This simple rule produces randomness through the mechanism of intrinsic randomness generation, and this randomness in turn leads to a pattern of growth that takes on an increasingly smooth more-or-less circular form.

And it takes only very simple rules to generate shapes that look like the villi and other corrugated structures one often sees in animals. … 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.

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. … The procedure takes the x and y coordinates of the square, and computes their remainders after division by 3 and 2 respectively.

In traditional logic there is always an operation of negation which takes any true statement, and makes it into a false one, and vice versa. … One can take negation to be the operation that interchanges black and white.

I had never expected it would take anything like as long, but I have discovered vastly more than I ever thought possible, and in fact what I have done now touches almost every existing area of science, and quite a bit besides. … For progress in science has mostly tended to take place

The rule takes the new gray level of each cell to be the fractional part of the average gray level of the cell and its neighbors multiplied by 3/2.

The rule used is of the same kind as on the previous page , but now takes the center cell to become black only if it has exactly 3 black neighbors.

But what happens if one takes for example a perfect single crystal—say a standard highly pure industrial silicon crystal—and breaks it? … Experience with systems from hand-held objects to engineering structures and earthquakes suggests that it can take a while for a crack to get started, but that once it does, the crack tends to move quickly and violently, usually producing a lot of noise in the process.

Work in the field of thermodynamics led to the idea that perhaps living systems could be defined by their ability to take disorganized material and spontaneously organize it—usually to incorporate it into their own structure. … The rule used is additive, and takes a cell to be black whenever an odd number of its neighbors were black on the step before (outer totalistic code 204).

Computational compression [and time] In the model for time described here, it is noteworthy that in a sense an arbitrary amount of underlying computation can take place between successive moments in perceived time.