The Central Limit Theorem leads to a self-similarity property for the Gaussian distribution: if one takes n numbers that follow Gaussian distributions, then their average should also follow a Gaussian distribution, though with a standard deviation that is 1/ √ n times smaller.

Direction makes little sense in 1D, but is meaningful in 2D.

For a cylinder, there are difficulties with boundary conditions at infinity, but the drag coefficient was nevertheless calculated by William Oseen in 1915 to be 8 π /(R (1/2 + Log[8/R] - EulerGamma)) . … A simple model due to Theodore von Kármán from 1911 predicts a relative spacing of π /Log[1 + √ 2 ] between vortices, and bifurcation theory analyses have provided some justification for some such result. … The drag coefficient remains almost constant at a value around 1 until R ≃ 3 × 10 5 , at which point it drops precipitously for a while.

(Ulam tried to construct a 1D analog, but ended up not with a cellular automaton, but instead with the sequences based on numbers discussed on page 908 .) … Quite disconnected from all this, even in the 1950s, specific types of 2D and 1D cellular automata were already being used in various electronic devices and special-purpose computers. … These systems turn out to be essentially 1D additive cellular automata (like rule 90) with a limited number of cells (compare page 259 ).

(Changes expand about 1.24 cells per step in rule 30, and about 1.17 in rule 45.)

On the right-hand edge, the first few periods that are seen are {1, 2, 2, 4, 8, 8, 16, 32, 32, 64, 64, 64, 64, 64, 128, 256} and in general the period seems to increase exponentially with depth. … The first n elements can be found efficiently using Module[{a = 1}, Table[First[IntegerDigits[ a, a = BitXor[a, BitOr[2a, 4a]]; 2, i]], {i, n}]] The sequence does not repeat in at least its first million steps, and I would amazed if it ever repeats, but as of now I know of no rigorous proof of this. ( Erica Jen showed in 1986 that no pair of columns can ever repeat, and the arguments on page 1087 suggest that neither can the center column together with occasional neighboring cells.)

For class 1 and 2 cellular automata, there are typically only a limited number of possible sequences of any length allowed. … Class 1 has h x = 0 and h t = 0 .

In class 1, information about initial conditions is always rapidly forgotten—for whatever the initial conditions were, the system quickly evolves to a single final state that shows no trace of them.

The first picture below shows an extreme example of a class 1 cellular automaton in which after just one step the only sequences that can occur are those that contain only black cells.

In the limit of large r , this number is approximately r 2 (1- k r 2 + …) where k turns out to be exactly proportional to the curvature.