Precursor Events and Penrose Tilings
(c) Robert Neil Boyd

Veil Nebula (HST)
Topology of Electromagnetism (R. N. Boyd)

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)