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Using a projection method of de Bruijn [Proc. K. Ned. Akad. Wet. Ser. A (1981), 43, 39-66], Whittaker & Whittaker [Acta Crysr (1988), A44, 105-112] obtained nonperiodic tilings of the plane with n-fold rotational symmetry, n=5, 7, 8, 9, 10 and 12. However, when their method was applied to the cases of 3-, 4-and 6-fold rotational symmetry it produced periodic tilings. This might be taken as circumstantial evidence that 3-, 4-and 6-fold rotational symmetry is incompatible with nonperiodicity. It is demonstrated that they are compatible by constructing quasiperiodic tilings with only 3-, 4-and 6-fold rotational symmetry. This approach uses basic eigenvalue methods of matrix theory. AbstractThe effects of polarization on X-ray multiple diffraction are investigated experimentally. Polarized and unpolarized incident beams are used in multiple 0108-7673/91/050502-09503.00 diffraction experiments for GaAs and Ge crystals. All the three-beam diffractions in a 360 ° azimuthal rotation of the crystals are analyzed for diffracted intensities and phase determination, based on the article by Chang