"Mercator" Projection for 4D Earth?

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"Mercator" Projection for 4D Earth?

Postby PatrickPowers » Fri Aug 10, 2018 12:36 am

The Mercator projection is a stereographic projection onto a cylinder. It was popular because it is a conformal mapping. It preserves angles. This was useful for navigation. It told a sailor what compass bearing to use. They weren't so concerned about distances.

Sailors were able to measure latitude but not longitude. So they would use the Mercator map to get a compass bearing. They would use that to deliberately sail to the east or west of their destination. Let's say east. Then once they got to the correct latitude they would then sail due west to their destination. At least that was the idea. If you tried to sail directly to your destination and missed, then you didn't know whether to go east or west.

So what would be the equivalent for a 4D Earth? I'm hoping already knows a conformal mapping of the 3-sphere to a 3D plane.
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Re: "Mercator" Projection for 4D Earth?

Postby wendy » Mon Aug 13, 2018 12:22 pm

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.
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Re: "Mercator" Projection for 4D Earth?

Postby PatrickPowers » Wed Aug 15, 2018 1:49 am

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.


I've been working with a planet where the periods of rotation are 12 hours and 48 hours.

The magnetic field of our Earth is generated by heat escaping the planet and the Coriolus force. This force is strongest at the poles and zero at the Equator. It seems to me that a planet with equal periods of rotation on each plane -- a Clifford rotation -- would have a weak magnetic field. The Cor force would be equally strong everywhere, so the net result would be zero. There would likely be a local field, but this would not be much use for navigation I would think.

A 12/48 planet would have the Coriolus force strongest near the slow pole and 1/4 the strength near the fast pole. A compass would be a disc that tends to align with the fast plane of rotation. (This would work on the real 3D Earth too, it's just impractical.) You can have a second perpendicular disc attached to it that MIGHT align itself with something useful, but I tend to think that the 2nd stationary point would be too indistinct to be of much practical use. It might take a serious study to get the real answer though.

So the compass would tell you where East and West were and some idea of your latitude, but not necessarily where North was. So you are right, a Mercator might not be that useful. Navigation would be more of a challenge than on 3D Earth. I'll have to think more about how they could find North with stars or Sun or Moon. It seems there should be some way. I have to get out of here in 10 minutes, so it will have to be later.

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.
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Re: "Mercator" Projection for 4D Earth?

Postby d023n » Wed Aug 15, 2018 3:51 am

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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Re: "Mercator" Projection for 4D Earth?

Postby wendy » Wed Aug 15, 2018 7:08 am

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.
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Re: "Mercator" Projection for 4D Earth?

Postby d023n » Wed Aug 15, 2018 7:23 am

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.


Hey, Wendy. You brought this point up in the 4D orbits thread, but I still do not understand the justification for it. Where is the torque coming from exactly? If there really is some sort of bleed between the 2 planes of rotation, that would mean that a 4D object made to spin in just 1 plane of rotation would, after some time, be found to be spinning more slowly in that plane and to be spinning in the other originally stationary plane. This would violate the conservation of angular momentum of each plane.
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Re: "Mercator" Projection for 4D Earth?

Postby PatrickPowers » Wed Aug 15, 2018 10:48 pm

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.


The magnetism thing is confusing because the Heaviside vector calculus that everyone uses works only in three dimensions. So first you have to recast magnetism in Clifford's geometric algebra, which gives the same results but will work in any number of dimensions. Instead of the troublesome cross product the magnetic field is thought of as a bivector field. In 4D it is then a field of two perpendicular bivectors. A bivector is an ordered pair that defines a plane along with a magnitude.

On the 4D planet North is toward the closest point on one of the poles, South is toward the closest point on the other pole. I think a compass wouldn't tell you that. It would tend to align with the faster plane of rotation. This is actually what goes on on our 3D Earth, it just isn't thought of that way. On the 3D Earth knowing East-West is enough to determine North, on 4D Earth more info is needed.

There would be torque on regions of the planet, but isn't that just the Coriolus effect? The surface of our familiar Earth is being torqued all the time, it is different in different regions, but this doesn't seem to consume any energy. On 4D Earth every point on the surface moves with constant speed that depends on the latitude, just like on 3D Earth.

You are quite right, the Sun would move in a complicated way on a 4:1 planet. The height in the sky would be the sum of two sine waves. So you need to take into account the magnitudes, periods, and amplitudes of each wave, plus the phase between the two. Most of these numbers depend on the location on the planet.
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Re: "Mercator" Projection for 4D Earth?

Postby Klitzing » Thu Aug 16, 2018 8:52 am

Speaking of 2 different wave functions and "the phase" between them:
who tells, that those 2 wave functions need to have the same frequencies?
Double-rotations generally would have different ones within either subspace, ain't they?

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Re: "Mercator" Projection for 4D Earth?

Postby wendy » Fri Aug 17, 2018 7:14 am

Two rotations at different frequencies produce a beat. You only talk of phases when the frequency is identical.

As to the rotations.

If you take a thing on a string, you can whirl it around you. The string pulls on your hand, this is matched by the outward centropedal force. This points to the centre of the circle the rotation makes. On earth, the rotation of the earth would point to different points on the polar axis. In a double-rotation, the rotation can be treated as the sum of two single-rotation vectors.

In a clifford rotation, the two rotations amount to r (sin t) w² + r (cos t) w², which makes a vector that points to the centre, as r w² (w is the angular velocity, aka omega). r is the radius, and t is the torus of rotation.

In a general rotation, with velocities v and w, you get r v² (sin t) + r w² (cos t). Because v and w are not equal, the vector representing centropedal force is not pointing to the centre of the glome.

