Tiles

Preview Smallest Pisot (dual)
Smallest Pisot (dual)

Finite Rotations Euclidean Windowed Tiling Polytopal Tiles Polytopal Windowed Tiling Canonical Substitution Tiling Parallelogram Tiles

Preview Socolar
Socolar

In connection with physical quasicrystals, the most interesting 2dim tilings are based on 5-, 8-, 10- and 12-fold rotational symmetry. This 12-fold tiling was studied thoroughly in [Soc89], where J. Socolar described the generating substitution as well as the local matching rules and the cut and project scheme, As well as the Penrose Rhomb tilings (5- resp. 10-fold) and the Ammann-Beenker tilings (8-fold), it allows a decoration by Ammann bars (see [GS87]).

Euclidean Windowed Tiling Polytopal Windowed Tiling Polytopal Tiles Parallelogram Tiles Canonical Substitution Tiling Mld Class Shield And Socolar

Preview Socolar's 7-fold
Socolar's 7-fold

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Sphinx
Sphinx

A classical example of a substitution with inflation factor 2. It arises from the well-known related rep-tile. It is not easy to see that this one is limitperiodic. This was shown in [LM01] , thus this one is a cut and project tiling, and therefore pure point diffractive. The prototile is not mirror symmetric. It occurrs in two versions in the tiling. The colours indicate if a tile is left- or right-handed.

With Decoration Finite Rotations P Adic Windowed Tiling Polytopal Tiles Rep Tiles Self Similar Substitution

Preview Sphinx-9
Sphinx-9

A variant of the well known Sphinx tiling. The tile (sphinx) is a rep-tile with 9 tiles, as well as with 4 tiles.

Finite Rotations P Adic Windowed Tiling Polytopal Tiles Rep Tiles Self Similar Substitution

Preview Sqrt6-triangles
Sqrt6-triangles

A substitution rule with inflation factor $\sqrt{6}$, using three triangles as prototiles. Like the pinwheel tilings, one of the first examples showing statistical circular symmetry, but being flc w.r.t. Euclidean motions.

Infinite Rotations Finite Local Complexity Polytopal Tiles Self Similar Substitution

Preview Square Chair
Square Chair

MLD to the more popular chair tiling, this version allows a simple translation into a coloured lattice: Replace each square of type i (1,2,3, or 4) with its midpoint, and assign to it colour i. Then each set of all points of colour i is a model set with internal p-adic space with p=2. This was first shown in [BMS98], a general framework is given in [LMS03].

With Decoration Finite Rotations P Adic Windowed Tiling Polytopal Tiles Self Similar Substitution Parallelogram Tiles Rhomb Tiles Mld Class Chair

Preview Square Triangle Pinwheel Variant
Square Triangle Pinwheel Variant

Preview Square-triangle
Square-triangle

The substitution tilings which are most relevant as models for physical quasicrystals are 5-fold, 8-fold, 10-fold and 12-fold symmetric ones. In the 5-fold (resp. 10-fold) case, there are the variations of the Penrose rhomb tilings and the Tuebingen triangle, for the 8-fold case there are the Ammann-Beenker tilings. These examples use a minimal set of prototiles, in the sense that the number of prototiles is equal to the algebraic degree of the corresponding inflation factor.

Preview Squeeze
Squeeze

One possible version of a substitution rule with a free parameter: The upper tip of the three triangles can be shifted arbitrarily in horizontal direction, the result is always a 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 Starry Night
Starry Night

A substitution tiling with six trapezoidal prototiles. The substitution rule is given for only five of the six tiles. The sixth tile (yellow) is substituted by nothing. The discoverer gives credits to Veit Elser for suggesting the shape of the tiles.

Preview T2000
T2000

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

Preview Tangram
Tangram

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Tetris
Tetris

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Tetris Faktor 2
Tetris Faktor 2

Preview Tetris Faktor 3
Tetris Faktor 3

Preview Thue Morse
Thue Morse

A classic. A lot of detail can be found in [JS99].

One Dimensional Self Similar Substitution

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 Trapezotriangular
Trapezotriangular

Preview Treasure
Treasure

Denote the elements of the field $F_4$ by $\{0, 1, w, w + 1\}$, where $w$ satisfies the following equation with coefficients in $F_2: w^2 + w + 1 = 0$. Treasure is a recurrent double sequence defined by $a(i, 0) = a(0, j) = 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z) = w x + y + w z + 1$. This recurrent double sequence can be also obtained using a system of substitutions of type 2 -> 4 with 12 rules, as it follows.

