4D planets (split from "Rings in 4d")

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.

Re: 4D planets (split from "Rings in 4d")

Postby wendy » Fri Jul 31, 2015 12:46 pm

Should not be too hard, it's just mental arithmetic. Even without equi-partition, you would get something pretty close, simply because there would be a generous amount of transverse acceleration, and hence torque to transfer the energy.

But a divergence of 1% would just produce a wobble, and you pretty much get seasons.
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Re: 4D planets (split from "Rings in 4d")

Postby Teragon » Sat Aug 01, 2015 10:13 am

wendy wrote:Even without equi-partition, you would get something pretty close, simply because there would be a generous amount of transverse acceleration, and hence torque to transfer the energy.


Ok, you have to explain that. Transversal to what?
It highly depends on the exact structure of the solar system. Firstly, why should planets and moons always arrange in a way that the slower angular momentum of each body tends to be kept or speed up, while the faster angular momentum tends to be slowed down. It would be hard to demonstrate that it's possible at all, still less that any solar system will arrange that way. Secondly, if both angular momenta eventually align, their speeds will simply diverge again, because the tidal forces remain largely the same. Tidal forces do not take energy from angular momentum A and transfer it to B. That would violate the conservation of angular momentum. They take energy from angular momentum A and transfer it to the orbital angular momentum (or vice versa in more seldom cases). There's no way planets in general come close to Clifford rotation! Some may, some may not.
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Re: 4D planets (split from "Rings in 4d")

Postby fallfromgrace » Sat Aug 01, 2015 9:04 pm

It seems that single rotation is possible in cases where equatorial orientation matches orbital orientation, otherwise the chances would be that double rotation happens.
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Re: 4D planets (split from "Rings in 4d")

Postby wendy » Sun Aug 02, 2015 7:21 am

In four and five dimensions, there are two orthogonal modes of rotation, and hence two degrees of freedom. In this discussion, w provides the transverse force.

A sphere rotating at different frequencies on the polars, can be thought as rotating at speeds c+w and c-w, where c is a clifford rotation, and w is the wobble. The total momentum is then 2mc, the total energy is c²+w². The momentum is kept as w -> 0, the energy dissapates as friction, or heat. A modest value of w, would tend to become heat at 2cw/j, where j is the mechanical equivalent of heat. For a value of w = 100 kms, and c = 1666 kms, we get in the order of 3°C of heating in the ground.

At the local level, c provides a centrifugal force which is in line with gravity, and w comes as a force perpendicular to direction of rotation and gravity (since they are both accelerations), and this force tends to be a turning force in the ground. It is this force that vanishes as heat.
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Re: 4D planets (split from "Rings in 4d")

Postby Teragon » Sun Aug 02, 2015 8:37 pm

Force is acceleration times mass, w is a velocity, so there's some explanation needed.

A force perpendicular to the direction of rotation and to the direction of gravity is not determined at all:
There's a whole plane of possible vectors left, all of which are the same distance to the polar equator at the solar equator. As the planet shape is symmetric under rotations in the polar plane, it's impossible that there is a net force.

A rigid body isn't able to disperse its angular momentum on it's own. Through atmospheric circulation or mantle convection it is, although it's a very slow process and it applies to both rotations. I can't see why the faster rotation should lose so much more energy than the slower one that both angular velocities approach in finite time through this process (even though I don't know).

If equatorial rotation starts at 2000 km/h and polar rotation at 1000 km/h, quite a large difference, but possible, how long would it take according to your calculation for both velocities to align?

When Clifford rotation is present, the natural shape of a planet because of the gravity field and the centrifugal force is not a ellipsoid anymore, it's the S3. It came to my mind that in such a state, a planet could be turned around freely by tidal forces, because the angular momentum points in any direction and is conserved, no matter on which latitude some place on it lies. That means when a planet reaches a very fine balance between the two rotational velocities, it may experience an additional cycle, where places on the planet travel through the different climate zones... because of the moon!
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Re: Rings in 4d

Postby PatrickPowers » Wed Dec 02, 2015 3:09 am

quickfur wrote: completely disregard the solar plane altogether, and consider a planet in double rotation alone. Let's assume that both rotations proceed at equal rates (it seems reasonable to assume that tidal forces or something along those lines would tend to equalize both rotations over time). Then there is a 2D toroidal sheet wrapped around the planet's surface, which experiences the greatest total velocity. This sheet corresponds with the ridge of the duocylinder. By analogy with a 3D planet's equator, which constitutes the points with greatest velocity, wouldn't it be reasonable to call this sheet the "equator"? Then the two interlocked circles where one of the rotations is not felt would be the rotational poles. They will not be stationary, but they will be the points on the planet with the least velocity.



