The same is true of fields of parallel sand dunes, as well as of almost-circular structures such as the 40-mile-diameter impact crater in Manicouagan, Canada (highlighted by an annular lake) and the 30-mile-diameter Richat structure in the Sahara desert of Mauritania.

But unlike for integers the same turns out to be true even for infinite lists of real numbers.

• Is there an assignment of truth values to variables that makes a given Boolean expression true?

Real algebra is also not universal (see page 1153 ), and the same is for example true for finite fields—but not for arbitrary fields.

If one starts from more than a single non- 0 element, then it is still true that a nested pattern will be produced if f is both associative and commutative.

It is also possible to represent values of cells as True and False .

For odd n this is inevitably true for any lattice with mirror symmetry.

And as known since the 1960s, the same is true for expanding universes.

In the specific case a = 4 , however, it turns out that by allowing more sophisticated mathematical functions one can get a complete formula: the result after any number of steps t can be written in any of the forms Sin[2 t ArcSin[ √ x ]] 2 (1 - Cos[2 t ArcCos[1 - 2 x]])/2 (1 - ChebyshevT[2 t , 1 - 2x])/2 where these follow from functional relations such as Sin[2x] 2  4 Sin[x] 2 (1 - Sin[x] 2 ) ChebyshevT[m n, x]  ChebyshevT[m, ChebyshevT[n, x]] For a = 2 it also turns out that there is a complete formula: (1 - (1 - 2 x) 2 t )/2 And the same is true for a = -2 : 1/2 - Cos[(1/3) ( π - (-2) t ( π - 3 ArcCos[1/2 - x]))] In all these examples t enters essentially only in a t .

The magnitude of this gap turns out to be given by With[{d = Conjugate[c], r = 1 - Abs[c] 2 }, Which[Im[c] < 0, d, Im[c]  0, 0, Re[c] > 0, With[{n = Ceiling[ π /2/Arg[c]]}, Im[c(1 - d n )/(1 - d)] + Im[c d n (1 + d)]/r], Arg[c] > 3 π /4, Im[c + c 2 ]/r, True, Im[c] + Im[c 2 + c 3 ]/r]] The picture below shows the region for which the gap is positive, corresponding to trees which are not connected.