A substitution rule that gives rise to an aperiodic tiling with dense tile orientations and 12-fold rotational symmetry. The inflation factor is $\sqrt{5+2\sqrt{3}}$. The tiling has finite local complexity with respect to rigid motions.

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

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 …

A substitution rule that gives rise to an aperiodic tiling with dense tile orientations and 5-fold rotational symmetry. The inflation factor is $\sqrt{6+\sqrt{5}}$. The tiling has finite local complexity with respect to rigid motions.

A classic simple substitution rule with Rauzy Fractal:

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

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 …

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 …

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. …

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 …

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.

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

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 …

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.

One of several substitution tilings found and submitted in February 2016 by L. Andritz with fivefold symmetry using tiles inspired by Islamic Girih patterns.

Despite the similarities the tiling is different to the “Tie and Navette” tiling as discussed in [Lue1990] and [LL1994] .

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

One of several substitution tilings found by L. Andritz with fivefold symmetry using tiles inspired by Islamic Girih patterns.

One of several substitution tilings found by L. Andritz with fivefold symmetry using tiles inspired by Islamic Girih patterns.

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 ) …

A substitution for the aperiodic hat monotile found in 2022 and published in [SMCG2024] , see also [BGS2023] . The “metatiles” in the first paper have some free parameter. Most of these parameters yield a substitution that is not selfsimilar. Here we show the selfsimilar version of the substitution …

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 …

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 …

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 $l$ , …

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

Find here the vector graphic.

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 …

A more or less obvious variant of the chair substitution.

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 …

One of several substitution tilings found by L. Andritz using similar right-angled quadrilaterals.

One of several substitution tilings found by L. Andritz using similar right-angled quadrilaterals.

One of several substitution tilings found by L. Andritz using similar right-angled quadrilaterals.

One of several substitution tilings found by L. Andritz using similar right-angled quadrilaterals.

One of several substitution tilings found by L. Andritz using similar right-angled quadrilaterals.

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.

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.

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 …

A volume hierarchic version of Conch.

The tiling shares a mld-class with the Penrose Tilings, e.g. Penrose Rhomb, Penrose kite-dart and Penrose Pentagon boat star). The inflation factor is the square of the golden mean $(\frac{\sqrt{5}}{2} + \frac{1}{2})^{2} = \frac{\sqrt{5}}{2} + \frac{3}{2} = 2.618033988\ldots$. In contrast to the …

The ochre tile - the “crown” - is an amazingly versatile tile to play with. This substitution is one possible substitution using it.

Find here a vector graphic.

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.

10-fold cyclotomic aperiodic tiling with dense tile orientations (CAST DTO).

8-fold cyclotomic aperiodic tiling with dense tile orientations (CAST DTO).

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 …

A relative of the Ente tiling. In contrast to the Ente tiling here the boundary of the prototile has infinite length, even though it consists of line segments (and accumulation points) only.

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 …

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 …

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.

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 …

The substitution tiling was derived from a mosaic at the Darb-i Imam Shrine in Isfahan, Iran. While the shrine dates back from 1453, [Lau2018] argues that the mosaic was created much later between 1715 - 1717.

The tiling relies on the regular decagon and two hexagons and has individual dihedral …

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, …

A simple limitperiodic tiling with two tiles.

In [RaSa98] Radin and Sadun constructed 3-dimensional tilings associated with subgroups of SO(3) generated by two rational rotations. One of the tilings was called “dite and kart”. The two-dimensional corresponding tiling is shown here (taken from Fig 1 in [RaSa98]).

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.

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 …

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

The decorative lining shows here how the prototile gets turned and mirrored for this example. The two rules are actually exactly the same. For decoration the horizontal tile was colored purple.

Tiling submitted by Andrew Hudson. The scaling factor is the smallest PV number, the Plastic Number which is a root of the polynomial $x^3 - x - 1 = 0$.

A natural variant of the simpler half-hex tiling.

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, …

A relative of the Squiral tiling. The boundary of the prototile has infinitely many line segments, but it has still finite length. A relative of this tiling is Dampflok, where the boundary of the prototile has infinite length, even though it consists of line segments (and accumulation points) only.

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

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.

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.

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.

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.

The FASS-curve of the pentagon bases on an aperiodic substitution tiling with four substitution rules and appropriate decorations. The substitution tiling was derived from the Robinson Triangle Tiling. Its inflation factor is the golden mean $\frac{\sqrt{5}}{2} + \frac{1}{2} = 1.618033988\ldots$.