Hmm. If both rotations proceed at the same rates, then every point on the surface rotates at the same rate. The period of rotation of every point is the same, and the distance traveled during each rotation is also the same for every point.
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Re: Rings in 4d

Postby PatrickPowers » Wed Dec 02, 2015 3:09 am

quickfur wrote: completely disregard the solar plane altogether, and consider a planet in double rotation alone. Let's assume that both rotations proceed at equal rates (it seems reasonable to assume that tidal forces or something along those lines would tend to equalize both rotations over time). Then there is a 2D toroidal sheet wrapped around the planet's surface, which experiences the greatest total velocity. This sheet corresponds with the ridge of the duocylinder. By analogy with a 3D planet's equator, which constitutes the points with greatest velocity, wouldn't it be reasonable to call this sheet the "equator"? Then the two interlocked circles where one of the rotations is not felt would be the rotational poles. They will not be stationary, but they will be the points on the planet with the least velocity.



Hmm. If both rotations proceed at the same rates, then every point on the surface rotates at the same rate. The period of rotation of every point is the same, and the distance traveled during each rotation is also the same for every point.
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Re: 4D planets (split from "Rings in 4d")

Postby PatrickPowers » Wed Dec 02, 2015 4:11 am

I became interested in maps of a 4D planet. A map would be 3D, so it would be possible to create that shape in our world.

I assumed that the periods of the two rotations would be different so it would be easy for inhabitants to identify the planes of rotation. From a point on the plane of rotation objects in the sky move in simple curves.

Each point on the surface of a 4D planet can be described with three coordinates. There are two longitudes, one for each plane of rotation, precisely analogous to longitude on Earth. Latitude is somewhat different from what we know here, since a 4D planet would not have a northern and southern hemisphere. There would be 90 unsigned degrees of latitude instead of +-90 as we have.*

To make the map one unrolls both equators/poles into a line, just as is done on Earth maps. In the map each equator is perpendicular to the other like a +. The equators do not intersect because they are separated in the third dimension, which we'll call height. The height is 90/360 = 1/4 of the length of either equator.

At each equator the width of the map is zero. As the latitude increases away from an equator, the width of the map grows proportional to the sine of the latitude. The length of the map shrinks proportional to the cosine of the latitude.

I couldn't visualize this shape, so I made some. I've given them away, so I don't have one now. No one seems to find the shape familiar. It has an unusual symmetry.

I believe it is a 4D sinusoidal projection, with the property that areas of surface features are preserved.

Image

It would be possible to make an animation of a rotating planet in such a projection. Every point in the projection moves in a straight line as the planet rotates. This makes it clear that it is a rigid rotation.

----

*If x is the shortest distance to one equator/pole and y the shortest distance to the other, then latitude is arctan(x/y).
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Re: 4D planets (split from "Rings in 4d")

Postby wendy » Fri Dec 04, 2015 1:06 am

Even if the planet were spinning at a constant velocity, you will still get three discrete dimensions, which are easy to tell.

In essence, they correspomd to

1) longitude, that is, time zones.
2) climatude, that is, hot and cold. This goes from 0 to 90, as the earth.
3) annatude, that is, season-zones, where in 3D, the N pole is 6 months different to the S pole, in 4D the full year exosts.

These make in the main, a 2p * p * 1 prism, which leads to 2p^2.
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Re: 4D planets (split from "Rings in 4d")

Postby PatrickPowers » Fri Dec 04, 2015 4:22 pm

But you are assuming that the planet orbits a sun. I read that in 4D orbits are not stable, so this is not a possibility. Stars only.

