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(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 .)
The function Nand[a_, b_] := Not[And[a, b]] used in the main text turns out to be universal for any k .
The default form of evaluation for recursive functions implemented by all standard computer languages (including Mathematica) is the so-called leftmost innermost scheme, which attempts to find explicit values for each f[k] that occurs first, and will therefore never notice if f[k] in fact occurs only in the combination f[k] - f[k] .
If the function f depends explicitly on time, then two equations suffice.
Except when e = 0 , the equation has no solution in terms of standard mathematical functions.
But traditional mathematics, with its emphasis on reducing everything to a small number of continuous numerical functions, has rather little to offer along these lines.
And if one computes even a function like 1/x almost any digit in x will typically have an effect on almost any digit in the result, as the pictures on the facing page indicate.
But as soon as one also puts in an abstract function or relation with more than one argument, one gets universality.
But usually these examples have involved rather sophisticated and obscure mathematical constructs—most often functions that are somehow set up to grow extremely rapidly.
And indeed my guess is that the essential features of all sorts of intricate structures that are seen in living systems can actually be reproduced with remarkably simple rules—making it for example possible to use technology to repair or replace a whole new range of functions of biological tissues and organs.