Hopf Fibration Stereogram (Divergent)
This animation is a stereogram, so if you view the left image with your left eye and the right image with your right eye, it should produce the illusion of being 3D. Focusing your eyes properly for this effect can take some practice. If you're unsure how to do this, I would recommend viewing on a small screen (like a smart phone) and start with it very close to your eyes and gradually move it away until it comes into focus.
If you prefer crossing your eyes to view stereograms, visit this link for a convergent version of this video: Hopf Fibration Stereogram (Convergent)
http://en.wikipedia.org/wiki/Hopf_fib...
This animation was created by first subdividing the vertices of an icosahedron to generate points on the unit 2-sphere. These points are rotated in 3-dimensional space and then mapped into 4-dimensional space as circles via the Hopf map.
To visualize how points on the 2-sphere correspond to circles under the Hopf map, I recommend this excellent video: Hopf fibration -- fibers and base
These circles are then rotated and then projected back into 3-dimensional space with stereographic projection.
The color is assigned based on the initial spherical coordinates of each point and remains constant throughout the video.
The specific rotations were obtained by generating a (smoothed) random walk on the 9 dimensional manifold RP³×S³×S³, where RP³ corresponds to the possible rotations on the input points (SO(3)), and S³×S³ corresponds to the set of rotations on the output circles (SO(4)).
This was created with F# and Mathematica, and then rendered in Unity 4.5.
If you prefer crossing your eyes to view stereograms, visit this link for a convergent version of this video: Hopf Fibration Stereogram (Convergent)
http://en.wikipedia.org/wiki/Hopf_fib...
This animation was created by first subdividing the vertices of an icosahedron to generate points on the unit 2-sphere. These points are rotated in 3-dimensional space and then mapped into 4-dimensional space as circles via the Hopf map.
To visualize how points on the 2-sphere correspond to circles under the Hopf map, I recommend this excellent video: Hopf fibration -- fibers and base
These circles are then rotated and then projected back into 3-dimensional space with stereographic projection.
The color is assigned based on the initial spherical coordinates of each point and remains constant throughout the video.
The specific rotations were obtained by generating a (smoothed) random walk on the 9 dimensional manifold RP³×S³×S³, where RP³ corresponds to the possible rotations on the input points (SO(3)), and S³×S³ corresponds to the set of rotations on the output circles (SO(4)).
This was created with F# and Mathematica, and then rendered in Unity 4.5.
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