I can imagine the inhabitants coming up with a non-orthogonal system based on important features. There are infinitely more non-orthogonal systems then orthogonal ones, so the Bayesian prior is to have non-orthogonal. Maybe it would be close to orthogonal, close enough for most purposes, so it is never worth while to correct it to a logical system. Maybe the circles would be chosen to follow an important coast line or mountain range, like the Pacific coast on Earth which is close to a straight line.

People tend to evolve illogical systems, like the British measuring and monetary schemes. The Ptolemaic orbits were used for quite some time, and there was resistance to change.

So I'm not at all confident that inhabitants would use a sinusoidal projection for a map. Such is rare on Earth. The Mercator projection is the most common, because it was the most useful for sailing with a compass. What sort of map would aid navigation on a 4D planet? I don't know what the magnetic field would be like, so I have no guess.
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Re: 4D planets (split from "Rings in 4d")

Postby wendy » Sat Dec 05, 2015 11:50 am

Even in the absence of a sun, the people on the planet can construct a lattitude sphere, and a longitude, according to the rising and setting of distant stars.

But one can always in the absence of better physics or maths, assume the presence of a sun. It helps enormously in figuring out things. Maybe the field is driven by a pair of point forces on a star, or by some other mechanism. We still don't understand gravity in 3d, although we use the radiant flux model of Netwon et al, this is the first approximation, and all that we need to get to the planets and asteroids.
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Re: 4D planets (split from "Rings in 4d")

Postby granpa » Sat Dec 19, 2015 6:36 pm

Keiji wrote:
Eric B wrote:Marp and Garp? Never heard of that one! I imagine that is ana and kata changed into global circles. A second kind of lattitude or longitude?


Using the definitions from Alkaline's glossary:
  • In 3D, a planet has an equator. The direction which the planet rotates in is called east; the opposite direction is west.
  • In 4D, the analogous equator is called the solar equator - it is the one most aligned with the plane of its orbit around its star. The planet still rotates around this equator, and the terms east and west retain the same meaning.
  • There's another equator for 4D planets, perpendicular to the solar equator, which is called the polar equator. The planet rotates around this equator independently from the other one, and the direction which the planet rotates in this equator is called marp. The opposite direction is garp.
  • This leaves only one axis not yet defined - so we can define this axis to be perpendicular to both east/west and marp/garp, and we call the directions in this axis north and south. In 3D, we could define north as being 90 degrees counter-clockwise from the direction east; in 4D we could define it as the direction of the vector n = e×m where e and m are vectors pointing in the east and marp directions respectively (the cross product works here because the surface of a 4D planet is 3D).

In 4d there is no axis of rotation
There are 2 planes of rotation at right angles to each other
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Re: 4D planets (split from "Rings in 4d")

Postby PatrickPowers » Tue Feb 20, 2018 7:13 am

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Re: 4D planets (split from "Rings in 4d")

Postby PatrickPowers » Tue Feb 20, 2018 7:32 am

wendy wrote:
The force of rotation (corilous) is not well understood in 4d, since the cross product has not been epanded.


It can be done with Clifford algebra, which works in any number of dimensions. Instead using the 1D dual space, in this case of 4D Coriolis force one would explicitly mention all three vectors.

However I don't understand the Coriolis force even in 3D, so I can't be of much help.
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Re: 4D planets (split from "Rings in 4d")

Postby PatrickPowers » Tue Feb 20, 2018 8:23 am

wendy wrote:We still don't understand gravity in 3d, although we use the radiant flux model of Netwon et al, this is the first approximation, and all that we need to get to the planets and asteroids.


Stable orbits exist only with the inverse square gravity law. I am told that the "smart money" believes that in 4D gravity would follow in inverse cube law, which has no stable orbits.

Simple magnets follow an inverse cube law. If you try you will quickly get a sense that it isn't possible to get a free floating magnet to orbit another magnet. They either deflect one another's path slightly or spiral in quickly to collision.
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Re: Rings in 4d

Postby PatrickPowers » Tue Mar 06, 2018 2:33 pm

"[*]A planet has a single toric equator. This contains the points of greatest rotational velocity on the planet."

The points of greatest angular velocity lie at a latitude of arctan(a'2/b'2) where a and b are the periods of rotation of the poles. The latitude of the equator is 45 degrees or pi/4. So the equator has the greatest angular velocity only for Clifford rotations.
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