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Hopf Fibration by bugman123 Hopf Fibration by bugman123
The Hopf map is a special transformation invented by Heinz Hopf that maps to each point on the ordinary 3D sphere from a unique circle of points on the 4D sphere. Taken together, these circles form a fiber bundle called a Hopf Fibration. If you apply a 4D to 3D stereographic projection to the Hopf Fibration, you get a beautiful 3D torus called a Clifford Torus composed of interlinked Villarceau circles. By applying 4D rotations to the Hopf Fibration, you can transform the Clifford Torus into a Dupin cyclide or you can turn it inside-out. As it turns out, there is a single point of rotation in 2D, there are 3 axes of rotation in 3D (x, y, z), and there are 6 planes of rotation in 4D (xy, xz, xw, yz, yw, zw).
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Submitted on
April 14, 2011
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