Ergodicity is a wide-spread phenomenon in the study of Riemannian manifolds. A quick sequence of examples, from simple to complicated, illustrates this point. All of the systems mentioned below have been proved to be ergodic via rigorous formal proofs. The irrational rotation of a circle is ergodic: the orbit of a point is such that eventually, every other point in the circle is visited. Such rotations are a special case of the interval exchange map. The beta expansions of a number are ergodic: beta expansions of a real number are done not in base-"N", but in base-formula_95 for some formula_96 The reflected version of the beta expansion is tent map; there are a variety of other ergodic maps of the unit interval. Moving to two dimensions, the arithmetic billiards with irrational angles are ergodic. One can also take a flat rectangle, squash it, cut it and reassemble it; this is the previously-mentioned baker's map. Its points can be described by the set of bi-infinite strings in two letters, that is, extending to both the left and right; as such, it looks like two copies of the Bernoulli process. If one deforms sideways during the squashing, one obtains Arnold's cat map. In most ways, the cat map is prototypical of any other similar transformation.