In the bottom picture each line represents one complete step of evolution, including one update of each cell. … Somewhat different results are typically obtained if one allows different updating orders. … Sequential cellular automata have a similar relationship to ordinary cellular automata as implicit updating schemes in finite difference methods have to explicit ones, or as infinite impulse response digital filters have to finite ones.

The cellular automaton rule used is a 4-color totalistic one with code 1004600.

Operator systems One can generalize symbolic systems by having rules that define transformations for any Mathematica pattern. Often these can be thought of as one-way versions of axioms for operator systems (see page 1172 ), but applied only once per step (as /. does), rather than in all possible ways (as in a multiway system)—so that the evolution is just given by NestList[#/.rule &, init, t] .

Spherical networks One can construct networks of constant positive curvature by approximating the surface of a sphere—starting with a dodecahedron and adding hexagons. … (Note that in C 60 one of the connections at each node is always a double chemical bond, since carbon has valence 4.)

And if one is looking at objects like letters, it helps that one has learned them at many different sizes. … With good lighting and good eyesight the textures in the picture can still be distinguished at a distance of 5 feet, where each square covers only one cell.

The first point to make is that the issues I am studying are rather different from the ones that are traditionally studied in the mathematics of these systems. … In particular, it is usually assumed that performing some standard mathematical operation, such as taking a square root, cannot have a significant effect on the system one is studying. … Indeed, we will see later in this book that large classes of digit sequences can be considered equivalent with respect to computational operations, but these classes are quite different ones from those that are considered equivalent with respect to mathematical operations.

Models of microscopic fracture Two kinds of models have traditionally been studied: ones based on looking at arrays of atoms, and ones based on continuum descriptions of materials. … A slightly better model, related to the one in the main text, is that the bonds between atoms are identical, and act like springs which break when they are stretched too far.

Flatten[IntegerDigits[ Append[2 - With[{w = Floor[Log[3, 2n]]}, IntegerDigits[n - (3 w + 1 - 1)/2, 3, w]], 3], 2, 2]] (e) Fibonacci encoding. Instead of decomposing a number into a sum of powers of an integer base, one decomposes it into a sum of Fibonacci numbers (see page 902 ). This decomposition becomes unique when one requires that no pair of 1's appear together.

Numerical analysis [of PDEs] To find numerical solutions to PDEs on a digital computer one has no choice but to make approximations. … The pictures below show as one example what happens with the diffusion equation when the cells have size dt in time and dx in space. … For equations where one can come at least close to having explicit algebraic formulas for solutions, it has often been possible to prove that a certain discretization procedure will yield correct results.

One reason is that which sperm fertilizes a given egg is determined by random details of sperm and fluid motion. … In most cells, say in humans, there are two versions of all 23 chromosomes—one from the father and one from the mother. But when meiosis forms egg and sperm cells they get only one version of each.