Finite Rotations

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

Finite Rotations Polytopal Tiles Parallelogram Tiles Rhomb Tiles Harrisss Rhomb

Preview Imbalanced orientations
Imbalanced orientations

Finite Rotations Polytopal Tiles Self Similar Substitution

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 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 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 Minitangram
Minitangram

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

Finite Rotations Polytopal Tiles Self Similar Substitution

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 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 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 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 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-2-2
Pinwheel-2-2

Finite Rotations Polytopal 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 Rhomb square oktagon
Rhomb square oktagon

Finite Rotations Polytopal Tiles

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 Smallest Pisot (dual)
Smallest Pisot (dual)

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

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 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 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 Tangram
Tangram

Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Tetris
Tetris

Finite Rotations 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 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 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