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})) + …

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 …

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

A fractalized version of the Penrose Kite and Dart tilings. Unlike Kite and Dart, this version is a selfsimilar substitution.

A selfsimilar version of a substitution with Penrose rhombs, but without reflections. These tilings are mld to the Penrose Rhomb tilings, even though they lack their mirror symmetry.

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 …

This tiling is a generalization of the Godreche-Lancon-Billard Binary first derived by T. Hibma and later worked out in detail by S. Pautze. All interior angles are integer multiples of $\frac{\pi}{n}$. For $n=5$ it is identical to the Godreche-Lancon-Billard Binary tiling with 2 prototiles. For odd …

A tiling resembling Islamic Girih patterns but using 14-fold symmetry rather than 8- or 10- or 12-fold. Its inflation factor is $1 + \cos(\frac{\pi}{14}) \csc(\frac{\pi}{7}) + 2 \cos(\frac{3 \pi}{14}) \csc(\frac{\pi}{7}) = 6.850855...$ which is a unit but not a PV number. It uses 11 prototiles …

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 …

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 …

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 …

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.

The Gosper Curve is a FASS-curve which can be derived by a substitution tiling with one substitution rule and appropriate decorations. The inflation factor $q$ is $sqrt(7)$.

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.

A pretty simple aperiodic monotile found in 2022 and published in [SMCG2024] . The tile is a tridecagon (13-gon) built from 16 basic triangles with edge lengths $1,2, \sqrt{3}$. The shape alone ensures that every tiling of the plane you can build from this tile will be aperiodic. In this sense it …

The original Heighway Dragon Curve as described in [gar1967] , can be derived by a substitution tiling with one substitution rule and appropriate decoration. However, it is not a FASS-curve because it is not self avoiding. With the results in [pau2021] it is possible to derive a substitution tiling …

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 was probably the best example of an aperiodic monotile before the discovery of the Hat tiling. This decorated hexagonal tile, …

The Hilbert Curve is one of the earliest FASS-curves. The original algorithm in [hil1891] bases on one substitution rule and an additional rule which describes how the substitutes have to be connected. As briefly mentioned in [pau2021] it is also possible to create the Hilbert Curve by a …

An artistic version of the 3-fold version of the 4-fold Hofstetter 4fold tilings. For details see there.

An artistic version of the Hofstetter 4fold tilings. For details see there.

A decorated version of the Hofstetter 4fold tilings. This version can be generated by a proper substitution rule. It was shown in [FH15] that this version is aperiodic, as well as limitperiodic. For more details see Hofstetters 4-fold.

The Viennese Artist Hofstetter Kurt found a new iteration method that produces aperiodic tilings. The idea is illustrated in the figure below: one starts with a single square tile of edge length one. A second square tile of the same size is put below the first one, such that its SW corner is …

An artistic version of the 6-fold version of the 4-fold Hofstetter 4fold tilings, inspired by islamic patterns. For details see there.

Tiling submitted by Andrew Hudson.

The scaling factor is $\frac{\sin(\frac{3\pi}{n})}{\sin(\frac{\pi}{n})}+\frac{\sin(\frac{2\pi}{n})}{\sin(\frac{\pi}{n})}$.

All tiles appear in one chirality only, so markings were omitted.

Find here a vector graphic.

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 …

An one-dimensional substitution rule that uses an infinite number of prototiles.

The inflation factor is $2.5$.

The substitution rules are given by:

$T_{0}\rightarrow T_{0},T_{1}$

$T_{1}\rightarrow T_{0},T_{0},T_{2}$

$T_{2}\rightarrow T_{0},T_{1},T_{3}$

$T_{k}\rightarrow T_{0},T_{k-1},T_{k+1}$ …

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, …

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 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 $x$).

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

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).

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 …

A simple substitution rule, generating tilings which don’t possess flc. The fractally shaped tiles make it a selfsimilar-substitution. Despite the fractal apperance, the dimension of the boundary of the prototile is one almost everywhere: the boundary of the tile consists of lines almost everywhere …

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 …

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 …

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 …

The substitution $a \rightarrow bcc, b \rightarrow ba, c \rightarrow bc$ is a member of the MLD class of the [Kolakoski-(3,1) sequence] (/substitution/kolakoski-3-1/). As the reversed substitution generates the same hull, it is mirror symmetric. The scaling factor $\lambda \approx $ 2.20557 is the …

The substitution $a \rightarrow abcc, b \rightarrow a, c \rightarrow bc$ is a member of the MLD class of the [Kolakoski-(3,1) sequence] (/substitution/kolakoski-3-1/). 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 …

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 …

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.

