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  "body": "\n## Summary\n\nThe Schwarz triangle function is the ratio of two independent solutions of the [[hypergeometric differential equation|W8VE25R5A7]].\nFor suitable real parameters it is a [[conformal map|0YYT36R4YD]] from the upper half-plane onto a triangle bounded by circular arcs or line segments.\nThe angles are $\\pi\\lambda$, $\\pi\\mu$, $\\pi\\nu$, where $\\lambda$, $\\mu$, $\\nu$ are the exponent differences.\nIts inverse functions, when the angles are $\\pi/p$, $\\pi/q$, $\\pi/r$, are automorphic functions for triangle groups.\nWhen all three angles are zero, the inverse function is the modular lambda function.\n\n## Explanations\n\n### L1 human\n\nA ratio of two solutions of a second-order linear equation forgets the scaling of the solutions and keeps only their shape.\nFor the hypergeometric equation, this ratio maps the upper half-plane onto a curved triangle, whose sides are arcs of circles.\nThe three vertices are the images of $0$, $1$ and $\\infty$, and the angles there are fixed by the parameters $a$, $b$, $c$.\nWhen the angles are of the form $\\pi/n$, reflecting the triangle across its sides again and again tiles a disc, the plane or the sphere.\nThe case of the sphere is how Schwarz found the algebraic hypergeometric functions.\n",
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