Both of these examples are now cases of a field called non-commutative geometry. The specific examples of von Neumann algebras and are known as non-commutative measure theory and non-commutative topology, respectively. Non-commutative geometry is not merely a pursuit of generality for its own sake and is not just a curiosity. Non-commutative spaces arise naturally, even inevitably, from some constructions. For example, consider the non-periodic Penrose tilings of the plane by kites and darts. It is a theorem that, in such a tiling, every finite patch of kites and darts appears infinitely often. As a consequence, there is no way to distinguish two Penrose tilings by looking at a finite portion. This makes it impossible to assign the set of all tilings a topology in the traditional sense. Despite this, the Penrose tilings determine a non-commutative and consequently they can be studied by the techniques of non-commutative geometry. Another example, and one of great interest within differential geometry, comes from foliations of manifolds. These are ways of splitting the manifold up into smaller-dimensional submanifolds called "leaves", each of which is locally parallel to others nearby. The set of all leaves can be made into a topological space. However, the example of an irrational rotation shows that this topological space can be inacessible to the techniques of classical measure theory. However, there is a non-commutative von Neumann algebra associated to the leaf space of a foliation, and once again, this gives an otherwise unintelligible space a good geometric structure.