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  "body": "\n## Summary\n\nA resistance form is a quadratic energy form on functions on a set which behaves like the energy of an electrical network.\nIt was introduced by Jun Kigami.\nIt gives a common frame to energies on finite networks, on intervals and on fractals such as the Sierpiński gasket.\nEach resistance form defines a metric, the resistance metric.\nIt generalizes the [[effective resistance|XAE5YJ3H68]] between two nodes of a network.\nUnder mild conditions a resistance form is also a [[Dirichlet form|MH20Q2N1AH]].\nSo it gives a Laplacian, a heat equation and a diffusion process on the set.\n\n## Explanations\n\n### L1 human\n\nThink of a network of wires that joins some points, and put a voltage $u(x)$ at each point $x$.\nThe energy of $u$ is the heat produced by the currents, a sum of terms $c \\, (u(x) - u(y))^2$ over the wires.\nA resistance form keeps the main features of this energy but allows infinitely many points, such as all points of a fractal.\nThe effective resistance between two points $x$ and $y$ is then the largest value of $(u(x) - u(y))^2$ among voltages of energy $1$.\nOn the interval $[0, 1]$ with energy $\\int_0^1 u'(s)^2 \\, ds$, this resistance between $x$ and $y$ is just $\\abs{x - y}$, like a uniform wire.\n",
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