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  "body": "\n## Statement\n\nLet $S^2 \\subseteq \\R^3$ be the unit [[sphere|T62369RFH6]], let $\\iota \\colon S^2 \\to \\R^3$ be the inclusion and let $\\alpha \\colon S^2 \\to \\R^3$ be the map $\\alpha(x) = -x$.\nThen there is a smooth map $H \\colon S^2 \\times [0, 1] \\to \\R^3$ such that each $H_t = H(\\cdot, t)$ is an [[immersion|6RW68JGBHC]], $H_0 = \\iota$ and $H_1 = \\alpha$.\n\n## Remarks\n\nSuch a family $H$ is a regular homotopy that turns the sphere inside out: the outward normal of $\\iota$ corresponds to the inward normal of $\\alpha$.\nThe surfaces $H_t$ may pass through themselves, but they may never develop a crease or a pinch.\nStephen Smale proved this as a consequence of his classification of immersions of spheres: regular homotopy classes of immersions $S^2 \\to \\R^3$ correspond to $\\pi_2(SO(3))$, which is the trivial group.\nThe proof was not constructive; explicit eversions were later found, for example by Bernard Morin, and by William Thurston with the method of corrugations.\nThe analogue for circles fails: by the Whitney–Graustein theorem, immersions $S^1 \\to \\R^2$ are classified by the turning number, and the standard circle and its reversal have turning numbers $1$ and $-1$.\n\n## Explanations\n\n### L1 human\n\nImagine a sphere made of a material that can pass through itself but cannot be creased.\nThe theorem says that such a sphere can be turned inside out by a continuous motion.\nThe result is surprising, because the obvious attempt, pushing two opposite caps through each other, leaves a sharp ring that cannot be smoothed away.\n\n### L2 human\n\nSphere eversion is the standard example of the power of the Smale–Hirsch theorem and of the homotopy principle.\nIt shows that a regular homotopy can connect immersions that look very different.\nThe only invariants of immersions $S^2 \\to \\R^3$ up to regular homotopy are homotopy-theoretic, and here they vanish.\nIts failure for circles in the plane shows that the outcome depends on the dimensions through the relevant homotopy group.\n",
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