The Mandelbrot set The pictures below show Julia sets produced by the procedure of taking the transformation z  {Sqrt[z - c], -Sqrt[z - c]} discussed above and iterating it starting at z = 0 for an array of values of c in the complex plane.

The number with run lengths corresponding to successive integers (so that the n th digit is Mod[Floor[1/2 + Sqrt[2n]], 2] ) turns out to be (1 - 2 1/4 EllipticTheta[2, 0, 1/2] + EllipticTheta[3, 0, 1/2])/2 , and appears at least not to be algebraic.

In 1D and 3D, the value at the origin quickly becomes exactly 0; in 2D it is given by 1-t/Sqrt[t 2 -1] , which tends to zero only like -1/(2t 2 ) (which means that a sound pulse cannot propagate in a normal way in 2D).

The pictures show all the distinct maximal cases that exist for a 7 ×7 grid, corresponding to possible circles with diameters Sqrt[m 2 +n 2 ] .

One can start from the fact that the volume density in any space is given in terms of the metric by Sqrt[Det[g]] . … Inverse[#2], RotateLeft[ Range[TensorRank[t]]]] &, t, Reverse[gl]] Laplacian[f_] := Inner[D, Sqrt[Det[g]] (Inverse[g] . Map[ ∂ # f &, p]), p]/Sqrt[Det[g]] In general the series in r may not converge, but it is known that at least in most cases only flat space can give a result that shows no correction to the basic r d form.

.}  Length[k]]]] There are a total of 2 m Fibonacci[m+2] black cells in the pattern obtained up to step 2 m , implying fractal dimension Log[2, 1 + Sqrt[5]] .

Central Limit Theorem Averages of large collections of random numbers tend to follow a Gaussian or normal distribution in which the probability of getting value x is Exp[-(x - μ ) 2 /(2 σ 2 )] / (Sqrt[2 π ] σ ) The mean μ and standard deviation σ are determined by properties of the random numbers, but the form of the distribution is always the same.

The formulas for local curvature as a function of arc length for each set of pictures are as follows: 1 (circle); s (Cornu spiral or clothoid); s 2 ; 1/Sqrt[s] (involute of circle); 1/s (logarithmic or equiangular spiral); 1/s 2 ; Exp[-s 2 ] ; Sin[s] ; s Sin[s] .

The amount of this time dilation is given by the classic relativistic formula 1/Sqrt[1-v 2 /c 2 ] , where v/c is the ratio of the speed of the clock to the speed of light.

In the first case shown, the total number of elements obtained doubles at every step; in the second case, it follows a Fibonacci sequence, and increases by a factor of roughly (1+Sqrt[5])/2 ≃ 1.618 at every step.