The digits that lie directly below and to the left of the original 1 at the top of the pattern correspond to the whole number part of each successive number (e.g. 3 in 3.375), while the digits that lie to the right correspond to the fractional part (e.g. 0.375 in 3.375). And instead of looking explicitly at the complete pattern of digits, one can consider just finding the size of the fractional part of each successive number.

So as an example the picture below shows a rule that determines the new gray level for a cell by just adding the constant 1/4 to the average gray level for the cell and its immediate neighbors, and then taking the fractional part of the result. … A continuous cellular automaton whose rule adds the constant 1/4 to the average gray level for a cell and its immediate neighbors, and takes the fractional part of the result.

Exact iterates [in iterated maps] For any integer a the n th iterate of x  FractionalPart[a x] can be written as FractionalPart[a n x] , or equivalently 1/2 - ArcTan[Cot[a n π x]]/ π .

And when the material is cut and stacked, the effect on the number is then to extract its fractional part. … In general, a point at position x on a particular step will move to position FractionalPart[2x] on the next step.

As an example, the n th digit of Log[2] in base 2 is formally given by Round[FractionalPart[2 n Sum[2 -k /k, {k, ∞ }]]] . And in practice the n th digit can be found just by computing slightly over n terms of the sum, according to Round[FractionalPart[ Sum[FractionalPart[PowerMod[2, n - k, k]/k], {k, n}] + Sum[2 n - k /k, {k, n + 1, n + d}]]] where several values of d can be tried to check that the result does not change.

As discussed on page 155 , each cell here can have any gray level between 0 and 1, and at each step the gray level of a given cell is determined by averaging the gray levels of the cell and its two neighbors, adding the specified constant, and then keeping only the fractional part of the result.

The rule takes the new gray level of each cell to be the fractional part of the average gray level of the cell and its neighbors multiplied by 3/2.

This list can then be updated using CCAEvolveStep[f_, list_List] := Map[f, (RotateLeft[list] + list + RotateRight[list])/3] CCAEvolveList[f_, init_List, t_Integer] := NestList[CCAEvolveStep[f, #] &, init, t] where for the rule on page 157 f is FractionalPart[3#/2] & while for the rule on page 158 it is FractionalPart[# + 1/4] & .

In connection with his study of continued fractions Carl Friedrich Gauss noted in 1799 complexity in the behavior of the iterated map x  FractionalPart[1/x] . Beginning in the late 1800s there was number theoretical investigation of the sequence FractionalPart[a n x] associated with the map x  FractionalPart[a x] (see page 903 ), notably by G.

This will happen whenever FractionalPart[Log[b, a[n]]] is uniformly distributed, which, as discussed on page 903 , is known to be true for sequences such as r n (with Log[b, r] irrational), n n , n!