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  "body": "\n## Summary\n\nA contact manifold is an odd-dimensional [[smooth manifold|F45VQ47GPV]] with a field of hyperplanes.\nThis field is as far as possible from being tangent to a family of hypersurfaces.\nContact geometry is the odd-dimensional sister of symplectic geometry.\nLevel sets of energy in phase space, unit cotangent bundles and spaces of first-order jets carry natural contact structures.\nIt is the setting of geometric optics, thermodynamics and Legendrian knot theory.\n\n## Explanations\n\n### L1 human\n\nAt each point of a three-dimensional space, pick a plane through that point.\nIf the planes fit together as the tangent planes of a stack of surfaces, the field is integrable.\nA contact structure is the opposite, a field of planes that twists everywhere.\nIn $\\R^3$ the planes spanned by $\\partial_y$ and $\\partial_x + y\\,\\partial_z$ form such a field: moving in the $y$ direction, the planes turn about the $y$-axis.\nNo surface can be tangent to these planes on an open set.\n",
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