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  "body": "\n## Statement\n\nLet $S$ be a set of points with a family of subsets of $S$, called lines, such that:\n\n1. any two distinct points lie on exactly one line;\n2. every line has at least three points;\n3. (Veblen's axiom) for distinct points $a, b, c, d$, if the lines $ab$ and $cd$ meet, then the lines $ac$ and $bd$ meet.\n\nA subspace of $S$ is a set of points that contains the whole line through any two of its distinct points.\nSuppose that the longest chains $S_0 \\subsetneq S_1 \\subsetneq \\cdots \\subsetneq S_n = S$ of nonempty subspaces have finite length $n$, and that $n \\ge 3$.\nThen there is a [[division ring|0EA5BTA0CX]] $D$ such that $S$ is isomorphic to the [[projective space|ZT9A6JK8QF]] $PG(n, D)$.\nIts points are the $1$-dimensional subspaces of the [[vector space|000AYPA9AJ]] $D^{n+1}$, and its lines are the $2$-dimensional subspaces.\nA point lies on a line when it is contained in it.\nThe division ring $D$ is unique up to isomorphism.\n\n## Remarks\n\nThe key step is that Desargues's theorem holds in every projective space of dimension at least $3$.\nGiven two triangles in perspective in a plane of $S$, one lifts the figure out of the plane.\nThere the theorem follows by intersecting planes, and a projection brings it back.\nDesargues's theorem then lets one define an addition and a multiplication on the points of a line minus one point, which form a division ring, and coordinates follow.\n\nThe hypothesis $n \\ge 3$ is necessary.\nFor $n = 2$ the axioms describe [[projective planes|DANFNN8C4Y]], and there are planes such as the [[Moulton plane|3V9GYYXM1H]] that are not of the form $PG(2, D)$ for any division ring $D$.\nIf $S$ is finite, $D$ is a finite division ring, hence a field by [[Wedderburn's little theorem|0RRNVX1VKA]], so $S \\cong PG(n, q)$ for a prime power $q$.\nThe division ring $D$ is commutative if and only if Pappus's theorem holds in the planes of $S$.\n\n## Explanations\n\n### L1 human\n\nOne can describe projective geometry by a few rules about points and lines, with no numbers at all.\nThe theorem says that as soon as the space has at least three dimensions, those rules force the space to come from linear algebra.\nThe points are the lines through the origin of a vector space, perhaps over a skew field such as the quaternions.\nIn two dimensions this fails, and there are many exotic planes.\n\n### L2 human\n\nThe theorem is the reason why the synthetic and the analytic approach to projective geometry agree in dimension at least $3$.\nIt combines with the [[fundamental theorem of projective geometry|GKV4RVXBRE]], which describes the isomorphisms between such spaces through semilinear maps.\nIt explains why the theory of non-classical geometries concentrates on planes, where coordinatization needs [[planar ternary rings|Q0NXXFADT9]].\nIt is also a model for later results, such as the classification of polar spaces and buildings of higher rank by Tits.\n",
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