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