Sub Rosa, A System of Quasiperiodic Rhombic Substitution Tilings with n-Fold Rotational Symmetry

Abstract

In this paper we prove the existence of quasiperiodic rhombic substitution tilings with 2n-fold rotational symmetry, for any n. The tilings are edge-to-edge and use ⌊ n 2 ⌋ rhombic prototiles with unit length sides. We explicitly describe the substitution rule for the edges of the rhombuses, and prove the existence of the corresponding tile substitutions by proving that the interior can be tiled consistently with the given edge substitutions.

Cyclotomic Aperiodic Substitution Tilings

“…He kindly asks for the readers indulgence and hopes that the content and the sketched ideas herein are helpful for further research despite possible formal issues. The recent publications of G. Maloney [8], J. Kari and M. Rissanen [47] and T. Hibma [40,42], who found similar results, demanded a response on short notice.…”

Tiling approach for the description of the sevenfold symmetry in quasicrystals

“…Indeed, one can start with ''roses'' [12,34] resembling Buddhist mandalas, inflate them with a suitable factor to ensure that two pairs of opposite vertices of the inflated T 1 n rhombuses coincide with the vertices of two alternative types, and find the way to fill the rest. However, even though it sounds very easy, it actually requires significant efforts.…”

“…On one the hand, our motivation was to simplify the rules, and on the other hand, to reproduce at least the central core of the known tiling [8]. Thus, the basic set of prototiles is formed by only five different rhombuses (compare with [8,27,28,33,34,40]). The inflation/deflation rules are presented in Fig.…”

“…For example, the rhombic tiling proposed by Harriss [27] was characterized by very clear inflation rules but lacked the repetitive sevenfold patches, even if seven or fourteen rhombuses were initially arranged into a star. Kari and Rissanen [34] discussed two types of local environments, which they referred as roses, and offered an original and very attractive solution, though the assumption that all vertices of initial rhombuses had equivalent local environments led to an enormously large inflation factor.…”

“…Different inflation factors have been found to be compatible with the sevenfold symmetry. The resulting tilings were with and without repeatedly appearing heptagonal patterns [24][25][26][27][28][29][30][31][32][33][34][35]. For example, the rhombic tiling proposed by Harriss [27] was characterized by very clear inflation rules but lacked the repetitive sevenfold patches, even if seven or fourteen rhombuses were initially arranged into a star.…”