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So is it in fact possible to get formulas for the colors of squares that involve only such functions?
… The succession of polynomials above can be obtained by expanding the generating functions 1/(1 - (1 + x) y) and 1/(1 - (1 + x + x 2 ) y) . … GegenbauerC is a so-called orthogonal polynomial—a higher mathematical function.

(Thus it is impossible with ruler and compass to construct π and "square the circle" but it is possible to construct 17-gons or other n -gons for which FunctionExpand[Sin[ π /n]] contains only Plus , Times and Sqrt .)

Like ordinary algebraic functions, Boolean functions can also be represented by a variety of kinds of formulas. … In general any given function will allow many DNF representations; minimal ones can be found as described below. Writing a Boolean function in DNF is the rough analog of applying Expand to a polynomial.

Locally isotropic growth
A convenient way to see what happens if elements of a surface grow isotropically is to divide the surface into a collection of very small circles, and then to expand the circle at each point by a factor h[x, y] . … The pictures below show results for several growth rate functions; in the last case, the function is not harmonic, and the surface cannot be drawn in the plane without tearing. … Harmonic growth rate functions can potentially be obtained from the large-time effects of a chemical subject to diffusion.

With material where parts can locally expand, but cannot change their shape, page 1007 showed that a 2D surface will remain flat if the growth rate is a harmonic function. The Riemann mapping theorem of complex analysis then implies that even in this case, any smooth initial shape can grow into any other such shape with a suitable growth rate function.

With evolution functions f i and f j the requirement for the emulation to work is
f j [a j ] InverseFunction[ ϕ ][f i [ ϕ [a j ]]]
In the main text the encoding function is taken to have the form Flatten[a /. rules] —where rules are say {1 {1, 1}, 0 {0, 0}} —with the result that the decoding function for emulations that work is Partition[ ã , b] /. … 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. … Various questions about encoding functions ϕ have been studied over the past several decades in coding theory.

In general the pattern of probabilities for changes can be thought of as being somewhat like a Green's function in mathematical physics—though the nonadditivity of most cellular automata makes this analogy less useful. … 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.

First, the definitions at the top of page 774 must be used to expand out various pieces of notation. … Note, however, that in predicate logic the expressions that appear on each side of any rule are required to be so-called well-formed formulas (WFFs) consisting of variables (such as a ) and constants (such as 0 or ∅ ) inside any number of layers of functions (such as + , · , or Δ ) inside a layer of predicates (such as or ∈ ) inside any number of layers of logical connectives (such as ∧ or ⇒ ) or quantifiers (such as ∀ or ∃ ). (This setup is reflected in the grammar of the Mathematica language, where the operator precedences for functions are higher than for predicates, which are in turn higher than for quantifiers and logical connectives—thus yielding for example few parentheses in the presentation of axiom systems here.)