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I was reading some article on angular momentum.

And at some point, the author started talking about the symmetric objects and axis of symmetry. Now I am wondering if the author means the geometrical symmetry or the symmetry in physics. For an example, if we take a uniform rod of length L, then the center of mass lies at the geometric center and the geometric axis of symmetry is same as the physical axis of symmetry. But if the mass is not distributed uniform on the length of the rod, then the center of mass is not at the geometric center and geometric axis of symmetry is not same as the physical axis of symmetry. Similar argument can be applied to any other geometrical symmetric object which might have mass evenly/unevenly distributed. I am little confused here. Wikepedia's article on symmetry probably means geometric/mathematical symmetry. Can somebody please elaborate ?