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  "body": "\n## Summary\n\nThe Feigenbaum–Cvitanović functional equation is $g(x) = -\\alpha \\, g(g(x/\\alpha))$, for an even function $g$ with $g(0) = 1$ and a constant $\\alpha$.\nIts solution is the fixed point of the renormalization operator that doubles the period of a unimodal map and rescales space.\nThe constant $\\alpha$ that comes with the solution is the second of the [[Feigenbaum constants|F1EHT2CPHC]], $\\alpha \\approx 2.5029$.\nThe equation explains why the [[period-doubling cascade|6J921PEDN2]] looks the same at every scale, for a large class of maps.\n\n## Explanations\n\n### L1 human\n\nTake a map with one hump, such as $x \\mapsto 1 - \\mu x^2$, and look at its second iterate near the top of the hump.\nAfter shrinking the picture by a fixed factor and flipping it upside down, it looks almost like the original map.\nThe equation describes a map that reproduces itself exactly under this operation.\nThe shrinking factor of that map, about $2.5$, then shows up in the period-doubling route to chaos of many different systems.\n",
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