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On the drawing below is a hexagonal lattice. For the basis vectors one can choose either the set of arrows in black or the set in yellow. The intersection coordinates of the plane in green seems to be the same regardless of choosing the black or the yellow basis. Why is that? My teacher said it is due to the 6-fold symmetry of the lattice while for me it seems like sheer accident.

I know of course that a hexagonal lattice has 6-fold symmetry, which for me means that I can rotate it by 60 degrees about any point and it will look the same. But how do I translate this symmetry property into a rotation of 60 degress of the basis vectors?

I know this question might be obvious to you, but I am having a hard time exactly seeing how to apply symmetry properties. How do YOU intuitively see it?

I know of course that a hexagonal lattice has 6-fold symmetry, which for me means that I can rotate it by 60 degrees about any point and it will look the same. But how do I translate this symmetry property into a rotation of 60 degress of the basis vectors?

I know this question might be obvious to you, but I am having a hard time exactly seeing how to apply symmetry properties. How do YOU intuitively see it?