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 …

A tiling with 7-fold symmetry and a lot of locally 7-fold symmetric patches. There are only three tile shapes, but nine different prototiles. The inflation factor is a PV number: $2+2\cos\left(\frac{\pi}{7}\right)+2\cos\left(\frac{2\pi}{7}\right) = 5.04891733952231\ldots$ which is the largest root of $x^{3}-6x^{2}+5x-1$.

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.

J. Millar discovered a set of tilings with patches of dihedral symmetry $D_2n$ and inflation multiplier $\sqrt{2 + 2 \cos(\frac{\pi}{n})}$, which is the same inflation multiplier as of the Generalized Godreche-Lancon-Billard Binary.

All interior angles of all prototiles are integer multiples of …

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

The Monnier Trapezium and Diamond tiling uses two prototiles, a trapezium and a rhomb. The inflation multiplier is $2$. By changing the chiralities of the prototiles within the first level supertiles several further variants can be derived.

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, …

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 $x^4 - x^3 + 1 = 0$.

A volume hierarchic version of Nautilus

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] .

The Nischke-Danzer-Deltoid 6-fold-2-2 was discussed and derived in [ND96] .

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:

A variant of the Pinwheel tiling without infinitely many rotations.

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.

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, …

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.

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.

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, …

The Peano Curve is one the earliest known FASS-curves. The original algorithm in [pea1890] bases on one substitution rule and an additional rule which describes how the substitutes have to be connected. As briefly mentioned in [pau2021] it is also possible to create the Peano Curve by a …

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

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 …

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: …

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.

The Robinson Triangle substitution is just one out of several possibilities to play with the triangle versions of the Penrose tilings. Another one is the Tübingen Triangle substitution. Here is an example where the obtuse triangle is the smaller prototile, and the acute triangle is the larger one. …

The Robinson Triangle substitution is just one out of several possibilities to play with the triangle versions of the Penrose tilings. Another one is the Tübingen Triangle substitution. Here is an example where the obtuse triangle is the smaller prototile, and the acute triangle is the larger one. …

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).

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.

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 …

Find the vector graphic here

The pinwheel tiling has several straight forward variants. Here is one with 13 tiles.

Find here the vector graphic

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 …

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 …

A variation of the Robinson Triangle substitution where the tiles are decorated in order to produce striking visual effects. The idea came from the design of Aperiodic 2012 in Cairns, Australia. Other variants can be found in psychedelic-penrose-variant-2, psychedelic-penrose-variant-3, …

A variation of the Robinson Triangle substitution where the tiles are decorated in order to produce striking visual effects. The idea came from the design of Aperiodic 2012 in Cairns, Australia. Other variants can be found in psychedelic-penrose-variant-1, psychedelic-penrose-variant-3, …

A variation of the Robinson Triangle substitution where the tiles are decorated in order to produce striking visual effects. The idea came from the design of Aperiodic 2012 in Cairns, Australia. Other variants can be found in psychedelic-penrose-variant-1, psychedelic-penrose-variant-2, …

A variation of the Robinson Triangle substitution where the tiles are decorated in order to produce striking visual effects. The idea came from the design of Aperiodic 2012 in Cairns, Australia. Other variants can be found in psychedelic-penrose-variant-1, psychedelic-penrose-variant-2, …

A variation of the Robinson Triangle substitution where the tiles are decorated in order to produce striking visual effects. The idea came from the design of Aperiodic 2012 in Cairns, Australia. Other variants can be found in psychedelic-penrose-variant-1, psychedelic-penrose-variant-2, …

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 …

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

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 …

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 …

Find the vector graphic here

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 …

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 …

A substitution rule that gives rise to a non-periodic tiling $\mathcal{T}_{17}$ with $17$-fold dihedral symmetry. The substitution factor is $\mu_{17}=1/(2\sin(\pi/34))$.

The tiling $\mathcal{T}_{17}$ is obtained by assigning orientations to relevant tiles in the 1-order supertiles to ensure the …

The substitution rule is similar to Say-awen 17-fold. It is invariant under $21-$fold dihedral symmetry and has infinite local complexity. The substitution factor is $\mu_{21}=1/(2\sin(\pi/42))$, which is non-Pisot with minimal polynomial $p_{21}(x)=x^6-8x^5+8x^4+6x^3-6x^2-x+1$.

