Without employing ad hoc assumptions, various equations and solutions for plane problems of one-dimensional quasicrystals are deduced systematically. A method for the exact solution of three-dimensional equations is presented under homogeneous and nonhomogeneous boundary conditions. The equations and solutions are used to construct the refined theory of thick plates for both an in-plane extensional deformation regime and a normal or shear surface loading. With this method, the refined theory can now be explicitly established from the general solution of quasicrystals and the Lur’e method. In two illustrative examples of infinite plates with a circular hole, it is shown that explicit expressions of analytical solutions can be obtained by using the refined theory.