Properties [of addition continuous cellular automata] At step t the background is FractionalPart[a t] .

If the temperature of any cell exceeds 1, then only the fractional part is kept, as in the systems on page 158 , representing the consumption of latent heat in the boiling process.

Given a sequence of length n , an approximation to h can be reconstructed using Max[MapIndexed[#1/First[#2] &, FoldList[Plus, First[list], Rest[list]]]] The fractional part of the result obtained is always an element of the Farey sequence Union[Flatten[Table[a/b, {b, n}, {a, 0, b}]]] (See also pages 892 , 932 and 1084 .)

The picture on the facing page shows what happens with a slightly more complicated rule in which the average gray level is multiplied by 3/2 , and then only the fractional part is kept if the result of this is greater than 1.

It is then possible to find special values of u (an example is 0.166669170371...) which make the first digit in the fractional part of u (3/2) n always nonzero, so that Mod[u (3/2) n , 1] > 1/6 .

In the special case a = 4 , it turns out that replacing x by Sin[ π u] 2 makes the mapping become just u  FractionalPart[2 u] , revealing simple shift map dependence on the initial digit sequence.

Thus for example if x is an integer with n digits then evaluating Sin[x] or FractionalPart[x c] requires respectively finding π or c to n -digit precision.

In 1962, however, Edward Lorenz did a computer simulation of a set of simplified differential equations for fluid convection (see page 998 ) in which he saw complicated behavior that seemed to depend sensitively on initial conditions—in a way that he suggested was like the map x  FractionalPart[2x] .