On the top right these forms are then ranked roughly from coarsest to finest. … If all the basic forms are included, then the original image is faithfully reproduced. … All the pictures on the facing page , however, were generated from the specific ordering of basic forms shown on the previous page .

Class 4 systems are then in the middle: for the activity that they show neither dies out completely, as in class 2, nor remains at the high level seen in class 3. … In some respects it is not surprising that among all possible cellular automata one can identify some that are effectively on the boundary between class 2 and class 3.

But at an overall level, all the patterns have exactly the same basic nested structure. … The first picture at the top of the facing page shows what happens if one starts with 1 and then successively multiplies by 2 at each step. … All these patterns ultimately have the same overall nested form.

The pictures at the top of the next page show a few more examples. And what we see is that in all cases there is obvious regularity in the patterns produced. … In all cases the overall patterns obtained can be seen to have a very regular nested form.

And sometimes particular ideas and results in this book will just benefit from the emphasis of having a whole paper or book or website devoted to them. … And indeed from almost every page of this book there are all sorts of new questions that emerge. … On the website associated with this book I plan to maintain a list of questions that I believe are of particular interest.

Yet if one sets up elements on a grid it is straightforward to allow the replacements for a given element to depend on its neighbors, as in the picture at the top of the next page . And if one does this, one immediately gets all sorts of fairly complicated patterns that are often not just purely nested—as illustrated in the pictures on the next page . … The details of each pattern are different, but in all cases the patterns have a nested overall structure.

. • (b) All strings of length n containing exactly one black cell are produced—after at most 2n - 1 steps. • (c) All strings containing even-length runs of white cells are produced. • (d) The set of strings produced is complicated. The last length 4 string produced is , after 16 steps; the last length 6 one is , after 26 steps. • (e) All strings that begin with a black element are produced. • (f) All strings that end with a white element but contain at least one black element, or consist of all white elements ending with black, are produced. … Those of length n appear after at most 3n - 3 steps. • (l) The same strings as in (k) are produced, taking now at most 2n + 1 steps. • (m) All strings beginning with a black element are produced, after at most 3n + 1 steps. • (n) The set of strings produced is complicated, and seems to include many but not all that do not end with . • (o) All strings that do not end in are produced. • (p) All strings are produced, except ones in which every element after the first is white.

The three networks at the top of the previous page were laid out precisely so as to make this the case respectively for one, two and three-dimensional space. … But now if one tried to lay out all these nodes in one dimension it is inevitable that the network would have to bulge out in order to fit in all the nodes. … For the other two networks at the top of the previous page similar arguments can be given.

All of these curves have fairly simple, essentially repetitive forms. … The top row shows three trigonometric functions. … In all cases the curves shown have fairly simple repetitive forms.

In many species the outer surface of the shell is covered by a kind of skin known as the periostracum, and in most cases this skin is opaque, thereby obscuring the patterns underneath until long after the animal has died.