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The sets of numbers that can be obtained by applying elementary functions like Exp , Log and Sin seem in various ways to be disjoint from algebraic numbers. … One can also ask what numbers can be generated by integrals (or by solving differential equations). … Integrals of rational functions over regions defined by polynomial inequalities have recently been discussed under the name "periods".

[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 ). When problems are originally stated as differential equations, results in terms of integrals ("quadrature") are sometimes considered exact solutions—as occasionally are convergent series.

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 ]}} .… (Versions of this were noticed by Richard Feynman in the 1940s in connection with his development of path integrals, and were pointed out again several times in the 1980s and 1990s.)