In each case the n th element appears at coordinates Sqrt[n] {Cos[n θ],Sin[n θ]} .

The pictures on the right below show Sin[1/2 π a[t, n]] 2 for these functions (equivalent to Mod[a[t, n], 2] for integer a[t, n] ).

As discovered by Joseph Fourier around 1810, this is satisfied for basis functions such as Sin[2 n π x]/ √ 2 .

So in the case of quantum mechanics one can consider having each new block be given by {{Cos[ θ ],  Sin[ θ ]}, {  Sin[ θ ], Cos[ θ ]}} .

The whole procedure can be represented using a mathematical formula that involves either functions like Mod or more traditional functions like Sin .

(Thus it is impossible with ruler and compass to construct π and "square the circle" but it is possible to construct 17-gons or other n -gons for which FunctionExpand[Sin[ π /n]] contains only Plus , Times and Sqrt .)

Simple case [of three-body problem] The position of the idealized planet in the case shown satisfies the differential equation δ tt z[t]  -z[t]/(z[t] 2 + (1/2 (1 + e Sin[2 π t] ) 2 ) 3/2 where e is the eccentricity of the elliptical orbit of the stars ( e = 0.1 in the picture).

This can be determined either from Mod[a, 2] or equivalently from (1 - (-1) a )/2 or Sin[ π /2 a] 2 .

The arrangement of triangles at step t can be obtained from a substitution system according to With[{ ϕ = GoldenRatio}, Nest[# /. a[p_, q_, r_]  With[{s = (p + ϕ q) (2 - ϕ )}, {a[r, s, q], b[r, s, p]}] /. b[p_, q_, r_]  With[{s = (p + ϕ r) (2 - ϕ )}, {a[p, q, s], b[ r, s, q]}] &, a[{1/2, Sin[2 π /5] ϕ }, {1, 0}, {0, 0}], t]] This pattern can be viewed as generalizations of the pattern generated by the 1D Fibonacci substitution system (c) on page 83 .

Such a circle has area 2 π a 2 (1 - Cos[r/a]) = π r 2 (1 - r 2 /(12 a 2 ) + r 4 /(360a 4 ) - …) In the d -dimensional space corresponding to the surface of a (d + 1) -dimensional sphere of radius a , the volume of a d -dimensional sphere of radius r is similarly given by d s[d] a d Integrate[Sin[ θ ] d - 1 , { θ ,0, r/a}] = s[d] r d (1 - d (d - 1) r 2 /((6 (d + 2))a 2 + (d (5d 2 - 12d + 7))r 4 /((360 (d + 4))a 4 ) …) where Integrate[Sin[x] d - 1 , x] = -Cos[x] Hypergeometric2F1[1/2, (2 - d)/2, 3/2, Cos[x] 2 ] In an arbitrary d -dimensional space the volume of a sphere can depend on position, but in general it is given by s[d] r d (1 - RicciScalar r 2 /(6(d + 2)) + …) where the Ricci scalar curvature is evaluated at the position of the sphere.