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Substitution discrete plane tilings with $2n$-fold rotational symmetry for odd n

Jarkko Kari, Victor H. Lutfalla

TL;DR

It is proved that the Sub Rosa substitution tilings with a 2n-fold rotational symmetry for odd n > 5 defined by Kari and Rissanen are not discrete planes – and therefore not cut-andproject tilings either.

Abstract

We study substitution tilings that are also discrete plane tilings, that is, satisfy a relaxed version of cut-and-projection. We prove that the Sub Rosa substitution tilings with a 2n-fold rotational symmetry for odd n greater than 5 defined by Kari and Rissanen are not discrete planes, and therefore not cut-and-project tilings either. We then define new Planar Rosa substitution tilings with a 2n-fold rotational symmetry for any odd n, and show that these satisfy the discrete plane condition. The tilings we consider are edge-to-edge rhombus tilings. We give an explicit construction for the 10-fold case, and provide a construction method for the general case of any odd n.

Substitution discrete plane tilings with $2n$-fold rotational symmetry for odd n

TL;DR

It is proved that the Sub Rosa substitution tilings with a 2n-fold rotational symmetry for odd n > 5 defined by Kari and Rissanen are not discrete planes – and therefore not cut-andproject tilings either.

Abstract

We study substitution tilings that are also discrete plane tilings, that is, satisfy a relaxed version of cut-and-projection. We prove that the Sub Rosa substitution tilings with a 2n-fold rotational symmetry for odd n greater than 5 defined by Kari and Rissanen are not discrete planes, and therefore not cut-and-project tilings either. We then define new Planar Rosa substitution tilings with a 2n-fold rotational symmetry for any odd n, and show that these satisfy the discrete plane condition. The tilings we consider are edge-to-edge rhombus tilings. We give an explicit construction for the 10-fold case, and provide a construction method for the general case of any odd n.

Paper Structure

This paper contains 18 sections, 32 theorems, 93 equations, 22 figures, 1 table.

Table of Contents

  1. Keywords:
  2. Acknowledgments.
  3. Introduction
  4. Settings
  5. Rhombus Tiling.
  6. Substitution.
  7. Lifting to $\mathbb{R}^n$.
  8. Discrete plane.
  9. $n$-fold symmetric tilings.
  10. Substitution discrete planes
  11. Sub Rosa substitution tilings
  12. Construction
  13. Lifting in $\mathbb{R}^n$
  14. Tileability conditions
  15. Planar Rosa: substitution discrete planes with $2n$-fold rotational symmetry
  16. ...and 3 more sections

Key Result

Theorem 1

The Sub Rosa tilings for odd $n>5$ are not discrete plane tilings.

Figures (22)

  • Figure 1: Example of rhombus tiling.
  • Figure 2: An example of 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 images of the internal edges coloured. On the right the patch obtained by gluing the images of the three tiles along the images of the shared edges.
  • Figure 3: The Chair substitution, an example of an edge-hierarchic substitution on only one prototile up to translations and rotations.
  • Figure 4: The Penrose substitution, an example of a vertex-hierarchic substitution on two rhombus tiles up to translations and rotations.
  • Figure 5: The Sub Rosa 5 substitution on the $\tfrac{\pi}{5}$ rhombus, an example of a vertex-hierarchic substitution where the images of all edges are identical up to rotations and translations. The edgeword is 131131 where 1 and 3 indicate the narrow and the wide rhombuses, respectively.
  • ...and 17 more figures

Theorems & Definitions (62)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Definition 1: Elementary matrices
  • Lemma 1
  • ...and 52 more