Stereoscopic 4D Klein Bottle
62.5 هزار بار بازدید -
11 سال پیش
-
This animation is a stereogram,
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.
The 3D image you see is the "shadow" of a four dimensional Klein bottle that has no self-intersection, which needs a minimum of four dimensions to achieve. A Klein bottle is a surface which is similar to a Mobius strip in that it only has one side, except the Klein bottle is completely closed; it has no edges. If you think of the Mobius strip as an annulus (flat cylinder) that twists in the third dimension, you can think of the Klein bottle as a torus (donut shape) that twists in the fourth dimension.
The first half of the animation shows a straight orthogonal projection into 3D space, and the second half shows a stereographic projection by normalizing all points to the unit 3-sphere and then projecting into R^3 with the usual stereographic projection map.
Although it appears that the Klein bottle is changing shape, it is actually just rotating in four dimensional space. This is similar to when you rotate a 3D object and look at its 2D shadow. Although the object stays the same shape, its shadow looks like it is deforming.
Angular velocity for rotations was determined by a random walk in R^6 (there are 6 planes of rotation in 4D space), and primary color was determined by a random walk in the unit cube (for RGB values).
This was made in Mathematica 9.0.
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The music is "Let Yourself Be Huge" by Cloudkicker, available under the Creative Commons Attribution license.
http://cloudkickermusic.com/album/let...
http://creativecommons.org/licenses/b...
http://cloudkickermusic.com/
http://blog.cloudkickermusic.com/
SoundCloud: bmsharp
The 3D image you see is the "shadow" of a four dimensional Klein bottle that has no self-intersection, which needs a minimum of four dimensions to achieve. A Klein bottle is a surface which is similar to a Mobius strip in that it only has one side, except the Klein bottle is completely closed; it has no edges. If you think of the Mobius strip as an annulus (flat cylinder) that twists in the third dimension, you can think of the Klein bottle as a torus (donut shape) that twists in the fourth dimension.
The first half of the animation shows a straight orthogonal projection into 3D space, and the second half shows a stereographic projection by normalizing all points to the unit 3-sphere and then projecting into R^3 with the usual stereographic projection map.
Although it appears that the Klein bottle is changing shape, it is actually just rotating in four dimensional space. This is similar to when you rotate a 3D object and look at its 2D shadow. Although the object stays the same shape, its shadow looks like it is deforming.
Angular velocity for rotations was determined by a random walk in R^6 (there are 6 planes of rotation in 4D space), and primary color was determined by a random walk in the unit cube (for RGB values).
This was made in Mathematica 9.0.
-------------------------------------------------------------------------
The music is "Let Yourself Be Huge" by Cloudkicker, available under the Creative Commons Attribution license.
http://cloudkickermusic.com/album/let...
http://creativecommons.org/licenses/b...
http://cloudkickermusic.com/
http://blog.cloudkickermusic.com/
SoundCloud: bmsharp
11 سال پیش
در تاریخ 1392/06/02 منتشر شده
است.
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