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In this paper, we disscuss one-dimensional hexagonal quasicrystals and infinite elasticity plane with single periodic of second fundamental problem by the condition basical hypothesis. We use the plane elasticity of complex variables method and Hilbert kernel integral formula proving the onedimensional hexagonal quasicrystals and infinite elasticity plane with single periodic of second fundamental problem, and obtaining the solution is not only existence but also unique. And in the proving process, we provide the specific arithmetic. At last, we offer an example to verify.