= Chiral hexagonal amplitude equations
{title2=$\dot A_i=\mu A_i+\alpha A_{i+1}^*A_{i+2}^*-A_i(a|A_i|^2+b|A_{i+1}|^2+c|A_{i+2}|^2)$}
Three Fourier wavevectors sum to zero and form a hexagonal star. Translations give each amplitude the phase of its own wavevector; cyclic rotations permute them, and a half-turn conjugates them. These symmetries permit the resonant quadratic term and force its coefficients to be real. A spatial reflection would interchange the two cross-couplings and require $b=c$, but chiral patterns need not have that extra symmetry. The equations describe rolls, hexagons, phase locking and oscillatory changes in dominant orientation.
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