Veil Nebula (HST)

Precognitive Tilings and Quasicrystals (R. N. Boyd)

Aperiodic Tilings in 2, 3, and 4 dimensions can be thought of as Irrational Slices of an 8-dimensional E8 Lattice and its sublattices, such as E6. (Tony Smith)

Quasicrystals and Geometry (Marjorie Senechal)

Tiling spaces are Cantor set fiber bundles (Lorenzo Sadun, R. F. Williams) -- We prove that fairly general spaces of tilings of R^d are fiber bundles over the torus T^d, with totally disconnected fiber. This was conjectured (in a weaker form) in [W3], and proved in certain cases. In fact, we show that each such space is homeomorphic to the d-fold suspension of a Z^d subshift (or equivalently, a tiling space whose tiles are marked unit d-cubes). The only restrictions on our tiling spaces are that 1) the tiles are assumed to be polygons (polyhedra if d>2) that meet full-edge to full-edge (or full-face to full-face), 2) only a finite number of tile types are allowed, and 3) each tile type appears in only a finite number of orientations. The proof is constructive, and we illustrate it by constructing a ``square'' version of the Penrose tiling system.

What are Penrose Tiles? (Eugenio Durand)