wendy wrote:While Mercartor preserves angles, it does so at the expense of many other features. The 'conformal' projection is the stereographic, which exists in 4d also. If you are looking for a map of the planet, rather than just a conformal mapping, then it is necessary to understand how the planet rotates, and what the navigator might see.
There are three axies, representing longitude (east-west), lattitude (north-south and two others). If the planet is in equal rotation, then one can reasonably determine the lattitude by way of a sky-sphere, marked out with the brightest stars. This sphere is placed perpendicular to the east-west axis, and the surface is orientated so that a ray through the sphere will point to the given star, when that star cumulates.
When longitude and lattitude is taken to account, each point on the sky-sphere corresponds to a full great circle on the earth, the 4-earth rotates without disturbing this order. The sun moves through the sky throughout the year, which means that it is overhead on a different great circle every day. The shape of this on the sky-sphere is a circle, representing on the planet-surface a torus.
The torus is wrapped around and equidistant from a circle, which we shall call the south circle, and opposite this is the north circle. As on the earth, we have the south representing the place of the sun (ie hot or humid country), and the north representing the colder climates. These are represented as the south and north poles on the sky-sphere, but are full lines of longitude.
The lines that run from the north pole to the south pole on the sky-sphere represent the parts where the sun will cumulate through the year. Where it crosses the places where the sun reaches the zenith or directly above, are the tropics. In 3d we have only the tropics of cancer and capricorn, but the 4d earth has the full complement: tropic of leo and tropic of aries etc.
The same distance from the north pole on the sky-sphere represents the artic torus. Unlike 3d, the artic and antartic are connected, so it is possible that penguins will have to cope with polar bears. On a given day, the sun will hug the horizon all day on the artic circle, but further north, it will rise and set as normal.
The fifth and sixth directions represent an axis of seasons. Just as longitude marks out time zones, this axis marks out the season-zones. In 3d earth, this is a two-directional pointer, which points to seasons in the north, and six months ahead in the south. This axis represents the full gammit of seasons.
If you place all this together, the unfolded glome becomes a tetrahedron, with the south and north circles becoming opposite edges. Any given slice between these becomes a rectangle, where the east-west lines run parallel to one diagonal, and the calendra (year-zones), run the opposite direction. The climata is the axis between the top and bottom.
If one supposes an east-west axis (longitude) first, the effect is to twist the tetrahedron so that the top and bottom become parallel. This will in the long run, turn the tetrahedron into a sphere-prism. If the calendra is now stretched rectangularly, you will end up with a box or rectangular prism, whose three sides represent the separate longitude, calendra and climata coordinates. The distortion of the surface is that at the south pole, the cells formed by the crossing of longitude and calendra will line up in one direction, and at the north pole, they will have rotated to the opposite direction.
Mercartor relies on being able to reliably measure north-south through a compass. This is transferred to a map, where the angles of the compass are preserved. It then becomes of some interest on how one might determine directions in four dimensions to get this to work.
PatrickPowers wrote:All that being said, if we do have a Mercator my main question is what sort of surface to project onto. For the Mercator it is a cylinder. We can't use the 4D version that has a 3D ball as its cross section. A torus of some sort seems like the way to go. I was confused for some time, because a 4D torus is often considered to be 2D. That won't work. It's got to be a projection onto a 3D surface that easily unfolds/uncurls to flat. I was hoping someone would know what that was.
wendy wrote:I would doubt that having a 1:4 ratio of rotation would work. These are pairs of modes of energy, and the equi-partition would cause them to speed up and slow down, such that something like 2:2 would be the order of the day. In essence, there is a lot of torque being applied to the surface, to effect the two rotations, and this would cause tides (aka earthquakes), in the rock. It would be a good deal harder to calculate how the sun would work, if the solar cycle is not aligned to the rotations of the earth.
wendy wrote:It depends really how you imagine that your magnetic field is going to work. Mercartor is relevant because there is a point that it normally points to (magnetic north/south). For example, if the magnetism is a gradient between two circles on the sphere then the compass would point in the direction of gradient, ie the shortest distance between the two circles.
If one were to suppose that the magnetic field arises from moving charge, and that such charge are thrown towards two opposite circles, the remaining question is how do we make these opposite circles align with the climata-rings. If they don't, then one has something like in 3d, where the rotation-poles still exist, but the magnetic poles are at cairo etc.
I would doubt that having a 1:4 ratio of rotation would work. These are pairs of modes of energy, and the equi-partition would cause them to speed up and slow down, such that something like 2:2 would be the order of the day. In essence, there is a lot of torque being applied to the surface, to effect the two rotations, and this would cause tides (aka earthquakes), in the rock. It would be a good deal harder to calculate how the sun would work, if the solar cycle is not aligned to the rotations of the earth.
wendy wrote:In the general case, there is an additional transverse force, which seeks to move things relative to each other. This is a tidal effect, where the rocks are placed under an unbalanced (and moving) force.
d023n wrote:PatrickPowers wrote:All that being said, if we do have a Mercator my main question is what sort of surface to project onto. For the Mercator it is a cylinder. We can't use the 4D version that has a 3D ball as its cross section. A torus of some sort seems like the way to go. I was confused for some time, because a 4D torus is often considered to be 2D. That won't work. It's got to be a projection onto a 3D surface that easily unfolds/uncurls to flat. I was hoping someone would know what that was.
A cylinder is just a 2-torus that hasn't been fully wrapped up, or a 1-torus prism; so I think that a 3-torus that hasn't been fully wrapped up, or a 2-torus prism, would work here.
For the 3D Mercator, this results in a 2D rectangular map that operates like a normal torus in the East-West directions but where the North and South poles have become stretched into the long edges of the rectangle.
For the 4D Mercator, the result would be a 3D square prism map where each square slice corresponds to a 2-torus of the 3-sphere. The middle square slice corresponds to the only symmetrical 2-torus, which, like the equator of the 3D Mercator, would be the only undistorted portion of the map, while the 2 square faces of the prism correspond to the 2 degenerate 2-tori, which are the most distorted portions of the map, being 1D regions stretched out into 2D regions.
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