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Note (b) for Intelligence in the Universe

The local positions of stars are generally assumed to be random. 88 constellations are usually named—quite a few presumably already identified by the Babylonians and Sumerians around 2000 BC.

Note (a) for Undecidability and Intractability

Density of difficult problems There are arguments that in an asymptotic sense most instances chosen at random of problems like limited-size PCP or tiling will be difficult to solve.

Note (g) for Evolution of Networks

[Networks generated by] random replacements As indicated in the note above, applying the second rule (T1, shown as (b) on page 511 ) at an appropriate sequence of positions can transform one planar network into any other with the same number of nodes. The pictures below show what happens if this rule is repeatedly applied at random positions in a network.

Note (e) for Systems Based on Constraints

For 3×3 templates with two colors extensive randomized searches have failed to discover examples where non-repetitive patterns are forced to occur.

Note (d) for Irreversibility and the Second Law of Thermodynamics

With completely random initial conditions, there are great fluctuations, but a typical period is around 2 n/3 .

Note (e) for Auditory Perception

With this setup my experience is that both repetitive and random sequences tend to seem quite monotonous and dull.

Note (a) for Recursive Sequences

At least up to n = 10 6 terms, it increases roughly like 13.5n , but as shown below the fluctuations seem random.

Note (b) for Origins of Discreteness

Percolation A simple example of a phase transition studied extensively since the 1950s involves taking a square lattice and filling in at random a certain density of black cells.

Note (b) for Fluid Flow

The result of this is that in most situations patterns of shocks generated have a fairly simple geometrical structure, with none of the randomness of turbulence.

Note (f) for The Notion of Reversibility

If one starts with completely random initial conditions then class 2 and class 3 behavior are often seen.