And it takes only very simple rules to generate shapes that look like the villi and other corrugated structures one often sees in animals. … One is furrowing or tearing of tissue through a loss of adhesion between cells. … If one looks at the sequence of events that take place in a Curves obtained by varying the local curvature according to definite rules as one goes from one end to the other.

But it turns out that one can actually also recognize curvature in the basic structure of a network. … But what if one replaces some of these hexagons by pentagons? One still has a fundamentally two-dimensional surface.

But how can one be sure that there really is absolutely no easy way to do this? … But as a practical matter one can say that not only have direct attempts to find easy ways to deduce the key in rule 30 failed, but also—despite some considerable effort—little progress has been made in solving any of various problems that turn out to be equivalent to this one. … So in a simple case if one has an array of black and white squares, what one would typically look for is a formula that takes the numbers which specify the position of a particular square and from these tells one whether the square is black or white.

But one of the consequences of the Principle of Computational Equivalence is that it does not. … But as soon as one tries to deal with systems whose behavior is anything but fairly simple one finds that this idealization breaks down, and it becomes necessary to consider perception and analysis as explicit processes in their own right. If one studies systems in nature it is inevitable that both the evolution of the systems themselves and the methods of perception and

So whether or not one chooses to say that the sequence is truly random, it is, as far as one can tell, at least random for all practical purposes. … Another issue is that if one always ran the cellular automaton from page 315 with the particular initial condition shown there, then one would always get exactly the same sequence of 0's and 1's. But by using different initial conditions one can get completely different

Given two nodes in a network, one can always identify a shortest path from one to the other that goes along a sequence of individual connections in the network. … Examples include familiar ones like inverse-square gravitational fields around massive objects, as well as unfamiliar ones like gravitational waves. … So this means that to know what will happen even in phenomena primarily associated with gravity one typically has to know all sorts of properties of matter.

Yet quite how this might work will certainly depend on what feature of the network one tries to look at. It has always been a major issue in quantum theory just how one tells what is happening with a particular particle like an electron. … And in fact just one interaction will certainly not be enough.

what happens if one divides an image into a collection of nested squares, but imposes a lower limit on the size of these squares. … So can one do better at maintaining the quality of the image? … But if one drops some of the later forms—thereby reducing the number of weights that have to be specified—one gets only an approximation to the image.

So if one receives the encrypted message (c), how can one recover the original message (a)? If one knows the encrypting sequence (b) then it is straightforward. … So what kind of procedure might one use to get an encrypting sequence from a key?

And it does this by successively eliminating cases that do not apply, until eventually only one case remains. … But the process itself is not too difficult to understand, and indeed it works in much the way one might expect of a practical electronic logic circuit. … And after all three stripes have passed, only one of the 8 cases ever survives, and this case is then the one that gives the new color for the cell.