With Decoration
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 …
With Decoration Finite Rotations Polytopal Windowed Tiling Canonical Substitution Tiling Parallelogram Tiles Self Similar Substitution Mld Class Ammann
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 …
With Decoration Finite Rotations Polytopal Windowed Tiling Canonical Substitution Tiling Rhomb Tiles Mld Class Ammann Matching Rules
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
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 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 …
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
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 …
Finite Local Complexity Polytopal Tiles Self Similar Substitution With Decoration Limitperiodic FASS_curve
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 …
Finite Local Complexity Polytopal Tiles Self Similar Substitution With Decoration Limitperiodic FASS_curve
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 …
With Decoration Finite Local Complexity Polytopal Tiles Self Similar Substitution Mld Class Pinwheel
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
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 …
Finite Local Complexity Polytopal Tiles Self Similar Substitution With Decoration Limitperiodic FASS_curve
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 …
With Decoration Finite Local Complexity Saduns Generalised Pinwheels Polytopal Tiles Self Similar Substitution Mld Class Pinwheel Infinite Rotations
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 …
With Decoration Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Polytopal Tiles Mld Class Shield and Socolar Matching Rules
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 …
With Decoration Finite Rotations P Adic Windowed Tiling Polytopal Tiles Rep 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
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