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.

Finite Local Complexity Polytopal Tiles Self Similar Substitution Infinite Rotations

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

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.

Finite Local Complexity Polytopal Tiles Self Similar Substitution Infinite Rotations

A classic simple substitution rule with Rauzy Fractal:

One Dimensional Euclidean Windowed Tiling Self Similar Substitution Polytopal 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 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

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

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

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

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

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] .

Finite Rotations Model Set Polytopal Tiles Self Similar Substitution Finite Local Complexity Girih

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

Finite Rotations Polytopal Tiles Self Similar Substitution Finite Local Complexity Girih

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

Finite Rotations Polytopal Tiles Self Similar Substitution Finite Local Complexity Girih

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

Finite Rotations Polytopal Tiles Self Similar Substitution Finite Local Complexity Girih

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 rule. This tiling was used to derive the cut-and-project description of the hat tiling in [BGS2023] .

Aperiodic Monotile Self Similar 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 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

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.

Self Similar Substitution Infinite Rotations

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

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

A more or less obvious variant of the chair substitution.

Finite Rotations Polytopal Tiles Self Similar Substitution

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

A volume hierarchic version of Conch.

Finite Rotations Euclidean Windowed Tiling Self Similar Substitution

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

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

Finite Local Complexity Polytopal Tiles Self Similar Substitution Infinite Rotations

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

Finite Local Complexity Polytopal Tiles Self Similar Substitution Infinite Rotations

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 Matching Rules

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

Substitution tiling with isosceles triangles as prototiles allow several variations: For each tile in the first order supertiles, one can choose whether it is a left-handed or a right-handed version. By playing around with these possibilities, one obtains this variant from Danzer’s 7-fold.

Finite Rotations Polytopal Tiles Self Similar Substitution Generalized Robinson Triangles

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

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 symmetry ‘$D_{10}$‘. It was published in [LS2007] but the complete set of substitution rules can be found in [Ten2008] .

Finite Local Complexity Finite Rotations Polytopal Tiles Self Similar Substitution Girih

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

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

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.

Finite Rotations Polytopal Tiles Polyomio Tiling Rep Tiles Self Similar Substitution

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

Self Similar Substitution Plastic Number Infinite Rotations

Finite Rotations Self Similar Substitution

Finite Rotations Self Similar Substitution

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

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 $\frac{\sqrt{5}}{2} + \frac{1}{2} = 1.618033988\ldots$.

Polytopal Tiles Self Similar Substitution With Decoration FASS_curve

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

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

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

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

Self Similar 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.

Self Similar Substitution

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 altogether, 10 of them showing $D_2$-symmetry and one showing $D_{14}$-symmetry. This shows that all resulting tilings have local patches with 14-fold symmetry, and that the hull contains tilings with global 14-fold symmetry.

Finite Local Complexity Finite Rotations Polytopal Tiles Self Similar Substitution Girih

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 Infinite Rotations

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

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 $sqrt(7)$.

Finite Local Complexity Polytopal Tiles Self Similar Substitution With Decoration Limitperiodic FASS_curve

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

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 which generates a Heighway Dragon FASS-Curve without disturbing self similarity. In detail the decoration on the proto tile is shifted away from the corners in different ways.

Finite Local Complexity Polytopal Tiles Self Similar Substitution With Decoration Limitperiodic FASS_curve

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, 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.

Aperiodic Monotile Self Similar Substitution

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 substitution tiling with two substitution rules and appropriate decorations. The inflation factor $q$ is 2 and the lines are shifted slightly away from the center of the sides to illustrate the matching rules.

Finite Local Complexity Polytopal Tiles Self Similar Substitution With Decoration Limitperiodic FASS_curve

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.

Self Similar Substitution Finite Rotations

Find here a vector graphic.

Finite Rotations Polytopal Tiles Self Similar Substitution

An one-dimensional substitution rule that uses an infinite number of prototiles. The inflation factor is $2.5$. The substitution rules are given by: $T_{0}\rightarrow T_{0},T_{1}$ $T_{1}\rightarrow T_{0},T_{0},T_{2}$ $T_{2}\rightarrow T_{0},T_{1},T_{3}$ $T_{k}\rightarrow T_{0},T_{k-1},T_{k+1}$ $T_{\infty}\rightarrow T_{0},T_{\infty},T_{\infty}$ The corresponding substitution matrix can be written as: $1 2 1 1 1 1 1 1 1 ...$ $1 0 1 0 0 0 0 0 0 ...$ $0 1 0 1 0 0 0 0 0 ...$

One Dimensional Self Similar Substitution

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

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

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

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 (plus accumulation points). Unlike the Koch curve, for instance, or the fractal Dart and Kite prototiles, here the entire length of the boundary is finite. For more details, see Kenyon non-FLC.

Rep Tiles Self Similar Substitution

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

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

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

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

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

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

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 $x^{3}-6x^{2}+5x-1$.

Polytopal Tiles Self Similar Substitution Finite Local Complexity Rhomb Tiles Finite Rotations

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.

Finite Rotations Polytopal Tiles Self Similar Substitution Finite Local Complexity

Finite Rotations Polytopal Tiles Self Similar Substitution Rhomb Tiles Finite Local Complexity

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

Finite Rotations Polytopal Tiles Self Similar Substitution

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.

Finite Local Complexity Polytopal Tiles Self Similar Substitution With Decoration Limitperiodic

A volume hierarchic version of Nautilus

Finite Rotations Euclidean Windowed Tiling Self Similar Substitution

The Nischke-Danzer-Deltoid 7-fold-2-2 was discussed and derived in [ND96] but not shown. A figure with the tiling can be found in [Pau2017] .

