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 \(n>5\) 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. Our methods are to lift the tilings and substitutions to \({\mathbb {R}}^{{n}}\) using the lift operator first defined by Levitov, and to study the planarity of substitution tilings in \({\mathbb {R}}^{{n}}\) using mainly linear algebra, properties of circulant matrices, and trigonometric sums. For the construction of the Planar Rosa substitutions we additionally use the Kenyon criterion and a result on De Bruijn multigrid dual tilings.
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Arnoux, P., Furukado, M., Harriss, E., Ito, S.: Algebraic numbers, free group automorphisms and substitutions on the plane. Trans. Am. Math. Soc. 363(9), 4651–4699 (2011)
Arnoux, P., Ito, S.: Pisot substitutions and Rauzy fractals. Bull. Belg. Math. Soc. Simon Stevin 8(2), 181–207 (2001)
Beenker, F.P.M.: Algebraic theory of non-periodic tilings of the plane by two simple building blocks: a square and a rhombus. Report # 82-WSK-04. Eindhoven University of Technology (1982). https://research.tue.nl/en/publications/algebraic-theory-of-non-periodic-tilings-of-the-plane-by-two-simp
Bédaride, N., Fernique, T.: When periodicities enforce aperiodicity. Commun. Math. Phys. 335(3), 1099–1120 (2015)
Baake, M., Grimm, U.: Aperiodic Order. Vol. 1: A Mathematical Invitation. Encyclopedia of Mathematics and Its Applications, vol. 149. Cambridge University Press, Cambridge (2013)
Baake, M., Grimm, U.: Aperiodic Order, Vol. 2: Crystallography and Almost Periodicity. Encyclopedia of Mathematics and Its Applications, vol. 166. Cambridge University Press, Cambridge (2017)
de Bruijn, N.G.: Algebraic theory of Penrose’s nonperiodic tilings of the plane. I, II. Nederl. Akad. Wetensch. Indag. Math. 43(1), 39–52, 53–66 (1981)
Davis, P.J.: Circulant Matrices. A Wiley-Interscience Publication. Pure and Applied Mathematics. Wiley, New York (1979)
Fernique, T.: Multidimensional Sturmian sequences and generalized substitutions. Int. J. Found. Comput. Sci. 17(3), 575–599 (2006)
Frank, N.P.: A primer of substitution tilings of the Euclidean plane. Expo. Math. 26(4), 295–326 (2008)
Grünbaum, B., Shephard, G.C.: Tilings and Patterns. W.H. Freeman, New York (1987)
Harriss, E.: On Canonical Substitution Tilings. PhD thesis, University of London (2004). https://www.academia.edu/255299/On_Canonical_Substitution_Tilings
Harriss, E.O., Lamb, J.S.W.: Canonical substitutions tilings of Ammann–Beenker type. Theor. Comput. Sci. 319(1–3), 241–279 (2004)
Jolivet, T.: Combinatorics of Pisot Substitutions. PhD thesis, Université Paris Diderot & University of Turku (2013). https://jolivet.org/timo/docs/thesis_jolivet.pdf
Kari, J., Rissanen, M.: Sub Rosa, a system of quasiperiodic rhombic substitution tilings with \(n\)-fold rotational symmetry. Discrete Comput. Geom. 55(4), 972–996 (2016)
Kenyon, R.: Tiling a polygon with parallelograms. Algorithmica 9(4), 382–397 (1993)
Levitov, L.S.: Local rules for quasicrystals. Commun. Math. Phys. 119(4), 627–666 (1988)
Lutfalla, V.H.: An effective construction for cut-and-project rhombus tilings with global \(n\)-fold rotational symmetry. In: 27th IFIP WG 1.5 International Workshop on Cellular Automata and Discrete Complex Systems (Aix-Marseille 2021). OpenAccess Series in Informatics, vol. 90, # 9. Leibniz-Zent. Inform., Wadern (2021)
Lutfalla, V.H.: Substitution Discrete Planes. PhD thesis, Université Sorbonne Paris Nord (2021). https://hal.archives-ouvertes.fr/tel-03376430
Masáková, Z., Mazáč, J., Pelantová, E.: On generalized self-similarities of cut-and-project sets (2019). arXiv:1909.10753
Penrose, R.: The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 10, 266–271 (1974)
Sano, Y., Arnoux, P., Ito, S.: Higher dimensional extensions of substitutions and their dual maps. J. Anal. Math. 83, 183–206 (2001)
Shechtman, D., Blech, I., Gratias, D., Cahn, J.W.: Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53(20), 1951–1954 (1984)
Socolar, J.E.S.: Weak matching rules for quasicrystals. Commun. Math. Phys. 129(3), 599–619 (1990)
Acknowledgements
We wish to thank Thierry Monteil and Nicolas Bédaride for their help regarding billiard words, and also Thomas Fernique for his help and proofreading.
