Let \(\omega\) be a root of \(x^2 - x + 1 = 0\). Then \(\omega^2 = \omega - 1\), and \(\omega^3 = \omega(\omega - 1) = \omega^2 - \omega = (\omega - 1) - \omega = -1\). So \(\omega^3 = -1\), and \(\omega^6 = 1\). - Groen Casting