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  "body": "\n## Summary\n\nHyperbolic geometry is the [[non-Euclidean geometry|SPCYMFN5V2]] in which, through a point off a line, there pass at least two lines that do not meet it.\nIt can be realised on a surface of constant negative [[Gaussian curvature|SQVQ5DQT6W]].\nIn it, the angles of a triangle add up to less than $\\pi$, and the area of a triangle is determined by its angles.\nIt appears throughout mathematics, from complex analysis and number theory to the topology of surfaces and 3-manifolds.\n\n## Explanations\n\n### L1 human\n\nImagine a world where space spreads out faster than in a flat plane, like the frilly edge of a lettuce leaf.\nIn this world, through a point beside a line you can draw many lines that never meet it.\nTriangles are thin, and their angles add up to less than $180^\\circ$.\nIn the [[upper half-plane model|RHKSMPBSQ1]], the lines are vertical half-lines and half-circles that meet the horizontal axis at right angles.\n",
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