Single-crystal copper (Cu), whose atom arrangement is in the same direction and has no grain boundary, is widely used in defense technology, civil electronics and network communication. As a diamond turnable material, fan-shaped patterns appear on the machined surface, which affects the machined surface quality and the optical function it carries. Previous studies on the surface generation mechanism in single-point diamond turning (SPDT) of Cu were limited to experimental analysis, while there is a lack of fundamental understanding of the fan-shaped pattern generation mechanism and suppression method. In the present study, the different fan-shaped patterns, surface quality, cutting force and chip morphology of the typical crystal planes (100), (110) and (111) planes of Cu were studied by both theoretical and experimental analyses. A molecular dynamics (MD) simulation was conducted to present the fundamental generation mechanism of the fan-shaped patterns from atom arrangement directions and its angle change with the main cutting direction, while a cutting dynamics model was established to simulate the generation of fan-shaped patterns on the machined surface. Based on theoretical and experimental analysis, it was found that the atom density arrangement directions of Cu and its angle change with the main cutting direction of SPDT caused fluctuations in the friction coefficient, which further caused the vibration of the cutting system and generated the fan-shaped patterns. The SPDT of crystal planes (100) can achieve the best surface quality. The present research provides a fundamental understanding of fan-shaped pattern formation on the machined surface, and provides an instruction for machining Cu to obtain better surface quality.