Path of Stars in 6D

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Path of Stars in 6D

Postby PatrickPowers » Fri Dec 04, 2015 5:27 pm

Suppose one is on the surface of a sixD planet. What would the path of a star appear to be?

I think it would look roughly like what is known as a 3D lissajous orbit. One may find easily free software that generates them. Essentially they are three sine waves modulating a motion. One may imagine oneself in the center of the 3D figure.
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Re: Path of Stars in 6D

Postby Teragon » Sun Dec 06, 2015 9:48 pm

Interesting question. It would take six dimensions to depict the path as it is. In 2D and 3D the path is 2D, in 4D and 5D (double rotation) the path is already embedded in 4D. In 6D the path of the sun lies on the 6D equivalent of a Clifford torus (which has a 3D surface). Projected to 3D you can think of it as a spiral closed to a spiral closed to a circle. In reality the three planes of the three loops do not change their orientation in the course of the path. Every loop plane is entirely perpendicular to the other two planes. It must be so much more beautiful in 6D!
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Re: Path of Stars in 6D

Postby wendy » Mon Dec 07, 2015 3:26 am

In even dimensions, on the theory of equipartition, the stars go straight across the sky in great circles. There are no fancy curves here.
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Re: Path of Stars in 6D

Postby PatrickPowers » Mon Dec 07, 2015 4:28 pm

wendy wrote:In even dimensions, on the theory of equipartition, the stars go straight across the sky in great circles. There are no fancy curves here.


That is true. In the 3D lissajous orbit with all periods and amplitudes equal a circle is generated.
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Re: Path of Stars in 6D

Postby Teragon » Mon Dec 07, 2015 5:45 pm

wendy wrote:In even dimensions, on the theory of equipartition, the stars go straight across the sky in great circles. There are no fancy curves here.

But only in this very special case. (And according to several arguments an unlikely one.)
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Re: Path of Stars in 6D

Postby PatrickPowers » Mon Dec 07, 2015 6:40 pm

Teragon wrote:Interesting question. It would take six dimensions to depict the path as it is. In 2D and 3D the path is 2D, in 4D and 5D (double rotation) the path is already embedded in 4D. In 6D the path of the sun lies on the 6D equivalent of a Clifford torus (which has a 3D surface). Projected to 3D you can think of it as a spiral closed to a spiral closed to a circle. In reality the three planes of the three loops do not change their orientation in the course of the path. Every loop plane is entirely perpendicular to the other two planes. It must be so much more beautiful in 6D!


What I'm interested in is how the path would appear to an observer on the surface of a planet. On our Earth the path of every star appears roughly to be a circle. It appears to be a curve embedded in a 2D space.

In 4D the case where the two periods differ is the more interesting. At each pole one sees stars moving in great circles. Close to a pole it would be a helix with its axis that great circle. A 3D observer would presumably see its projection, which could be a great circle modulated by a sine wave. At the equator I'm not sure what the observed path is.

The circle modulated by a sine wave is obtainable with the 3D lissajous figures. That gave me an idea. Two of the dimensions are making a circle, so they are actually a single complex dimension. A second complex dimension would give paths on a torus. A third complex dimension perpendicular to the first two would give as good of a projection of a 6D orbit as one could hope for, I would think.
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Re: Path of Stars in 6D

Postby PatrickPowers » Mon Dec 07, 2015 7:02 pm

Teragon wrote:
wendy wrote:In even dimensions, on the theory of equipartition, the stars go straight across the sky in great circles. There are no fancy curves here.

But only in this very special case. (And according to several arguments an unlikely one.)


The Bayesian prior is that the equal period orbits are unlikely. But according to general arguments of physics the planet will converge to equal period orbits. In other words, orbits of differing periods are unstable.

I could get fancy and talk about entropy, but it boils down to this. A 4D planet rotating in only one plane will deform. This deformation contains potential energy. Objects "prefer" positions of less potential energy and will go there if they can. So the planet "wants" to become a sphere.
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