The three pictures on the facing page can always be described by explicitly giving a list of the colors of each of the 6561 cells that they contain.

And much as in the picture on page 817 , what one sees is that right at the beginning of the list there are several theorems that are identified as interesting.

Implementation [of proof example] Given the axioms in the form s[1] = (a_ ⊼ a_) ⊼ (a_ ⊼ b_)  a; s[2, x_] := b_  (b ⊼ b) ⊼ (b ⊼ x); s[3] = a_ ⊼ (a_ ⊼ b_)  a ⊼ (b ⊼ b); s[4] = a_ ⊼ (b_ ⊼ b_)  a ⊼ (a ⊼ b); s[5] = a_ ⊼ (a_ ⊼ (b_ ⊼ c_))  b ⊼ (b ⊼ (a ⊼ c)); the proof shown here can be represented by {{s[2, b], {2}}, {s[4], {}}, {s[2, (b ⊼ b) ⊼ ((a ⊼ a) ⊼ (b ⊼ b))], {2, 2}}, {s[1], {2, 2, 1}}, {s[2, b ⊼ b], {2, 2, 2, 2, 2, 2}], {s[5], {2, 2, 2}}, {s[2, b ⊼ b], {2, 2, 2, 2, 2, 1}}, {s[1], {2, 2, 2, 2, 2}}, {s[3], {2, 2, 2}}, {s[1], {2, 2, 2, 2}}, {s[4], {2, 2, 2}}, {s[5], {}}, {s[2, a], {2, 2, 1}}, {s[1], {2, 2}}, {s[3], {}}, {s[1], {2}}} and applied using FoldList[Function[{u, v}, MapAt[Replace[#, v 〚 1 〛 ] &, u, {v 〚 2 〛 }]], a ⊼ b, proof]

If the rules for a one-element-dependence tag system are given in the form {2, {{0, 1}, {0, 1, 1}}} (compare page 1114 ), the initial conditions for the Turing machine are TagToMTM[{2, rule_}, init_] := With[{b = FoldList[Plus, 1, Map[Length, rule] + 1]}, Drop[Flatten[{Reverse[Flatten[{1, Map[{Map[ {1, 0, Table[0, {b 〚 # + 1 〛 }]} &, #], 1} &, rule], 1}]], 0, 0, Map[{Table[2, {b 〚 # + 1 〛 }], 3} &, init]}], -1]] surrounded by 0 's, with the head on the leftmost 2 , in state 1 .

Linear and nonlinear systems A vast number of different applications of traditional mathematics are ultimately based on linear equations of the form u  m . v where u and v are vectors (lists) and m is a matrix (list of lists), all containing ordinary continuous numbers.

So I started a research center and a journal, published a list of problems to attack, and worked hard to communicate the importance of the direction I was defining.

In every block of 20 cells in the universal cellular automaton, these rules are encoded in a very straightforward way, by listing in order the outcomes for each of the 8 possible cases.

One can think of an axiom system—say one of those listed on pages 773 and 774 —as giving a set of constraints that any object it describes must satisfy.

(Z transform or generating function methods can be applied directly only for substitution systems with rules such as {1  list, 0  1 - list} .)

(An example is NestList[Mod[2 #, 1]&, N[ π /4, 40], 200] ; Map[Precision, list] gives the number of significant digits of each element in the list.)