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Rotation of the 3-sphere

A 3-sphere has two planes of rotation. The two planes are always at right angles to one another, and can rotate with completely different periods. I want to make an animation that shows this. It's been done, but this as far as I know is a different way.

The idea is analogous to animating a 2D map of the surface of a 2-sphere like the Earth. The image would move from west to east. As portions of the image move over the eastern boundary, they reappear in the west. So first we make a 3D map of the surface of the 3-sphere, then animate it to show the rotations.

The 3-sphere can be parameterized with polar coordinates as sin(x)e^it, cos(x)e^iu, with 0<=x<pi/2, -pi<=t,u<pi. Map to the three axies of R^3 with x, sin(x)t, and cos(x)u. One may think of this as specifying a rectangle for each value of x, with the rectangles stacked along the x axis. For each such rectangle the double 4D rotation causes the points to move in a diagonal direction. This direction is the same for every point in that rectangle. Each rectangle is topologically a torus, so movement over an edge means reentry over the opposite edge.

If I ever get around to making a computer graphic, I suppose I'll have the vertices of 4D Platonic polyhedral on the surface of the 4-sphere, with "straight" lines between the vertices. Rotate and display the results on the 3D map.

Could any recommend a graphics package that is capable of this? If I go to all this effort then I'll want to distribute it. For that it would be necessary to render a video.

Comments

  • 1.
    edited July 2014

    Hello

    Hopf Fiberation of 3-sphere

    It might take a few seconds to load, the controls are primitive but you get the idea

    The code is there for you to review, one could easily change the parametrization. One could add translucent torus...

    You could Export a movie or you could ray trace an animation, or you could export the geometries in polygon form and have another package render or ray trace.

    Dara

    Comment Source:Hello [Hopf Fiberation of 3-sphere](http://mathematica.lossofgenerality.com/2012/08/31/hopf-fibration/) It might take a few seconds to load, the controls are primitive but you get the idea The code is there for you to review, one could easily change the parametrization. One could add translucent torus... You could Export a movie or you could ray trace an animation, or you could export the geometries in polygon form and have another package render or ray trace. Dara
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