to search through these rules, trying each one in turn, and looking to see if it produces the behavior one wants. … Surely we cannot simply search through possible rules of certain kinds, looking for one whose behavior happens to fit what we see in physics?

The picture below shows an example of how successive steps in a particular evolution history can be recovered from a particular set of slices through the causal network derived from it. … An example of how the succession of states in an evolution history can be recovered by taking appropriate slices through a causal network.

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.

From its inception in classical times, through its development in the 1600s to 1800s, number theory was largely separate from other fields of mathematics. … And through these connections, sophisticated proofs of such results as Fermat's Last Theorem—open for 350 years—have been constructed. Long considered a rather esoteric branch of mathematics, number theory has in recent years grown in practical importance through its use in areas such as coding theory, cryptography and statistical mechanics.

Code 942 slices The following is the result of taking vertical slices through the pattern with a sequence of offsets from the center:

But it turns out that all that really seems to matter is that randomness is present: the mechanism through which it arises appears to be largely irrelevant. … But through the mechanism of intrinsic randomness generation, the behavior of the system exhibits considerable randomness.

Shown on a logarithmic scale, representations (b) through (e) (given here for numbers 1 through 500) all grow roughly linearly:

But superimposed on this effectively random background will then presumably also be some definite structures that persist through many updatings of the network. … As I discussed in the last two sections [ 14 , 15 ], causal invariance of the underlying rules implies that such structures should be able to move at a range of uniform speeds through the background.

Website A large amount of additional material related to this book and these notes will progressively be made available through the website www.wolframscience.com .

If a large amount of numerical data has been made up by a person this can be detectable through statistical deviations from expected randomness—particularly in structural details such as frequency of digits. … Such effects are usually first noticed through unexpected regularities in some detail of output.