For almost all of the systems discussed I have personally collected extensive data and samples, often over the course of many years, and sometimes in quite unlikely and amusing circumstances.

Limiting procedures [for cellular automata] Several different limiting procedures all appear to yield the same continuum behavior for the cellular automata shown here.

Consequences of an ultimate model [of physics] Even if one knows an ultimate model for the universe, there will inevitably be irreducible difficulty in working out all its consequences.

If anisotropy is present, however, then there can be all sorts of solutions—classified for example as having different Bianchi symmetry types. … But in all cases the structure is too simple to capture much that seems relevant for our present universe. … But in all cases they were found to be unstable—decaying into ordinary gravitational waves.

At each level there are many problems known that are complete at that level in the sense that all other problems at that level can be translated to instances of that problem using only computations at a lower level. (Thus, for example, all problems in NP can be translated to instances of any given NP-complete problem using computations in P.)

The main such systems and their dates of earliest known reasonably formalized use have been (see also page 901 ): positive integers (before 10,000 BC), rationals (3000 BC), square roots (2000 BC), other roots (1800 BC), all integers (600 AD, 1600s), decimals (950 AD), complex numbers (1500s, 1800s), polynomials (1591), infinitesimals (1635), algebraic numbers (1744), quaternions (1843), Grassmann algebra (1844), ideals (1844, 1871), octonions (~1845), Boolean algebra (1847), fields (1850s, 1871), matrices (1858), associative algebras (1870), axiomatic real numbers (1872), vectors (1881), transfinite ordinals (1883), transfinite cardinals (1883), operator calculus (1880s), Boolean algebras (1890), algebraic number fields (1893), rings (1897), p -adic numbers (1897), non-Archimedean fields (1899), q -numbers (1926), non-standard integers (1930s), non-standard reals (hyperreals) (1960), interval arithmetic (1968), fuzzy arithmetic (1970s), surreal numbers (1970s). … But in almost all cases the systems are set up so as to preserve as many theorems as possible—a notion that was for example made explicit in the Principle of Permanence discussed by George Peacock in 1830 and extended by Hermann Hankel in 1869.

Properties [of example symbolic system] All initial conditions eventually evolve to expressions of the form Nest[ ℯ , ℯ , m] , which then remain fixed. … For all initial conditions this depth seems at first to increase linearly, then to decrease in a nested way according to FoldList[Plus, 0, Flatten[Table[ {1, 1, Table[-1, {IntegerExponent[i, 2] + 1}]}, {i, m}]]] This quantity alternates between value 1 at position 2 j and value j at position 2 j - j + 1 .

Other so-called Fourier series in which the coefficient of Sin[m x] is a smooth function of m for all integer m yield similarly simple results. The pictures below show Sum[Sin[n 2 x]/n 2 , {n, k}] , where in effect all coefficients of Sin[m x] other than those where m is a perfect square are set to zero.

If all possible sequences of colors were allowed, then there would be k possibilities for what block could follow a given block, given by Map[Rest, Table[Append[list, i], {i, 0, k - 1}]] . … Given the network for a particular n , it is straightforward to see what happens when only certain length n blocks are allowed: one just keeps the arcs in the network that correspond to allowed blocks, and drops all other ones.

For k = 2 , r = 1 , it then turns out that all the reversible rules and their inverses have s = 1 . … For k = 3 , r = 1 , there are a total of 936 rules with this property: 576, 216 and 144 with s = 4 , 5 and 6 , and in all cases s = 3 .