Numbering scheme [for 2D constraints] The constraint numbered n allows the templates at Position[IntegerDigits[n, 2, 32], 1] in the list below.

The pictures below show as black squares all the quadratic residues for each successive m going down the page (the ordinary squares 1, 4, 9, 16, ... show up as vertical black stripes). If m is a prime p , then the simple tests JacobiSymbol[x, p]  1 (see page 1081 ) or Mod[x (p - 1)/2 , p]  1 determine whether x is a quadratic residue.

[History of] exact solutions Some notable cases where closed-form analytical results have been found in terms of standard mathematical functions include: quadratic equations (~2000 BC) ( Sqrt ); cubic, quartic equations (1530s) ( x 1/n ); 2-body problem (1687) ( Cos ); catenary (1690) ( Cosh ); brachistochrone (1696) ( Sin ); spinning top (1849; 1888; 1888) ( JacobiSN ; WeierstrassP ; hyperelliptic functions); quintic equations (1858) ( EllipticTheta ); half-plane diffraction (1896) ( FresnelC ); Mie scattering (1908) ( BesselJ , BesselY , LegendreP ); Einstein equations (Schwarzschild (1916), Reissner–Nordström (1916), Kerr (1963) solutions) (rational and trigonometric functions); quantum hydrogen atom and harmonic oscillator (1927) ( LaguerreL , HermiteH ); 2D Ising model (1944) ( Sinh , EllipticK ); various Feynman diagrams (1960s-1980s) ( PolyLog ); KdV equation (1967) ( Sech etc.); Toda lattice (1967) ( Sech ); six-vertex spin model (1967) ( Sinh integrals); Calogero–Moser model (1971) ( Hypergeometric1F1 ); Yang–Mills instantons (1975) (rational functions); hard-hexagon spin model (1979) ( EllipticTheta ); additive cellular automata (1984) ( MultiplicativeOrder ); Seiberg–Witten supersymmetric theory (1994) ( Hypergeometric2F1 ).

1D phenomena Among the phenomena that cannot occur in one dimension are those associated with shape, winding and knotting, as well as traditional phase transitions with reversible evolution rules (see page 981 ).

At least with the initial condition used here, despite considerable early apparent randomness, the differences in number of elements do repeat (shifted by 1) every 1071 steps.

In each case they start with the number 1, then successively multiply by the specified multiplier, keeping only the rightmost 31 digits in the base 2 representation of the number obtained at each step.

In 1D, a basic form a 〚 i 〛 is just a list. … Here a typical orthogonality property is Integrate[f[r, x] f[s, x], {x, 0, 1}]  KroneckerDelta[r, s] .

An example originally popular in the earth and environmental sciences is so-called mathematical morphology, based on "dilation" of data consisting of 0's and 1's with a "structuring element" σ according to Sign[ListConvolve[ σ , data, 1, 0]] (as well as the dual operation of "erosion").

Note that in cases like a = 0.475 there is some trace of a pattern at every step—but it only becomes obvious when it makes values wrap around from 1 to 0. The pictures below show successive colors of (a) the background (compare page 950 ) and (b) the center cell for each a = n/500 from 0 to 1 for the systems on page 159 .

Similar formulas in terms of n th roots have been known since the 1500s for equations with degrees n up to 4, although their LeafCount starting at n = 1 increases like 6, 25, 183, 718. … For degrees 5 and 6 it was shown in the late 1800s that EllipticTheta or Hypergeometric2F1 are sufficient, although for degrees 5 and 6 respectively the necessary formulas have a LeafCount in the billions.