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  "body": "\n## Example\n\nLet $q$ be a prime power, let $\\F_q$ be the [[finite field|0ZXZ82FTX2]] with $q$ elements, and let $n \\ge 1$.\nThe projective space $PG(n, q)$ is the [[projective space|ZT9A6JK8QF]] of $V = \\F_q^{n+1}$.\nIts points are the $1$-dimensional subspaces of $V$, and its lines are the $2$-dimensional subspaces.\nIn general its $k$-dimensional subspaces are the $(k+1)$-dimensional subspaces of $V$, with incidence given by inclusion.\n\nCount of points: $V$ has $q^{n+1} - 1$ nonzero vectors, and each $1$-dimensional subspace contains $q - 1$ of them, so $PG(n, q)$ has\n$$\\frac{q^{n+1} - 1}{q - 1} = q^n + q^{n-1} + \\cdots + q + 1$$\npoints.\nEach line has $(q^2 - 1)/(q - 1) = q + 1$ points.\n\nCheck for $PG(3, 2)$: it has $15$ points and each line has $3$ points.\nTwo distinct points span a unique line, so the number of lines is $\\binom{15}{2} / \\binom{3}{2} = 105 / 3 = 35$.\nIts planes are the $3$-dimensional subspaces of $\\F_2^4$, and there are $15$ of them, one for each nonzero linear form up to scaling.\n\nFor $n = 2$, the plane $PG(2, q)$ is a [[finite projective plane|2HWB7P1DZN]] of order $q$, and $PG(2, 2)$ is the Fano plane.\n\n## Explanations\n\n### L1 human\n\nTake the usual recipe for projective space, lines through the origin of a vector space, and use a finite field in place of the real numbers.\nThe result is a projective space with finitely many points, one of the basic objects of [[finite geometry|MCDZP31ZF3]].\nEvery axiom of projective geometry still holds, and all counts become explicit formulas in $q$.\nFor $q = 2$ and $n = 2$ one gets the seven points and seven lines of the Fano plane.\n",
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