Preview Triangle Crown
Triangle Crown

Preview Triangle Duo
Triangle Duo

Infinite Rotations Finite Local Complexity Polytopal Tiles Self Similar Substitution

Preview Triangle Duo (single mirror)
Triangle Duo (single mirror)

Preview Triangle Duo (twin mirror)
Triangle Duo (twin mirror)

Preview Tribonacci
Tribonacci

The three letter substitution rule analysed by G. Rauzy in [Rau82] . The Rauzy fractal for this tiling is the Rauzy fractal.

Euclidean Windowed Tiling One Dimensional Polytopal Tiles Self Similar Substitution

Preview Tribonacci Dual
Tribonacci Dual

Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogram Tiles Self Similar Substitution

Preview Trihex
Trihex

A simple rule to generate nonperiodic tilings with one prototile, a triangle with angles 30°, 60°, 90°. It looks pretty much periodic: the hexagonal patches cover 75% of the plane, and this part is clearly periodic. The triangles in between the hexagons destroy the periodicity. But, by the selfsimilarity of the tilings, one finds larger periodic subsets in the tiling, covering 93,75%, 98,44%… of the plane. Thus, the tiling is limitperiodic.

Preview Tritriangle
Tritriangle

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Tuebingen Triangle
Tuebingen Triangle

Beside the Penrose rhomb tilings (and its variations), this is a classical candidate to model 5-fold (resp. 10-fold) quasicrystals. The inflation factor is - as in the Penrose case - the golden mean, $\frac{\sqrt{5}}{2} + \frac{1}{2}$. The prototiles are Robinson triangles, but these tilings are not mld to the Penrose tilings. The relation is different: The Penrose rhomb tilings are locally derivable from the Tübingen Triangle tilings. These tilings were discovered and studied thoroughly by a group in Tübingen, Germany, thus the name [BKSZ90] .

Finite Rotations Polytopal Windowed Tiling Polytopal Tiles Self Similar Substitution

Preview Uberpinwheel
Uberpinwheel

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Vampire
Vampire

Denote the elements of the field $F_4$ by $\{0, 1, w, w + 1\}$, where $w$ satisfies the following equation with coefficients in $F_2: w^2 + w + 1 = 0$. Vampire is a recurrent double sequence defined by $a(i, 0) = a(0, j) = 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z) = w x + (w + 1) x^2 + (w + 1) y^2 + w z + (w + 1) z^2$.

Preview Viper
Viper

A simple rule with just one prototile that generates tilings with statistical circular symmetry. The image use two colours for the aesthetic effect only. The substitution do not even use reflections, so the tilings consist of ‘right-handed’ tiles only.

Infinite Rotations Polytopal Tiles Rep Tiles Self Similar Substitution

Preview Waltonchair
Waltonchair

A substitution rule with inflation factor $\sqrt2$ and two prototiles. This makes it a candidate for being a model set with p-adic internal space. This has not been checked so far, but in principle it is possible by the methods in [LMS03] or [FS].

Preview Watanabe Ito Soma 12-fold
Watanabe Ito Soma 12-fold

This tiling is one of the possible variants of a rule given in [WSI95]. Unfortunately the definition given there is not unique. Patches of this rule are not the same as the patches given in the paper. If anyone finds the correct orientations of the thin rhombs then this page will be updated. Watanabe Ito Soma 12-fold (variant) is another version of this rule (though not one given in the paper), with more symmetry.

Finite Rotations Polytopal Tiles

Preview Watanabe Ito Soma 12-fold (variant)
Watanabe Ito Soma 12-fold (variant)

This is a variant of Watanabe Ito Soma 12-fold, with more symmetry.

Finite Rotations Polytopal Tiles

Preview Watanabe Ito Soma 8-fold
Watanabe Ito Soma 8-fold

This tiling was originally introduced in [WSI87] , however the description given there admits several substitution rules. This is the version given explicitly in [WSI95] . This is an example of a cut and project with a mixed internal space, a product of Euclidean and $p$-adic spaces, namely $\mathbb{R}^2 \times \mathbb{Q}_2$.

Finite Rotations Model Set Rhomb Tiles

Preview Wheel Tiling
Wheel Tiling

There is a very simple rule to transform the wheel tiling into the shield tiling: Replace each edge in the tiling by an edge orthogonal to it, of equal length, such that the old and new edge intersect in their midpoints. Applying this rule to the wheel tiling yields the shield tiling and vice versa. This is a very simple example of tilings which are mld.

With Decoration Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Polytopal Tiles Mld Class Shield And Socolar