And from the limiting growth rate of this one can immediately get the Ricci scalar curvature—just as in the continuous case discussed above. To get the Ricci tensor one also needs a direction. But one can get this from a geodesic—which is in effect the analog of a straight line in the network.

The maps shown can be thought of as being made by taking an infinitely dense limit of the array of pictures on the facing page , but keeping only what one sees in each picture by looking through a peephole at a particular position relative to the original stem.

Note that if one just uses the original cellular automata rules, then with any nonzero probability of reversing the colors of cells, the patterns will be essentially destroyed. With more complicated cellular automaton rules, one can get behavior closer to the continuous cellular automata shown here.

To interpret or parse an expression in a context-free language one has to go backwards and find out which rules could be used to generate that expression. … = {}) &];]] ReverseRule[a_  b_, {i_}] := {___, {s[x___, b, y___], {u___}}, ___}  {s[x, a, y], {i, u}} /; FreeQ[s[x], s[a]] In general, there will in principle be more than one such list, and to pick the appropriate list in a practical situation one normally takes the rules of the language to apply with a certain precedence—which is how, for example, x + y z comes to be interpreted in Mathematica as Plus[x, Times[y, z]] rather than Times[Plus[x, y], z] . (Note that in practice the output from a parser for a context-free language is usually represented as a tree—as in Mathematica FullForm —with each node corresponding to one rule application.)

But if one applies multivariate elliptic or hypergeometric functions it was established in the late 1800s and early 1900s that one can in principle reach any algebraic number. One can also ask what numbers can be generated by integrals (or by solving differential equations). … One can also consider numbers obtained from infinite sums (or by solving recurrence equations).

Gradient descent [in constraint satisfaction] A standard method for finding a minimum in a smooth function f[x] is to use FixedPoint[# - a f'[#] &, x 0 ] If there are local minima, then which one is reached will depend on the starting point x 0 . It will not necessarily be the one closest to x 0 because of potentially complicated overshooting effects associated with the step size a .

And one might think that if the history of each universe corresponds to one path in a multiway system then the convergence of paths might represent interactions between universes.

And indeed, the pictures on the next two pages [ 202 , 203 ] show examples of what can happen if the rules are allowed to depend on the number of distinct nodes reached by following not just one but up to two successive connections from each node. With such rules, the sequence of networks obtained no longer needs to form any kind of simple progression, and indeed one finds that even the total number of nodes at each step can vary in a way that seems in many respects completely random.

But if one systematically examines possible constraints in the order shown on pages 214 and 215 , then it turns out that after examining more than 18 million of them, one finally discovers the system shown on the facing page .

At the outset, one might have thought that there would be just one definite mechanism for each type of simple behavior.