Perhaps the most peculiar element, however, is a diagram indicating the 21 cm transition in hydrogen—by showing two abstract quantum mechanical spin configurations represented in a way that seems completely specific to the particular details of human mathematics and physics.

Probably the simplest is a statement shown to be unprovable in Peano arithmetic by Laurence Kirby and Jeff Paris in 1982: that certain sequences g[n] defined by Reuben Goodstein in 1944 are of limited length for all n , where g[n_] := Map[First, NestWhileList[ {f[#] - 1, Last[#] + 1} &, {n, 3}, First[#] > 0 &]] f[{0, _}] = 0; f[{n_, k_}] := Apply[Plus, MapIndexed[#1 k^f[{#2 〚 1 〛 - 1, k}] &, Reverse[IntegerDigits[n, k - 1]]]] As in the pictures below, g[1] is {1, 0} , g[2] is {2, 2, 1, 0} and g[3] is {3, 3, 3, 2, 1, 0} . g[4] increases quadratically for a long time, with only element 3 × 2 402653211 - 2 finally being 0.

To get all the familiar properties of additivity one needs an addition operation that is associative ( Flat ) and commutative ( Orderless ), and has an identity element (white or 0 in the cases above)—so that it defines a commutative monoid.

(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.)

(An idealized soap film or other minimal surface extremizes the integral of the intrinsic volume element Sqrt[Det[g]] , without a RicciScalar factor.)

The MacArthur Fellowship that I received in May 1981 was an important element of personal support, and in fact it was a few months after this award that I made the decision to focus

Given n spins one can imagine using their 2 n possible configurations to represent each element of Range[m] .

If only one color of element ever appears this is the complete condition for a solution—and for r = 2 solutions exist if Apply[Times, d] < 0 and are then of length at least Apply[Plus[##]/GCD[##]&, Abs[d]] .

Typically the rules imagined for each element of such systems are however immensely more complicated than for any of the simple cellular automata I consider.