Under investigation are the soliton interactions for a (2[Formula: see text]+[Formula: see text]1)-dimensional nonlinear Schrödinger equation, which can describe the dynamics of a nonlinear photonic quasi-crystal or vortex Airy beam in a Kerr medium. With the symbolic computation and Hirota method, analytic bright N-soliton and dark two-soliton solutions are derived. Graphic description of the soliton properties and interactions in a nonlinear photonic quasicrystal or Kerr medium is done. Through the analysis on bright and dark one solitons, effects of the optical wavenumber/linear opposite wavenumber and nonlinear coefficient on the soliton amplitude and width are studied: when the absolute value of the optical wavenumber or linear opposite wavenumber increases, bright soliton amplitude and dark soliton width become smaller; nonlinear coefficient has the same influence on the bright soliton as that of the optical wavenumber or linear opposite wavenumber, but does not affect the dark soliton amplitude or width. Overtaking/periodic interactions between the bright two solitons and overtaking interactions between the dark two solitons are illustrated. Overtaking interactions show that the bright soliton with a larger amplitude moves faster and overtakes the smaller, while the dark soliton with a smaller amplitude moves faster and overtakes the larger. When the absolute value of the optical wavenumber or linear opposite wavenumber increases, the periodic-interaction period becomes longer. All the above interactions are elastic. Through the interactions, soliton amplitudes and shapes keep invariant except for some phase shifts.