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  "body": "\n## Summary\n\nA quasiconformal mapping is an orientation-preserving [[homeomorphism|0QT00PB5G2]] between plane domains that distorts infinitesimal circles into ellipses of bounded eccentricity.\nThe bound $K \\ge 1$ on the ratio of the axes is the maximal dilatation, and $1$-quasiconformal maps are exactly the [[conformal maps|0YYT36R4YD]].\nQuasiconformal maps are much more flexible than conformal maps, yet they keep many of their compactness and boundary properties.\nThey are a basic tool in Teichmüller theory, complex dynamics and the theory of Kleinian groups.\n\n## Explanations\n\n### L1 human\n\nA conformal map sends tiny circles to tiny circles.\nA quasiconformal map is allowed to squash tiny circles into tiny ellipses, as long as the ellipses never get too thin.\nThe number $K$ measures the worst ratio of the long axis to the short axis.\nFor example, the stretch $(x, y) \\mapsto (2x, y)$ of the plane turns every circle into an ellipse with axis ratio $2$, so it is $2$-quasiconformal.\nSuch maps can bend shapes in ways that conformal maps cannot, which makes them easy to build by hand.\n",
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