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  "body": "\n## Definition\n\nLet $P_1$ and $P_2$ be two [[planes|0V3XAMF3C1]] in $\\mathbb{R}^3$ that meet in a line. Let $n_1$ and $n_2$ be [[normal vectors|0WQJJCCTC8]] of $P_1$ and $P_2$, and let $\\cdot$ be the dot product.\nThe dihedral angle between $P_1$ and $P_2$ is the unique real number $\\varphi$ with $0 \\leq \\varphi \\leq \\pi / 2$ such that\n$$\\cos \\varphi = \\frac{|n_1 \\cdot n_2|}{\\|n_1\\| \\, \\|n_2\\|}.$$\n\n## Remarks\n\nThe value does not depend on the choice of the normal vectors. The absolute value is necessary because $-n_1$ is also a normal vector of $P_1$.\nFor two half-planes with a common edge line, the dihedral angle is in $[0, \\pi]$. Take a point $O$ on the edge. Let $u_1$ and $u_2$ be vectors that are perpendicular to the edge and point from $O$ into the two half-planes. The dihedral angle is the [[angle between $u_1$ and $u_2$|0ZEHZY85E6]].\nFor the planes $z = 0$ and $x = 0$, the normal vectors $(0, 0, 1)$ and $(1, 0, 0)$ have dot product $0$, so the dihedral angle is $\\pi / 2$.\nThe dihedral angles of a polyhedron are the angles between the planes of two faces with a common edge, measured inside the solid.\n\n## Explanations\n\n### L1 human\n\nOpen a book partly. The two covers are half-planes that meet along the spine. The dihedral angle is how far the book is open: $0$ when it is closed and $\\pi$ when it lies flat.\n",
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