Planar Rosa : a family of quasiperiodic substitution discrete plane tilings with $2n$-fold rotational symmetry

Abstract

We present Planar Rosa, a family of rhombus tilings with a $2n$-fold rotational symmetry that are generated by a primitive substitution and that are also discrete plane tilings, meaning that they are obtained as a projection of a higher dimensional discrete plane. The discrete plane condition is a relaxed version of the cut-and-project condition. We also prove that the Sub Rosa substitution tilings with $2n$-fold rotational symmetry defined by Kari and Rissanen do not satisfy even the weaker discrete plane condition. We prove these results for all even $n\geq 4$. This completes our previously published results for odd values of $n$.

Figures (15)

• Figure 1: Central fragments of canonical Sub Rosa $4$ and Planar Rosa $4$ tilings with $8$-fold rotational symmetry.
• Figure 2: Central fragments of canonical Sub Rosa $6$ and Planar Rosa $6$ tilings with $12$-fold rotational symmetry.
• Figure 3: The canonical Ammann-Beenker tiling: an example of edge-to-edge rhombus tiling with $4$ edge directions.
• Figure 4: An example of a combinatorial substitution: on the left the initial patch with the three internal edges coloured, in the middle the images of the three initial tiles with the image of the internal edges coloured also, and on the right the patch obtained by gluing the image of the three tiles along the image of the shared edges.
• Figure 5: The Ammann-Beenker substitution, an example of vertex-hierarchic substitution on two rhombus tiles up to translation and rotation.

Theorems & Definitions (35)

• Theorem 1
• Theorem 2
• Lemma 3
• Lemma 4
• Proposition 5: Sufficient conditions for planarity and for non-planarity kari2021
• Proposition 6
• Definition 7: Abelianized edgeword $\left[u\right]$
• Proposition 8: Expansion matrices
• Definition 9: Elementary matrix $EM_{n}(i)$
• Proposition 10: Decomposition of the expansion matrix