With appropriate values of parameters, these tests in practice tend to be at least somewhat independent, although in principle, if sufficient data were available, they could all be subsumed into basic block frequency and run-length tests. Of the sequences on page 594 , (a) through (d) as well as (f) fail every single one of the tests, (e) fails only the serial test, while (g) and (h) pass all the tests. (Failure is defined as a value that is as large or small as that obtained from the data occurring below a specified probability in the set of all possible sequences.)

Real textures The textures I consider in the main text are all based on arrays of discrete black and white squares.

The picture below shows in sorted order the configurations obtained at each successive step in the evolution of all 256 elementary cellular automata starting from a single black cell.

Note that the reason that the six arms of a single snowflake usually look the same is that all of them have grown in essentially the same environment.

Einstein equations In the absence of matter, the standard statement of the Einstein equations is that all components of the Ricci tensor—and thus also the Ricci scalar—must be zero (or formally that R ij = 0 ). But since the vanishing of all components of a tensor must be independent of the coordinates used, it follows that the vacuum Einstein equations are equivalent to the statement RicciTensor . e . e  0 for all timelike unit vectors e —a statement that can readily be applied to networks of the kind I consider in the main text. (A related statement is that the 3D Ricci scalar curvature of all spacelike hypersurfaces must vanish wherever these have vanishing extrinsic curvature.)

And so far as one can tell, almost all these kinds of numbers also have apparently random digit sequences.

Indeed, just like patterns produced by one-dimensional substitution systems on page 83 , all the patterns shown here ultimately have a simple nested structure.

But from what we have seen in this section such behavior appears to be quite rare: unlike many of the simple rules that we have discussed in this book, it seems that almost all simple constraints lead only to fairly simple patterns.

In a class 2 system with random initial conditions, a similar thing happens: since different parts of the system do not communicate with each other, they all behave like separate patterns of limited size.

As we will see on page 338 the presence of such patterns is particularly clear when there are equal numbers of black and white cells in the initial conditions—but how these cells are arranged does not usually matter much at all.