19, 13, 22, 15, 19, ... and is approximately . Chaos Rule 30 meets rigorous definitions of chaos proposed by Devaney and Knudson. In particular, according to Devaney's criteria, Rule 30 displays sensitive dependence on initial conditions (two initial configurations that differ only in a small number of cells rapidly diverge), its periodic configurations are dense in the space of all configurations, according to the Cantor topology on the space of configurations (there is a periodic configuration with any finite pattern of cells), and it is mixing (for any two finite patterns of cells, there is a configuration containing one pattern that eventually leads to a configuration containing the other pattern). According to Knudson's criteria, it displays sensitive dependence and there is a dense orbit (an initial configuration that eventually displays any finite pattern of cells). Both of these characterizations of the rule's chaotic behavior follow from a simpler and easy to verify property of Rule 30: it is left permutative, meaning that if two configurations and differ in the state of a single cell at position , then after a single step the new configurations will differ at cell . Applications Random number generation As is apparent from the image above, Rule 30 generates seeming randomness despite the lack of anything that could reasonably be considered random input. Stephen Wolfram proposed using its center column as a pseudorandom number generator (PRNG); it passes many standard tests for randomness, and Wolfram previously used this rule in the Mathematica product for creating random integers. Sipper and Tomassini have shown that as a random number generator Rule 30 exhibits poor behavior on a chi squared test when applied to all the rule columns as compared to other cellular automaton-based generators. The authors also expressed their concern that "The relatively low results obtained by the rule 30 CA may be due to the fact that we considered N random sequences generated in parallel, rather than the single one considered by Wolfram." Decoration The Cambridge North railway station is decorated with architectural panels displaying the evolution of Rule 30 (or equivalently under black-white reversal, Rule 135). The design was described by its architect as inspired by Conway's Game of Life, a different cellular automaton studied by Cambridge mathematician John Horton Conway, but is not actually based on Life. Programming The state update can be done quickly by bitwise operations, if the cell values are