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  "body": "\n## Summary\n\nThe f-vector of a [[simplicial complex|V8GZ0W1ZZ5]] or of a polytope lists the number of its faces of each dimension.\nThe h-vector is an invertible linear transform of the f-vector that often has simpler properties.\nIt is symmetric for simplicial spheres and nonnegative for many important complexes.\nThe [[Dehn–Sommerville equations|4WKHF5BCTA]] and the [[g-theorem|4CZMWQVC30]] are stated in terms of the h-vector.\n\n## Explanations\n\n### L1 human\n\nCount the vertices, edges, triangles, and so on of a shape built from simplices, and you get its f-vector.\nFor the surface of an octahedron the counts are $6$ vertices, $12$ edges and $8$ triangles.\nRewriting these numbers as an h-vector gives $(1, 3, 3, 1)$, which reads the same forwards and backwards.\nThis hidden symmetry is one reason to use the h-vector.\n",
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