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  "body": "\n## Statement\n\nLet $P$ be a [[convex polytope|300K6CB1KH]] of dimension $d \\ge 1$, with [[f-vector|872BAKDQQK]] $(f_0, \\dots, f_{d-1})$.\nThen\n$$\\sum_{i=0}^{d-1} (-1)^i f_i = f_0 - f_1 + f_2 - \\cdots + (-1)^{d-1} f_{d-1} = 1 - (-1)^d.$$\n\n## Remarks\n\nWith $f_{-1} = 1$ for the empty face and $f_d = 1$ for $P$, the relation reads $\\sum_{i=-1}^{d} (-1)^i f_i = 0$.\nOne proof is topological: the boundary of $P$ is homeomorphic to the sphere $S^{d-1}$, and its faces form a cell decomposition.\nSo the alternating sum is the [[Euler characteristic|RNDDWDAFJ5]] $1 + (-1)^{d-1}$ of the sphere.\nA combinatorial proof builds the boundary facet by facet in a shelling order and checks that the alternating sum changes correctly at each step.\n\nFor $d = 3$ the formula is [[Euler's polyhedron formula|09HT3P35YP]] $f_0 - f_1 + f_2 = 2$.\nFor $d = 4$ the alternating sum is $0$; the $4$-cube has $16 - 32 + 24 - 8 = 0$.\nFor a polygon with $n$ vertices, $n - n = 0 = 1 - (-1)^2$.\n\nIt is the only linear equation satisfied by the f-vectors of all $d$-polytopes, up to scaling.\nFor simplicial polytopes there are further linear relations, the [[Dehn–Sommerville equations|4WKHF5BCTA]].\n\n## Explanations\n\n### L1 human\n\nCount the faces of each dimension of a convex polytope and add them with alternating signs.\nThe answer does not depend on the polytope, only on its dimension: it is $2$ in odd dimensions and $0$ in even dimensions.\nIn three dimensions this is the classical rule vertices minus edges plus faces equals $2$.\n\n### L2 human\n\nThe relation $\\sum (-1)^i f_i = 1 - (-1)^d$ is the Euler–Poincaré relation.\nIt is the first constraint any candidate f-vector must meet, and in dimension $3$ together with the inequalities $f_2 \\le 2 f_0 - 4$ and $f_0 \\le 2 f_2 - 4$ it characterises the f-vectors of $3$-polytopes.\nIts proof by shellings was historically delicate: early arguments assumed that the boundary of a polytope can be shelled, which was proved only later by Bruggesser and Mani.\nThe same alternating sum, for general cell complexes, is the Euler characteristic of algebraic topology.\n",
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