  Rhombus Penrose tilings are tilings of the plane by two decorated rhombi such
that the decoration match at the junction between two tiles (like in a jigsaw
puzzle). In dynamical terms, they form a tiling space of finite type. If we
remove the decorations, we get, by definition, a sofic tiling space that we
here call geometrical Penrose tilings. Here, we show how to compute the
patterns of a given size which appear in these tilings by two different method:
one based on the substitutive structure of the Penrose tilings and the other on
their definition by the cut and projection method. We use this to prove that
the geometrical Penrose tilings are characterized by a small set of patterns
called vertex-atlas, i.e., they form a tiling space of finite type. Though
considered as folk, no complete proof of this result has been published, to our
knowledge.
