Similarity Dimension of Fractal Curves with Multiple Generators
We propose a definition for the similarity dimension of fractal curves with multiple generators.
math.MG↗
arXiv subjects
Stefan Pautze
Publications and source records attributed to Stefan Pautze.
Cyclotomic Aperiodic Substitution Tilings with Dense Tile Orientations
The class of Cyclotomic Aperiodic Substitution Tilings (CAST) whose vertices are supported by the $2n$-th cyclotomic field $\mathbb{Q}\left(ζ_{2n}\right)$ is extended to cases with Dense Tile Orientations (DTO). It is shown that every CAST with DTO has an inflation multiplier $η$ with irrational argument so that $\frac{kπ}{2n}\neq\arg\left(η\right)\notin\mathbb{πQ}$. The minimal inflation multiplier $η_{min.irr.}$ is discussed for $n\geqq2$. Examples of CASTs with DTO, minimal inflation multiplier $η_{min.irr.}$ and individual dihedral symmetry $D_{2n}$ are introduced for $n\in\left\{ 2,3,4,5,6,7\right\} $. The examples for $n\in\left\{ 2,3,4,5,6\right\} $ also yield finite local complexity (FLC).
math.MG↗
Cyclotomic Aperiodic Substitution Tilings
The class of Cyclotomic Aperiodic Substitution Tilings (CAST) is introduced. Its vertices are supported on the 2n-th cyclotomic field. It covers a wide range of known aperiodic substitution tilings of the plane with finite rotations. Substitution matrices and minimal inflation multipliers of CASTs are discussed as well as practical use cases to identify specimen with individual dihedral symmetry Dn or D2n, i.e. the tiling contains an infinite number of patches of any size with dihedral symmetry Dn or D2n only by iteration of substitution rules on a single tile.
math.MG↗