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  "body": "\n## Example\n\nLet $\\mathbb{O}$ be the [[octonions|1HRMT3DSX2]], the $8$-dimensional real division algebra that is alternative but not associative.\nThe *octonionic projective plane* $\\mathbb{O}P^2$, also called the Cayley plane, has the following points and lines.\n\n1. Affine points $(x, y)$ with $x, y \\in \\mathbb{O}$, points $(m)$ with $m \\in \\mathbb{O}$, and one more point $(\\infty)$.\n2. Lines $[m, b] = \\set{(x, y) : y = x m + b} \\cup \\set{(m)}$ for $m, b \\in \\mathbb{O}$, lines $[c] = \\set{(c, y) : y \\in \\mathbb{O}} \\cup \\set{(\\infty)}$ for $c \\in \\mathbb{O}$, and the line at infinity $\\set{(m) : m \\in \\mathbb{O}} \\cup \\set{(\\infty)}$.\n\nThis is a [[projective plane|DANFNN8C4Y]].\nThe key check concerns two affine points $(x_1, y_1)$ and $(x_2, y_2)$ with $x_1 \\ne x_2$.\nThey lie on a line $[m, b]$ exactly when $(x_1 - x_2) m = y_1 - y_2$ and $b = y_1 - x_1 m$.\nIn an alternative division algebra, $a m = c$ with $a \\ne 0$ has the unique solution $m = a^{-1} c$, since $a (a^{-1} c) = c$.\n\nIt is a [[Moufang plane|7F4HSNJJH8]], since it is coordinatized by the alternative division ring $\\mathbb{O}$.\nIt is not a [[Desarguesian plane|HPX85PA8NX]]: a Desarguesian plane is coordinatized by an associative division ring, and $\\mathbb{O}$ is not associative.\n\n## Remarks\n\nAs a topological space $\\mathbb{O}P^2$ is a compact manifold of dimension $16$.\nFor its natural Riemannian metric, its group of isometries is the compact Lie group $F_4$.\nIts group of continuous collineations is a real form of $E_6$.\nThere is no octonionic projective space of dimension $3$ or more.\nIndeed, [[Desargues's theorem|5Z19Y02MME]] holds in every projective space of dimension at least $3$, which would force the coordinates to be associative.\n\n## Explanations\n\n### L1 human\n\nThe real, complex and quaternionic numbers each give a projective plane, built from lines through the origin.\nThe octonions are a larger number system whose multiplication is not associative, so the usual construction with lines through the origin breaks down.\nStill, one can write down points and lines with octonion coordinates, and they obey the rules of a projective plane.\nThis plane is special: it satisfies a weak form of Desargues's theorem but not the full theorem, and there is no octonionic projective space of dimension $3$ or more.\n",
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