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  "body": "\n## Example\n\nFor $d \\ge 1$, the $d$-dimensional *hypercube* is\n$$C_d = [0,1]^d = \\set{x \\in \\R^d : 0 \\le x_i \\le 1 \\text{ for } i = 1, \\dots, d}.$$\nIt is a [[convex polytope|300K6CB1KH]]: it is the convex hull of the $2^d$ points with all coordinates in $\\set{0, 1}$, and these are its [[vertices|R7V3S9V891]].\nIts [[facets|DPDXN2FJC4]] are the $2d$ sets where one coordinate is fixed to $0$ or to $1$.\n\nA face of $C_d$ is obtained by fixing some coordinates to $0$ or $1$ and leaving the others free.\nA $k$-dimensional face leaves $k$ coordinates free, so the number of $k$-faces is\n$$f_k(C_d) = 2^{d-k} \\binom{d}{k}.$$\nFor $d = 3$ this gives the [[f-vector|872BAKDQQK]] $(8, 12, 6)$ of the ordinary cube.\n\n$C_d$ is a [[simple polytope|7E6FXTGH2Y]].\nAt a vertex $v$ the edges change exactly one coordinate of $v$, and there are $d$ coordinates, so every vertex lies in $d$ edges.\n\n## Remarks\n\nThe graph of $C_d$ is the hypercube graph: its vertices are the $0/1$ strings of length $d$, joined when they differ in one place.\nMany authors use $[-1, 1]^d$ instead; it has the same combinatorics and is centrally symmetric about $0$.\nFor $d = 4$ the hypercube is also called the tesseract, with $f$-vector $(16, 32, 24, 8)$.\n\n## Explanations\n\n### L1 human\n\nThe hypercube is the cube in any dimension.\nA segment is a $1$-cube, a square is a $2$-cube and the ordinary cube is a $3$-cube.\nTo get the next one, slide the previous cube one unit in a new direction and fill in the space it sweeps out.\nEach step doubles the number of corners, so the $d$-cube has $2^d$ corners.\n",
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