Maximal block compression If one has data that consists of a long sequence of blocks, each of length b , and each independently chosen with probability p[i] to be of type i , then as argued by Claude Shannon in the late 1940s, it turns out that the minimum number of base 2 bits needed on average to represent each block in such a sequence is h = -Sum[p[i] Log[2, p[i]], {i, 2 b }] . … If only one block occurs with nonzero probability then h  0 . … With this assumption one then finds that maximal compression occurs if a block of probability p[i] is represented by a codeword of length -Log[2, p[i]] .

Repetition in continuous systems A standard approach to partial differential equations (PDEs) used for more than a century is so-called linear stability analysis, in which one assumes that small fluctuations around some kind of basic solution can be treated as a superposition of waves of the form Exp[  k x] Exp[  ω t] . … But for some k one often finds that ω has an imaginary part. … But particularly when the original PDE is nonlinear one often finds that Im[ ω ] < 0 for some range of k —implying an instability which causes modes with certain spatial wavelengths to grow.

But at intermediate times one will see all sorts of potentially dramatic gullies that reflect the pattern of drainage, and the formation of a whole tree of streams and rivers. … If one imagines a uniform slope with discrete streams of water going randomly in each direction at the top, and then merging whenever they meet, one immediately gets a simple tree structure a little like in the pictures at the top of page 359 .

Considering the amount of mathematical work that has been done on partial differential equations, one might have thought that a vast range of different equations would by now have been studied. But in fact almost all the work—at least in one dimension—has concentrated on just the three specific equations on the facing page , together with a few others that are essentially equivalent to them.

But one of the central discoveries of this book is that this is not in fact the case, and that at least if one thinks in terms of programs rather than traditional mathematical equations, then even models that are based on extremely simple underlying rules can yield behavior of great complexity.

Second, complex behavior inevitably involves many elaborate details, and since different ones of these details may happen to be the deciding factors in the fates of individual organisms, it becomes very difficult for natural selection to act in a consistent and definitive way. … Fourth, if random mutations can only, say, increase or decrease a length, then even if one mutation goes in the wrong direction, it is easy for another mutation to recover by going in the opposite direction.

One might at the outset have thought that leaves would get their shapes through some mechanism quite unrelated to other aspects of plant growth. … The traditional intuition of biology would suggest that whenever one sees complexity—say in the shape of a leaf—it must have been generated for some particular purpose by some sophisticated process of natural selection.

The vertical black line on the left-hand side of the page represents in effect the original stem at each step, and the pictures are arranged so that the one which appears at a given position on the page shows the pattern that is generated when the tip of the right-hand new stem goes to that position relative to the original stem shown on the left. … To get some idea of this one can construct a kind of limit of the array on the next page in which the total number of pictures is in effect infinite, but only a specific infinitesimal region of each picture is shown.

One example is the arrangement of sequences of plant organs or other elements around a stem. … And instead what I strongly suspect is that the patterns are just inevitable consequences of a rather simple process of growth not unlike one that was already discussed, at least in general terms, nearly a century ago.

Away from mollusc shells, coiled structures—like branched ones—are not especially common in animals. … But the presence of many different kinds of parts is in the end one of the most obvious features of many animals.