The centropedal force in the direction of the glome-centre, is balanced by gravity. This means that things are not under any adverse or additional force that is not in gravity. 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.
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Re: "Mercator" Projection for 4D Earth?

Postby d023n » Sat Aug 18, 2018 7:04 am

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.


Interesting.

This actually lines up with what I have been saying about stable orbits being possible in higher dimensions, where the acceleration vector (and so on) gets deflected from pointing at the center of mass; that is to say, there is an additional transverse component as you mentioned just now.

In the case of non-orbital rotations though, I had not considered that this additional component would be gradually attenuated by the dispersive influences of the material that is generating the centripetal force, but it does seem to make sense now. Thank you for explaining that! ^_^

However, I do not think that there would be any such corrective mechanism for orbital rotations, meaning that, while planetary rotations would naturally tend toward isoclinic rotations, orbital rotations would retain any perturbations away from an isoclinic orbtial path. So, figuring out the yearly behaviors of the sun and background stars would be rather complicated indeed. Hmm.

---

I wonder just how long Patrick's 12/48 planet would take to stabilize and if a 1:4 ratio is enough to render the planet molten and lifeless. Also, the planet probably would not ever perfectly equalize and would retain some measure of non-isoclinism, meaning that surface would take more than a single rotation to return to the exact same orientation, which would actually make it a good deal harder, or perhaps just a slight bit more tedious, to calculate how the sun would work. Earthquakes and the internal circulation of planetary material would also likely constantly cause little perturbations, shifting how close the planet was to perfect isoclinism.

Heck, the presence of a moon would make all of this even more complicated. In fact, our own moon is increasing its orbital angular momentum by tapping the spin angular momentum of the Earth. This might actually serve as that corrective mechanism for orbital rotations that have been perturbed away from isoclinism, or it might be neutral, causing further deflections just as easily. Again, hmmm.
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Re: "Mercator" Projection for 4D Earth?

Postby PatrickPowers » Fri Apr 12, 2019 2:29 am

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.


Aha, I finally understand what you say. Each latitude torus of the 3-sphere can be unfolded without distortion to a rectangle. We can distort each of these rectangles into a square. These squares are all stacked upon one another. Then we get a cuboid or right cuboid with proportions 1x4x4. All the latitude lines are distorted to be straight.

Unlike our 3D world, the where a full Mercator map of the globe would be of infinite height.
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Re: "Mercator" Projection for 4D Earth?

Postby wendy » Sun Apr 14, 2019 1:08 pm

Mecator projections suppose you can get direction at any point. If we suppose you can measure against the day and year, then the climate is orthogonal to that.

The general projection using the directions given above, is to suppose that the world unfolds into a shape, represented by slicing a cylinder into a tetrahedron, where two poles are diameters of the circular bases, separated by layers of climata.

So suppose you have the tropics at the bottom, represented by the y-axis. You then rotate this axis to the left and right as you rise (as in a double spiral), until they meet again on the x-axis. The height in Z is the climata (from tropical to polar or 0 to 90 lattitude), the axis rotating left is your time axis, and the axis rotating right is your calendar axis.

At any given height, there is a rectangle that exactly folds into a given torus of the sphere.

This is the representation of a 4-sphere, roughly equal in shape to an orthogonal projection of the earth. In practice, you unroll the lines of lattitude to give a pi:1 ellipse. On earth, there is no year-axis, but the north and south roll through the year-cycle 6 months apart. In 4d, you have a full cycle of seasons, in climata.

This could unfold into a 3-block, of size 2pi : 2pi : 1/2, which leads to a surface of 2pi². This distortion has the effect of converting every rectangle into a square, by stretching in one axis.

From this, the projection would be distorted in the height, depending on whether you can separately detect N-S. But I think here that if climata is found by the orthogonal to the other two, there would not be an infinite dialation towards the poles.
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Re: "Mercator" Projection for 4D Earth?

Postby PatrickPowers » Thu Apr 18, 2019 2:16 pm

I think I finally have it figured out.

The reason the Mercator became so dominant is its usefulness for navigation. A line between two points on the map is the rhumb line. That means, measure the angle relative to north, set your compass to that angle, then follow that line.

The Mercator is a projection of the surface of the Earth onto an enclosing cylinder. Start with circle, next a perpendicular line through the center, then take the Cartesian product of the two sets. That's the enclosing cylinder.

In 4D, start with circle, next a perpendicular 2D plane through the center, then take the Cartesian product of the two sets. That's the enclosing cylinder.

I haven't proved that the result has rhumb lines. Maybe later.
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Re: "Mercator" Projection for 4D Earth?

Postby PatrickPowers » Wed Jul 10, 2019 10:04 pm

After a few months I realized that a Mercator projection of a 4D Earth is impossible.

What could be done is two separate Mercator maps, one for each plane of rotation. But such would be used by navigators only. Too much hassle for ordinary people for too little value.

BUT navigation on a 4D planet would be quite different. An Earthlike planet would have a radius of maybe one kilometer. People would be about 1cm high. Walking around the Earth in a month or driving in 24 hours would be quite feasible. Great circle routes become much more important.

Still working on it.
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Re: "Mercator" Projection for 4D Earth?

Postby PatrickPowers » Tue Jul 23, 2019 7:54 pm

So, a 4D Earth with 1km radius and people 1cm high. Rather absurd, but that's what we get. Then the horizon would be 4.5 meters from such a person. Proportional to their height it would be as if it were 900 meters to us. Needless to say, the curvature of the Earth would be quite evident. It would never be believed that such an Earth was flat.
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