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  "body": "\n## Statement\n\nLet $k$ be a [[field|04C5ZBG1WT]] with at least $3$ elements, and let $n \\ge 2$.\nLet $A$ and $B$ be [[affine spaces|09XNAPTJQ7]] of dimension $n$ over $k$, with spaces of translations $V$ and $W$.\nLet $f \\colon A \\to B$ be a bijection that maps every line of $A$ onto a line of $B$.\nThen there are a field automorphism $\\sigma$ of $k$ and a bijection $\\varphi \\colon V \\to W$ such that:\n\n1. $\\varphi(u + v) = \\varphi(u) + \\varphi(v)$ and $\\varphi(c v) = \\sigma(c) \\varphi(v)$ for all $u, v \\in V$ and $c \\in k$;\n2. $f(p + v) = f(p) + \\varphi(v)$ for all $p \\in A$ and $v \\in V$.\n\nIn particular, if $k = \\R$, then $f$ is an [[affine map|0JF0TVYGWT]].\n\n## Remarks\n\nA map with properties 1 and 2 is called semi-affine.\nThe proof first shows that $f$ maps planes to planes and hence parallel lines to parallel lines.\nParallelograms then go to parallelograms, which makes the induced map $\\varphi$ on vectors well defined and additive.\nComparing $f$ on the points of one line with its action on scalars defines $\\sigma$ and shows that it is a field automorphism.\nFor $k = \\R$ the only field automorphism is the identity, which gives the last sentence.\n\nThe hypothesis on $k$ is needed: over $\\F_2$ each line has exactly two points, so every bijection maps lines to lines.\nFor $n \\ge 3$ there are bijections of $\\F_2^n$ that are not affine, for example the map that swaps two basis vectors $e_1$ and $e_2$ and fixes every other point.\nIt is not affine, since it fixes $0$, $e_3$ and $e_1 + e_3$ but moves $e_1$.\nThe hypothesis $n \\ge 2$ is needed too: an affine line has only one line, so every bijection of it qualifies.\nOver $\\C$, the map $(z_1, \\dots, z_n) \\mapsto (\\bar z_1, \\dots, \\bar z_n)$ maps lines to lines but is not affine, so $\\sigma$ cannot be dropped.\n\n## Explanations\n\n### L1 human\n\nSuppose a bijection of the plane or of space sends every straight line to a straight line.\nOver the real numbers, the theorem says that it must then be an affine map: a linear map followed by a translation.\nSo keeping lines straight forces the map to keep ratios along lines and parallelism.\nOver other fields the map can also twist the scalars by a field automorphism, such as complex conjugation.\n\n### L2 human\n\nThe theorem shows that the incidence structure of points and lines determines the affine structure of $k^n$, up to the automorphisms of $k$.\nIt is the affine version of the [[fundamental theorem of projective geometry|GKV4RVXBRE]], and one can pass between the two by adding or removing a hyperplane at infinity.\nIt shows that the [[collineations|V7PRPN24CV]] of a real affine space are exactly the invertible affine maps.\nIt also explains why the automorphism group of a field enters the symmetry groups of geometries over that field.\n",
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