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  "body": "\n## Summary\n\nDiscrete differential geometry builds discrete counterparts of notions of differential geometry, such as curvature, geodesics and conformal maps.\nIt works with polyhedral surfaces, meshes and other finite structures.\nA good discrete notion keeps the key theorems of the smooth theory exactly, not only in the limit.\nThe angle defect at a vertex of a [[polyhedron|0325SGSJAR]] plays the role of [[Gaussian curvature|SQVQ5DQT6W]], and [[circle packings|S360JBEYKD]] serve as discrete conformal maps.\nThe field supplies the mathematics behind computer graphics, geometry processing and architectural design.\n\n## Explanations\n\n### L1 human\n\nSmooth surfaces are approximated on computers by surfaces made of flat triangles.\nDiscrete differential geometry asks how to define curvature and related ideas directly on such triangle meshes.\nAt a vertex of a polyhedral surface, the curvature is measured by how much the angles around the vertex fall short of a full turn $2\\pi$.\nAt a corner of a cube three right angles meet, so the defect is $2\\pi - 3 \\cdot \\frac{\\pi}{2} = \\frac{\\pi}{2}$.\nThe eight corners give a total of $4\\pi$, exactly as the Gauss–Bonnet theorem predicts for a sphere; this is [[Descartes's theorem|PT67C5ANSD]].\n",
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