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  "body": "\n## Statement\n\nLet $S \\subseteq \\R^3$ be a closed set such that $\\mathcal{H}^2(S \\cap K) \u003c \\infty$ for every compact set $K$, where $\\mathcal{H}^2$ is the $2$-dimensional [[Hausdorff measure|Q1TGZK8RN6]].\nLet $\\delta \u003e 0$, $C \u003e 0$ and $\\alpha \u003e 0$.\nSuppose that $S$ is almost minimizing in the sense of Almgren: for every Lipschitz map $\\phi \\colon \\R^3 \\to \\R^3$, with $W = \\set{x : \\phi(x) \\ne x}$ and $r = \\operatorname{diam}(W \\cup \\phi(W)) \u003c \\delta$,\n$$\\mathcal{H}^2(S \\cap W) \\le (1 + C r^\\alpha) \\, \\mathcal{H}^2(\\phi(S \\cap W)).$$\nThen every point of $S$ has a neighbourhood $U$ with the following property.\nThe set $S \\cap U$ is the image, under a $C^{1,\\beta}$ diffeomorphism with $\\beta \u003e 0$, of an open piece of one of these three cones:\n\n1. a plane;\n2. the cone $Y$: three closed half-planes with a common boundary line, meeting along it at angles of $120^\\circ$;\n3. the cone $T$ over the six edges of a regular [[tetrahedron|JFX8ASQS3E]] centred at the origin, whose six sectors meet in threes along four rays.\n\n## Remarks\n\nJean Taylor proved the theorem in 1976, using the regularity theory of Almgren.\nThe main step is to classify the minimizing cones.\nA tangent cone of $S$ at a point is a cone over a network of arcs of great circles on the unit sphere.\nThe minimizing condition leaves only the three cones above.\nThen an epiperimetric inequality shows that $S$ approaches its tangent cone at a rate that is a power of the radius, which gives the $C^{1,\\beta}$ parametrization.\nSoap films and clusters of soap bubbles are modelled by such almost-minimizing sets, so the theorem proves [[Plateau's laws|381RB634WS]] within this model.\nFor example, take a soap film on a wire frame shaped like the edges of a regular tetrahedron.\nIt consists of six flat triangles meeting at the centre, a piece of the cone $T$.\n\n## Explanations\n\n### L1 human\n\nSoap films can meet along edges and at corners, but only in a few ways.\nTake any surface that cannot reduce its area by small deformations.\nTaylor's theorem says that near each point it looks like a bent copy of one of three models.\nThe models are a plane, three sheets meeting along an edge at $120^\\circ$, and the tetrahedral corner where four such edges meet.\nThis turns the old experimental rules of Plateau into a theorem.\n\n### L2 human\n\nThe theorem is a model case of the classification of singularities in geometric variational problems.\nIts ingredients, the tangent cone classification and the epiperimetric inequality, have since been used for other problems, such as the regularity of free boundaries.\nThe almost-minimizing condition is flexible: it covers films spanning wires, bubbles enclosing fixed volumes, and films with small external forces.\n",
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