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  "body": "\n## Summary\n\nThe Dehn invariant of a [[polyhedron|0325SGSJAR]] combines the lengths of its edges with their [[dihedral angles|05HN08ZSRJ]] in a tensor product of abelian groups.\nIt does not change when a polyhedron is cut into pieces and the pieces are reassembled.\nMax Dehn used it in 1900 to solve [[Hilbert's third problem|81035XQ5HE]].\nA cube and a regular tetrahedron of the same volume have different Dehn invariants, so neither can be cut into the other.\n\n## Explanations\n\n### L1 human\n\nWhen you cut a solid into pieces, new edges appear.\nAround a new edge inside the solid, the angles of the pieces add up to a full turn or a half turn.\nThe Dehn invariant adds up \"edge length times angle\" over all edges, but counts angles only up to rational multiples of $\\pi$.\nWith this rule the new edges contribute nothing.\nA cube has only right angles, which are rational multiples of $\\pi$, so its invariant is $0$.\nA regular tetrahedron has the angle $\\arccos(1/3)$, which is not, so its invariant is not $0$.\n",
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