Basic theory [of cryptography] As was recognized in the 1920s the only way to make a completely secure cryptographic system is to use a so-called one-time pad and to have a key that is as long as the message, and is chosen completely at random separately for each message.

Random searches suggest that among s = 3 , k = 3 Turing machines roughly one in 25 million reproduce rule 60 in the same way as the machines discussed here.

Random packings of spheres typically have densities around 0.64 (compared to 0.74 for fcc).

Consider generating a long piece of text by picking at random from k letters and a space.

Finding layouts [for networks] One way to lay out a network g so that network distances in it come as close as possible to ordinary distances in d -dimensional space, is just to search for values of the x[i, k] which minimize a quantity such as With[{n = Length[g]}, Apply[Plus, Flatten[(Table[Distance[g, {i, j}], {i, n}, {j, n}] 2 - Table[ Sum[(x[i, k] - x[j, k]) 2 , {k, d}], {i, n}, {j, n}]) 2 ]]] using for example FindMinimum starting say with x[1, _]  0 and all the other x[_, _]  Random[] .

I had bought a copy of the physics textbook below, and had become very curious about the process of randomization illustrated on its cover.

Two sections from now I will discuss a rather striking potential example of this: if one looks at molluscs of various types, then it turns out that the range of pigmentation patterns on their shells corresponds remarkably closely with the range of patterns that are produced by simple randomly chosen programs based on cellular automata.

And if this in fact happens, then at least eventually the system will inevitably spend most of its time in states that seem quite random.

In the early parts of Chapter 7 we discussed various possible origins for the apparent randomness that we see in many natural systems.

But in fact—just as in so many other seemingly irreversible processes—all that is needed to preserve reversibility is that if one looks at sufficient details of the system there can be arbitrary and seemingly random behavior.