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  "body": "\n## Summary\n\nThe Wigner 6-j symbol $\\begin{Bmatrix} j_1 \u0026 j_2 \u0026 j_3 \\\\ j_4 \u0026 j_5 \u0026 j_6 \\end{Bmatrix}$ is a number attached to six spins.\nIt compares the two ways to couple three irreducible representations of [[SU(2)|E5P8AG6CK7]]: first the first two, or first the last two.\nIt is a sum of products of four [[3-j symbols|YJRJVZ998D]], and it is closely related to the Racah W-coefficient and to the [[Racah polynomials|P6JYNDX5DR]].\n6-j symbols are used in atomic and nuclear physics, and in the construction of invariants of 3-manifolds.\n\n## Explanations\n\n### L1 human\n\nThree spins can be added in two orders: $(j_1 + j_2) + j_3$ or $j_1 + (j_2 + j_3)$.\nBoth orders give a basis of the same space, and the change of basis between them is given by 6-j symbols.\nSo a 6-j symbol is the analogue, for representations, of the associativity of addition.\nIts six entries can be drawn on the edges of a tetrahedron, and the symbol has the symmetries of the tetrahedron.\n",
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