Schaad’s 7-fold is a variation of Madison’s 7-Fold, hence it shares many properties with it. It allows for tilings with global 7-fold symmetry and a lot of locally 7-fold symmetric patches. There are three tile shapes, but only seven instead of nine different prototiles. The inflation factor is a PV …

A substitution tiling using four different prototiles, as described by [LS12].

A substitution tiling using four different prototiles, as described by [LS12].

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] .

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].

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 …

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) …

The three letter substitution rule whose scaling is the smallest PV number, the Plastic Number which is 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:

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 …

Socolar found the basis for this tiling already in 1987, but recently added a substitution tiling. An interesting feature is that there exists a context-independent Ammann bar decoration of the tiles, similar to the one in the original Socolar 12-fold but with different relative phases, and hence a fairly simple set of matching rules.

A square-triangle tiling without mirroring.

The Spectre tiling was derived from the Hat tiling. While the aperiodic monotiles of the Hat tiling appear in two chiralities, the aperiodic monotiles of the Spectre tiling have the same chirality. As the Hat tiling this one is also not really a substitution tiling.

The patch was generated with the …

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 …

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

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.

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].

Nowadays (2019) there are several generalizations of a tile substitution. One was introduced already in Sad98 Even though his idea produces a welath of examples, most of these have not yet been well studied.

The idea is to give a rule how to substitute a prototile and apply this rule always to the …

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 …

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.

The inflation multiplier is the smallest PV number, the Plastic Number which is a root of the polynomial $x^3 - x - 1 = 0$.

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 …

This is the block version of the Squiral tiling. The graphic was created by Leif Borgstedt.

One of several 12-fold tilings by Peter Stampfli.

One of several 12-fold tilings by Peter Stampfli.

A 12-fold substitution tiling without squares (most other 12-fold tilings do use squares, search for “12-fold” to see them).

A substitution rule for a tiling with prototiles based on 12-fold dihedral symmetry. However, the tilings show only 6-fold dihedral symmetry, in contrast to the usual suspects related to 12-fold symmetry, like the shield tilings or the Socolar tilings.

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.

J. Kari and M. Rissanen derived a set of rhomb substitution tilings in [KR2016] with n-fold dihedral symmetry.

• All substitution rules have dihedral $D_{2}$ symmetry.
• All edges of the substitution rules are equal and also have dihedral $D_{2}$ symmetry.
• All interior angles of all prototiles are …

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

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

This is the 2D version of the one-dimensional Thue Morse tiling due to Shelomo Ben-Abraham et.al. The graphic was created by Leif Borgstedt.

An one-dimensional substitution rule that uses an infinite number of proto tiles and yields a transcendental inflation multiplier.

The inflation factor is approximately $2.7899$.

The substitution rules are given by:

$T_{0}\rightarrow T_{0},T_{1}$

$T_{1}\rightarrow 3T_{0},T_{2}$

$T_{2}\rightarrow …

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 …

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) = …

Nowadays (2019) there are several generalizations of a tile substitution. One was introduced already in Sad98 Even though his idea produces a welath of examples, most of these have not yet been well studied.

The idea is to give a rule how to substitute a prototile and apply this rule always to the …

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

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 …

A Chair variant.

A Chair variant.

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 …

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) = …

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.

Find here a vector graphic.

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].

This Wanderer tiling is the first of an infinite series of substitution tilings by Joan Taylor based on paper-folding sequences. It uses a single square tile with four distinct rotations and two distinct reflections. Here we use colours to distinguish left-handed (brown) from right-handed (white) …

This Wanderer tiling is one in an infinite series of substitution tilings by Joan Taylor based on paper-folding sequences. It uses a single square tile with four distinct rotations and two distinct reflections. Here we use colours to distinguish vertical (blue) from horizontal (ochre) tiles. In the …

The source of the tiling can be found in [WSI95] Fig. 2 (iii) and Fig. 3. Its inflation factor is $2+\sqrt{3}$ and it has finite local complexity with respect to rigid motions.

Unfortunately the corresponding substitution rules given in Fig. 2 (iii) of the paper are not unique. For some time the …

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

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 $\mathbb{R}^2 \times \mathbb{Q}_2$.

A reference to the shape of the trays in the Westend Cafeteria of Bielefeld University (before 2010).

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.