And the fact that the underlying rules can be so simple vastly expands the kinds of components that can realistically be used to implement them.

The positions of the black cells are given by (and this establishes the connection with the picture on page 117 ) Fold[Flatten[{#1 - #2, #1 + #2}] &, {0}, 2^DigitPositions[t]] DigitPositions[n_] := Flatten[Position[Reverse[IntegerDigits[n, 2]], 1]] - 1 The actual pattern generated by rule 90 corresponds to the coefficients in PolynomialMod[Expand[(1/x + x) t ], 2] (see page 1091 ); the color of a particular cell is thus given by Mod[Binomial[t, (n + t)/2], 2] /; EvenQ[n + t] .

For any additive or partially additive class 3 cellular automaton (such as rule 90 or rule 30) any change in initial conditions will always lead to expanding differences.

(Changes expand about 1.24 cells per step in rule 30, and about 1.17 in rule 45.)

In most cases, however, introducing these kinds of slightly more complicated encodings does not fundamentally seem to expand the set of rules that a given rule can emulate.

And as known since the 1960s, the same is true for expanding universes.

First, the definitions at the top of page 774 must be used to expand out various pieces of notation.

And with this assumption it is inevitable that if the universe in a sense expands at the speed of light, then regions on opposite sides of it can essentially never share any common history.

(When matter or a cosmological term is present one gets different solutions—that always expand or contract, and are much studied in cosmology.)

From this representation of Power the universal equation can be converted to a purely polynomial equation with 2154 variables—which when expanded has 1683150 terms, total degree 16 (average per term 6.8), maximum coefficient 17827424 and LeafCount 16540206.