Two ways of visualization of three-dimensional sphere surface in the Four-Dimensional Euclidean Space.
Sphere, a geometrical shape is too symmetric, flat and not eye-catching. Sphere in four dimensional space is even more difficult to imagine. In contrast with its simplicity in the mathematical description , the complicity of its visualization is not only purely impressive, but also suggestion of where the beauty comes from.
Seeing it in Hyper spherical coordinate system is a kind of analogy of 2-dimensional sphere surface in three-dimensional space. On the other hand, the formula in Hopf Coordinates System is more symmetric and looks like non-human way. More surprisingly, that is also invented by a man, Heinz Hopf.