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  "body": "\n## Statement\n\nLet $a, b, c$ be real, and assume that the [[monodromy of the hypergeometric equation|T8SBE28DM1]] with these parameters is irreducible.\nLet $\\lambda = \\abs{1 - c}$, $\\mu = \\abs{c - a - b}$ and $\\nu = \\abs{a - b}$ be the exponent differences at $0$, $1$ and $\\infty$.\nThen all solutions of the [[hypergeometric differential equation|W8VE25R5A7]] are algebraic functions of $z$ if and only if the triple $(\\lambda, \\mu, \\nu)$ belongs to Schwarz's list.\nHere the triple is taken up to permutation, after a standard normalization by sign changes and integer shifts.\nThis list consists of one infinite family, $(\\frac12, \\frac12, \\nu)$ with $\\nu$ rational, and $14$ further triples.\nOf these, $2$ have tetrahedral, $2$ octahedral and $10$ icosahedral monodromy.\nAmong them are $(\\frac12, \\frac13, \\frac13)$, $(\\frac12, \\frac13, \\frac14)$ and $(\\frac12, \\frac13, \\frac15)$.\n\n## Remarks\n\nAll solutions are algebraic exactly when the monodromy group is finite.\nThe ratio of two solutions, the [[Schwarz triangle function|XMD3Q49Z2C]], maps the upper half-plane onto a circular-arc triangle with angles $\\pi\\lambda$, $\\pi\\mu$, $\\pi\\nu$.\nThe monodromy group modulo scalars is generated by products of reflections in its sides.\nFinite monodromy forces the triangle to be spherical and the projective monodromy group to be a finite subgroup of $\\mathrm{PSL}_2(\\C)$: cyclic, dihedral, or the rotation group of a [[Platonic solid|017E5PRS1Q]].\nThe cyclic case is excluded by irreducibility, the dihedral case gives the infinite family, and the tetrahedral, octahedral and icosahedral groups give the remaining entries.\n\nExample: $(\\frac12, \\frac13, \\frac14)$ gives a spherical triangle with angles $\\frac{\\pi}{2}, \\frac{\\pi}{3}, \\frac{\\pi}{4}$, whose reflections tile the sphere in $48$ triangles, matching the symmetries of the octahedron.\n\nA necessary condition, for the normalized triple, is $\\lambda + \\mu + \\nu \u003e 1$, which says that the triangle is spherical.\nIrreducibility fails exactly when one of $a$, $b$, $c - a$, $c - b$ is an integer.\nThat case is treated separately, since then one solution is elementary and the other is an integral of it.\n\n## Explanations\n\n### L1 human\n\nUsually the hypergeometric function is transcendental: it cannot be defined by a polynomial equation with its variable.\nSchwarz asked when it is algebraic, like $\\sqrt{1 - z}$.\nHe found that this happens only for a short explicit list of parameter values, and the list matches the symmetry groups of the regular polyhedra.\n\n### L2 human\n\nThe proof combines the monodromy of the hypergeometric equation with the geometry of reflection groups on the sphere.\nIt is the first instance of a general question: when does a linear differential equation have a full set of algebraic solutions?\nBeukers and Heckman answered it for the higher generalized hypergeometric equations.\nThe same triangles, when $\\lambda + \\mu + \\nu \u003c 1$, lead instead to hyperbolic triangle groups and automorphic functions.\n",
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