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 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

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

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 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

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