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  "body": "\n## Summary\n\nThe dimension of an algebraic variety is the length of the longest chain of nested irreducible closed subsets in it.\nFor an affine variety it equals the Krull dimension of the coordinate ring.\nFor any variety it equals the transcendence degree of the function field over the base field.\nThese different descriptions agree, which makes dimension one of the most robust invariants in algebraic geometry: points have dimension $0$, curves $1$, surfaces $2$.\n\n## Explanations\n\n### L1 human\n\nInside a surface, a chain of ever smaller irreducible pieces can go surface, then a curve on it, then a point on that curve.\nThat chain has two steps, and the surface has dimension $2$.\nAnother way to see it is to count how many coordinates can be chosen freely.\nOn the sphere $x^2 + y^2 + z^2 = 1$ over $\\C$, two coordinates are free, and then the third has at most two possible values.\nSo the sphere is a surface, of dimension $2$.\n",
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