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  "body": "\n## Summary\n\nThe normal fan of a [[convex polytope|300K6CB1KH]] $P$ is the collection of its normal cones, one cone for each face.\nThe normal cone of a face collects the linear functionals that reach their maximum on $P$ at that face.\nThe normal fan records the combinatorics of $P$ in a dual form.\nWhen $P$ is full-dimensional, its vertices give the full-dimensional cones and its facets give the rays.\nFor a full-dimensional lattice polytope it is a rational fan, and it defines the projective [[toric variety|YWB981MEF2]] of the polytope.\n\n## Explanations\n\n### L1 human\n\nStand at a corner of a polytope and ask in which directions that corner is the farthest point of the polytope.\nThese directions form a cone, the normal cone of the corner.\nDoing the same for edges and the other faces cuts the space of directions into cones that fit together with no gaps or overlaps.\nFor a square centred at the origin, the four corners give the four quadrants and the four sides give the four half-axes.\n",
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