Polytopal Tiles Self Similar Substitution Finite Local Complexity Finite Rotations Nischke Danzer Deltoid

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

Polytopal Tiles Self Similar Substitution Finite Local Complexity Finite Rotations Nischke Danzer Deltoid Limitperiodic

One Dimensional Self Similar Substitution Polytopal Tiles

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 substitution tiling with two substitution rules and appropriate decorations. The inflation factor $q$ is 3 and the lines are shifted slightly away from the center of the sides to illustrate the matching rules.

Finite Local Complexity Polytopal Tiles Self Similar Substitution With Decoration Limitperiodic FASS_curve

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

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

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

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 Infinite Rotations

Find the vector graphic here

Finite Rotations Polytopal Tiles Self Similar Substitution

The pinwheel tiling has several straight forward variants. Here is one with 13 tiles. Find here the vector graphic

Finite Rotations Polytopal Tiles Self Similar Substitution

Finite Rotations Polytopal Tiles Self Similar Substitution

Finite Rotations Polytopal Tiles Self Similar Substitution

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

Finite Rotations Polytopal Tiles Self Similar Substitution

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

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, psychedelic-penrose-variant-4, and psychedelic-penrose-variant-5.

Finite Rotations Polytopal Tiles Self Similar Substitution

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, psychedelic-penrose-variant-4, and psychedelic-penrose-variant-5.

Finite Rotations Polytopal Tiles Self Similar Substitution

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, psychedelic-penrose-variant-4, and psychedelic-penrose-variant-5.

Finite Rotations Polytopal Tiles Self Similar Substitution

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, psychedelic-penrose-variant-3, and psychedelic-penrose-variant-5.

Finite Rotations Polytopal Tiles Self Similar Substitution

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, psychedelic-penrose-variant-3, and psychedelic-penrose-variant-4.

Finite Rotations Polytopal Tiles Self Similar Substitution

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

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

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 Generalized Robinson Triangles

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 existence of a patch invariant under $17$-fold dihedral symmetry in a supertile of the substitution. This patch of $34$ triangles equivalent to $T{17,15}$ whose $\pi/17$ vertices meet at a point serves as the seed for $\mathcal{T}_{17}$.

Finite Rotations Polytopal Tiles Self Similar Substitution Generalized Robinson Triangles

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 $p{21}(x)=x^6-8x^5+8x^4+6x^3-6x^2-x+1$.

Finite Rotations Polytopal Tiles Self Similar Substitution Generalized Robinson Triangles

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 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 $x^{3}-6x^{2}+5x-1$.

Polytopal Tiles Self Similar Substitution Finite Local Complexity Rhomb Tiles Finite Rotations

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

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

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:

One Dimensional Euclidean Windowed Tiling Self Similar Substitution Polytopal Tiles Plastic Number

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

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

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

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

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

Self Similar Substitution Polytopal Tiles Plastic Number

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 integer multiples of $\frac{\pi}{n}$. The minimal inflation facctor for this type of substitution tilings was discussed and derived in [Pau2017] . The example shown below is the tiling for $n=7$.

Polytopal Tiles Self Similar Substitution Finite Local Complexity Finite Rotations Rhomb Tiles

Finite Rotations Polytopal Tiles Self Similar Substitution

Finite Rotations Polytopal Tiles Self Similar Substitution

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

One Dimensional Self Similar Substitution

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: $T_{0}\rightarrow T_{0},T_{1}$ $T_{1}\rightarrow 3T_{0},T_{2}$ $T_{2}\rightarrow 2T_{0},T_{1},T_{3}$ $T_{3}\rightarrow T_{0},T_{2},T_{4}$ $T_{4}\rightarrow 2T_{0},T_{3},T_{5}$ $T_{5}\rightarrow T_{0},T_{4},T_{6}$ $T_{6}\rightarrow T_{0},T_{5},T_{7}$ $T_{k}\rightarrow (1+f\left(k\right))T_{0},T_{k-1},T_{k+1}$ with $f\left(k\right)$ as the Thue-Morse sequence. The corresponding substitution matrix can be written as: $1 3 2 1 2 1 1 2 2 ...$

One Dimensional Self Similar Substitution

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

Infinite Rotations Finite Local Complexity Polytopal Tiles Self Similar Substitution

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

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

Finite Rotations Polytopal Tiles Self Similar Substitution

A Chair variant.

Finite Rotations Self Similar Substitution

A Chair variant.

Finite Rotations Self Similar Substitution

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

Finite Rotations Polytopal Tiles Self Similar Substitution

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.

Infinite Rotations Polytopal Tiles Rep Tiles Self Similar Substitution

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) tiles. In the substitution rule the orientation of the tiles is indicated by a line in the interior of the tiles. In the large patch below these lines and all edges are omitted since the interesting feature are the patterns produced by white resp.

Limitperiodic Polytopal Tiles Rep Tiles Self Similar Substitution

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 substitution rule the orientation of the tiles is indicated by a line in the interior of the tiles, the chirality (left-handed vs right-handed) is indicated by a point.

Limitperiodic Polytopal Tiles Rep Tiles Self Similar Substitution

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 exact substitution rules which generate the square level-3-supertile in Fig. 3 of the paper remained unclear. Finally Alessandro Musesti and Maurizio Paolini submitted the correct set of substitution rules in January 2023 to us.

Finite Rotations Polytopal Tiles Self Similar Substitution Finite Local Complexity

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 Polytopal Tiles Self Similar Substitution Finite Local Complexity