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  "body": "\n## Definition\n\nLet $P$ be a [[polyhedron|0325SGSJAR]] in $\\R^3$.\nFor each edge $e$ of $P$, let $\\ell(e)$ be its length and let $\\theta(e)$ be the [[dihedral angle|05HN08ZSRJ]] of $P$ at $e$.\nThe *Dehn invariant* of $P$ is the element\n$$D(P) = \\sum_{e} \\ell(e) \\otimes \\theta(e)$$\nof the [[tensor product|F0HQXQ0HCD]] $\\R \\otimes_\\Z (\\R / \\pi\\Z)$ of [[abelian groups|082GHNPJ53]].\nThe sum runs over all edges of $P$.\n\n## Remarks\n\nExample: for the cube of side $a$, every dihedral angle is $\\pi/2$.\nSo each edge gives $a \\otimes \\pi/2 = (a/2) \\otimes \\pi = 0$, and $D(\\text{cube}) = 0$.\nA polyhedron with nonzero invariant: the regular tetrahedron of side $a$ has dihedral angle $\\arccos(1/3)$, which is not a rational multiple of $\\pi$.\nIts invariant $6a \\otimes \\arccos(1/3)$ is not $0$, since $\\R \\otimes_\\Z (\\R/\\pi\\Z)$ is isomorphic to the $\\Q$-vector space $\\R \\otimes_\\Q (\\R/\\Q\\pi)$.\nThe invariant is additive: if $P$ is cut into polyhedra $P_1, \\dots, P_k$, then $D(P) = D(P_1) + \\cdots + D(P_k)$.\n",
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