PatrickPowers wrote: The phase has a bigger effect on the climate than does latitude, so phase would be the dominant concept. Of the three primary directions, one would be the direction in which neither phase nor amplitude changes. Call it the iso or cis direction. The second would be the direction transphase or mutaphase, in which phase changes most rapidly with no change in amplitude. The third would be the direction transamp or mutaamp in which amplitude changes most rapidly but phase remains constant. This corresponds to our latitude. But who could forget the Pontiac Firebird Transam?
wendy wrote: Let#s look at a rotating 4D earth, using isoclinal rotation. This is what would happen by tidal effects if a non-isoclinal rotation was started.
mr_e_man wrote:wendy wrote: Let#s look at a rotating 4D earth, using isoclinal rotation. This is what would happen by tidal effects if a non-isoclinal rotation was started.
You keep saying that. But what happens to the angular momentum? The total cannot change. If it starts non-isoclinic, it stays non-isoclinic.
Are the "tidal effects" within the earth, or between the earth and the sun? I suppose the angular momentum could be traded between them until the sun is left-isoclinic and the earth is right-isoclinic, or vice-versa. But there shouldn't be a sun at all, because 4D orbits are unstable.
Does it go into water on the surface, causing gigantic whirlpools? Does it set the earth's interior flowing, causing magnetic fields?
wendy wrote:Let's look a little closer into 4d rotation.
All rotation, isoclinal or not, can be described as the join of two orthogonal rotations, in the wx and yz hedrices (ie 2-spaces). This motion will cause any star to be overhead in a given 'torus', because the relative ratios of r(wx) and R(yz) is unchanged, and a torus corresponds to R/r.
If the rotations are unequal, the rotation forms a helix or spring-shape in the torus, going around one way more than the other. The projection of this rotation into the xy space gives a Lissajour or Bowditch figure. It should be remembered that this motion has both radial and transverse accelerations, the radial acceleration is countered by gravity, but nothing is acting against the transverse rotation. In effect, the two rotations are a couple, and the interlinkage of the transverse rotation will cause them to fall into the same resonance or rate. This is why I hold that planets tend to isoclinal rotations. You also have to suppose that planets are not 'solid', but rather can be treated as liquid drops at that scale.
PatrickPowers wrote:As noted elsewhere, a hyperdimensional sphere has much more surface curvature per unit volume. Even a 4D planet would be quite noticeably curved locally. Great circle routes would prevail, as they would be significantly shorter than the loxodromes we get on the Mercator projection. So you would want a map in which great circle routes are straight lines. All and all, you would want a system that emphasizes great circles.
mr_e_man wrote:PatrickPowers wrote:As noted elsewhere, a hyperdimensional sphere has much more surface curvature per unit volume. Even a 4D planet would be quite noticeably curved locally. Great circle routes would prevail, as they would be significantly shorter than the loxodromes we get on the Mercator projection. So you would want a map in which great circle routes are straight lines. All and all, you would want a system that emphasizes great circles.
What about the gnomonic projection?
or the usual spherical coordinates? If we want the coordinate lines (where two of the three coordinates are constant) to be great circles, we can "diagonalize" the second type:
w = cos a cos(b + c)
x = cos a sin(b + c)
y = sin a cos(b - c)
z = sin a sin(b - c)
or in my notation,
x = (e1cos(b + c) + e2sin(b + c))cos a + (e3cos(b - c) + e4sin(b - c))sin a
PatrickPowers wrote:With the spherical you are mapping the 3sphere to a flat 3 ball. Wild!
mr_e_man wrote:Your argument here viewtopic.php?f=27&t=2325#p26238 is more clear to me. The centrifugal "force" from a non-isoclinic rotation has a component parallel to the surface of the earth. But that would just change the shape of the earth, not its angular momentum. In 3D, this causes the bulge at the equator, the deviation from a perfect sphere to an oblate spheroid. An irregularly-shaped object's angular velocity will precess around its constant angular momentum, so they're parallel on average. (By "parallel" I mean that the bivectors are proportional.) They're still non-isoclinic.
Return to Higher Spatial Dimensions
Users browsing this forum: No registered users and 4 guests