We argue that the collection of Penrose tilings of the plane generated by a dart and a kite is an object of an inherently noncommutative nature. This is exhibited in the obstructions present in classical approaches to obtaining information about the space and the appearance of a rich structure when viewed in a non-classical light. This very structure readily yields operator-valued functions on the space and an alternative view of it gives a topological invariant in the form of its dimension group.