In the second case, there is random input only in the initial conditions.

And what we found there was that it is crucial to think not in terms of the sizes of the numbers x , but rather in terms of their digit sequences represented in base 2. … Indeed in many ways the only real difference is that instead of The digit sequences of positions of points on successive steps in the two examples of kneading processes at the bottom of the previous page . … In general, a point at position x on a particular step will move to position FractionalPart[2x] on the next step.

And in general, in any system with definite rules that only ever visits a limited number of states, it is With each cell at each step having a gray level that is the average of its predecessor and its two neighbors the total amount of black is conserved, but eventually becomes spread uniformly throughout the system.

It is usually only at the tip of a stem that growth can occur, and much of the time all that ever happens is that the stem just gets progressively longer.

One of the general features of plants is that most of their cells tend to develop fairly rigid cellulose walls which make it essentially impossible for new material to be added inside the volume of the plant, and so typically force new growth to occur only on the outside of the plant—most importantly at the tips of stems.

So this means that given a sufficiently precise knowledge of the state of a physical system at the present time, it is therefore possible to deduce not only what the system will do in the future, but also what it did in the past.

One feature of causal networks is that they tell one not only what the consequences of a particular event will be, but also in a sense what its causes were.

And in almost all of these, there is no overlap not only within a single cluster, but also between different clusters.

(b) is ordinary binary or base 2 representation. … (e) uses a non-integer base derived from the Fibonacci sequence, with the property that a pair of black cells can appear only at the end of each number.

But for run-length encoding it turns out that ordinary base 2 digit sequences do not quite work. … A variant on this approach, illustrated in picture (e), uses a non-integer base in which pairs of black cells can occur only at the end of a number.