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Appendix
Appendix
Lemma A.1
(eigenvalues of the elementary matrices) Let n be an odd integer and \(i\in \{0,\dots , \lfloor {n}/{2}\rfloor -1\}\). The elementary matrix \(M_i(n)\) defined in Definition 4.1 has eigenspaces \(\Delta \) and \({\mathcal {E}}_{{j}}\) for \(0\leqslant j<\lfloor n/2\rfloor \) with eigenvalue \(\lambda _\Delta =0\) and
Proof
Let z be a complex number such that \(z^n=1\) and Z be the vector \((z^k)_{0\leqslant k < n}\). We have
With
we get that \({\mathcal {E}}_{{j}}\) is eigenspace of \(M_i(n)\) with eigenvalue \(\lambda _{i,j}(n)\),
With \(z=1\) we get that \(\Delta \) is an eigenspace with eigenvalue 0. \(\square \)
Lemma A.2
(trigonometric manipulations) Let k be an integer and \(i\in \{0,\dots , k-1\}\). Let us define \(C_{j,k}\) by
with \(\theta _{j,k}:={(2j+1)\pi }/({2(2k+1)})\). We have
Proof
Let us write \(\theta \) for \(\theta _{j,k}\) for the sake of simplicity.
\(\square \)
Lemma A.3
(the eigenvalue matrix \(N_{{n}}\) is orthogonal up to a scalar) Let n be an odd integer, then the matrix \(N_{{n}}\) from Definition 4.4 is orthogonal up to a scalar. More precisely, \((1/\sqrt{n})N_{{n}}\) is an orthogonal matrix, i.e.,
Proof
Let us first recall that n is an odd integer. Let us define three matrices A, B, and C by:
Let us first remark that A is a Discrete Cosine Transform matrix, sometimes called DCT-III, which is known to be orthogonal. From this we will prove that \(B\cdot B^\top = \mathrm{Id}_{\rfloor {n/2}\rfloor }\) and then that C is orthogonal.
Let us prove \(B\cdot B^\top = \mathrm{Id}_{\lfloor {{n}/{2}}\rfloor }\). Let us look at the j, k coefficient of \(B\cdot B^\top \) for \(0\leqslant j,k <n/2\):
Now from the fact that \(B\cdot B^\top = \mathrm{Id}_{\lfloor {{n}/ {2}}\rfloor }\) let us prove that C is orthogonal. Let us first remark that for \(0\leqslant i<\lfloor {{n}/{2}}\rfloor \) we have
This is due to the fact that n is odd, which implies that \(n=2\lfloor {{n}/{2}}\rfloor +1\). Let us also remark that for \(0\leqslant i,j<\lfloor {{n}/{2}}\rfloor \) we have
The first equality is just a reformulation of the definition, and for the second equality let us develop \(B_{i,n-1-j}\) as follows:
Now let us prove that \((C\cdot C^\top )_{j,k} =(B\cdot B^\top )_{j,k}\) for \(0\leqslant j,k < \lfloor {{n}/{2}}\rfloor \).
Hence C is orthogonal and \(N_{{n}}\) is orthogonal up to a scalar. \(\square \)
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Kari, J., Lutfalla, V.H. Substitution Discrete Plane Tilings with 2n-Fold Rotational Symmetry for Odd n. Discrete Comput Geom 69, 349–398 (2023). https://doi.org/10.1007/s00454-022-00390-z
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DOI: https://doi.org/10.1007/s00454-022-00390-z
Keywords
- Substitution tilings
- Discrete planes
- Cut-and-project tiling
- n-Fold symmetric tiling
- Quasiperiodic tilings
- Rhombus tiling