The Efficiency of Delone Coverings of the Canonical Tilings Τ ^*(a4) and Τ ^*(d6)

Abstract

This chapter is devoted to the coverings of the two quasiperiodic canonical tilings [1] Τ ^*(a4) [2] and Τ ^*(d6) ≡ Τ ^*(2f) [3, 4], obtained by projection [3] from the root lattices A4 and D6, respectively [5]. The projection from the “high-dimensional lattice” L onto the space of a quasiperiodic tiling, called “parallel space” and denoted by $\mathbb{E}_\parallel$, is defined by the representations of noncrystallographic groups [1]. We consider a canonical quasiperiodic tiling Τ ^*(l) [1] projected from a lattice L onto parallel space $\mathbb{E}_ \bot$ , whose coding window in perpendicular space $\mathbb{E}_\parallel$ is a projected Voronoi cell [5, 1] V _⊥, and whose tiles in parallel space are projected boundaries of Delone cells [1, 5] X ^*‖. These tiles are, in the case of Τ*^(a4), two golden triangles with edge lengths ➁, a standard length parallel to a 2-fold axis of an icosahedron, and τ^➁, where τ = (1 + √5)/2. The “2-fold direction” is defined only in 3-dimensional space, and indeed the decagonal tiling Τ ^*(a4) can be seen as a subtiling of the icosahedral tiling Τ ^*(d6) [3, 4]. The tiles of Τ ^*(d6) are six golden tetrahedra [3, 6] of edge lengths ➁ and τ ^➁, as above. The tiles are coded in perpendicular space $\mathbb{E}_ \bot$ by corresponding dual Voronoi boundaries projected onto $\mathbb{E}_ \bot$. We denote these projected boundaries by X⊥. The codings X⊥ are, in the case of Τ^*(a4) two rhombuses of the same shape as the prototiles of the Penrose tiling P2, with angles of 36° and 72° [2, 7]. In the case of Τ ^*(d6) the codings X⊥ are acute and obtuse rhombohedra and four pyramids [3] (see also Fig. 5.18).