Stripes on Penrose tilings

We study the existence of one-dimensional quasicrystal structures on the vertex set Lambda(P) of a Penrose tiling, in an arbitrary direction w is an element of C. If wR boolean AND Z[zeta] not equal 0, zeta = e(2 pi i/5), then Lambda(P) + wR is a discrete family of lines that has a one-dimensional quasicrystal structure. Conversely, if w not equal 0 and wR boolean AND Z[zeta] = 0, Lambda(P) + wR is a dense subset of C. We also have a weak analog of Kronecker's approximation theorem.