Note that for x of the form p π /q , the k = ∞ sum is just ( π /q/(2q)) 2 Sum[Sin[n 2 p π /q]/Sin[n π /(2q)] 2 , {n, q - 1}] The pictures below show Sum[Cos[2 n x], {n, k}] (as studied by Karl Weierstrass in 1872). … The curves below are approximations to Sum[Cos[2 n x]/2 a n , {n, ∞ }] .

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

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.

(Note that Nest[Sqrt[# + 2] &, 0, n]  2 Cos[ π /2 n + 1 ] .)

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 ]}} .

Chords Two pure tones played together exhibit beats at the difference of their frequencies—a consequence of the fact that Sin[ ω 1 t] + Sin[ ω 2 t]  2 Sin[1/2( ω 1 + ω 2 ) t] Cos[( ω 1 - ω 2 ) t] With ω ≃ 500 Hz , one can explicitly hear the time variation of the beats if their frequency is below about 15 Hz, and the result is quite pleasant.

And if these hit polarizing ("anti-glare") filters whose orientation is off by an angle θ , then the waves transmitted will have intensity Cos[ θ ] 2 . … In ordinary quantum theory, a straightforward calculation implies that the expected value of the product of the two measured spin values will be -Cos[ θ ] . … Yet as mentioned on page 1058 , actual experiments show that in fact the decrease with θ is more rapid—and is instead consistent with the quantum theory result -Cos[ θ ] .

In addition: • GoldenRatio is the solution to x  1 + 1/x or x 2  x + 1 • The right-hand rectangle in is similar to the whole rectangle when the aspect ratio is GoldenRatio • Cos[ π /5]  Cos[36 ° ]  GoldenRatio/2 • The ratio of the length of the diagonal to the length of a side in a regular pentagon is GoldenRatio • The corners of an icosahedron are at coordinates Flatten[Array[NestList[RotateRight, {0, (-1) #1 GoldenRatio, (-1) #2 }, 3]&, {2, 2}], 2] • 1 + FixedPoint[N[1/(1 + #), k] &, 1] approximates GoldenRatio to k digits, as does FixedPoint[N[Sqrt[1 + #],k]&, 1] • A successive angle difference of GoldenRatio radians yields points maximally separated around a circle (see page 1006 ).

For a = 8/3 , the solution can be written without Jacobi elliptic functions, and is given by 3 Sin[Sqrt[5/6] t] 2 /(2 + 3 Cos[Sqrt[5/6] t] 2 )

[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 ).