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  "body": "\n## Summary\n\nThe conformal modulus of a family of curves in the plane is the reciprocal of its [[extremal length|HX72KT9D0G]].\nIt is a conformal invariant.\nIt gives the modulus of a quadrilateral and of a ring domain, which classify these domains up to conformal equivalence.\nModuli are the basic quantities in the geometric definition of [[quasiconformal maps|SB8KG46NHP]], which distort them by a bounded factor.\n\n## Explanations\n\n### L1 human\n\nTwo annuli $\\set{r \u003c \\abs{z} \u003c R}$ and $\\set{r' \u003c \\abs{z} \u003c R'}$ are conformally equivalent exactly when $R/r = R'/r'$.\nSo one number, essentially $\\log(R/r)$, tells which annulus one has, and this is the modulus of the ring.\nThe conformal modulus of a curve family generalises this: many curves that are short give a large modulus, few long curves give a small one.\nFor the curves crossing a rectangle of width $a$ and height $b$ from left to right, the modulus is $b/a$.\n",
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