Map of a 4D Earth

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.

Map of a 4D Earth

Postby PatrickPowers » Tue Jul 23, 2019 8:38 pm

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.

My best guess is a map shaped like a particular type of croissant, curved so that the two ends touch.
Croissant.jpg
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I don't trust the attachment to work, so here's a link to the image. http://www.creationfood.ca/wp-content/u ... ssant1.jpg This is a disgustingly sugar glazed mass produced frozen croissant, but that can't be helped. The axis of the croissant has been bent into a circle which is within one plane of rotation we'll call Red, the maximum girth perpendicular to that axis is the other plane of rotation we'll call Blue.

The Blue plane of rotation isn't distorted at all on the map. On routes confined to Blue straight lines are indeed great circles.

The outer limits of the Red plane are stretched out, the inner limits squished.
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On a planet with an isoclinic rotation a compass is of no use for navigation. There is no such thing as north and south. Instead the basic concept is phase. At every location on the planet the movement of the sun is the sum of two sine waves with a given phase and amplitude that never changes. (The phase relative to the sun changes, but that's different. The relative phase between two locations never changes.) 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?

I like mixing languages and going with isophase, phasemorph, and transamp. Or iso, morph, and trans for short. Nyuk nyuk nyuk. Iso is sort of like East-West, trans is kind of like North-South, and morph is like nothing that is familiar. On a planet with isoclinic rotation, going steadily any one of these three directions always makes a great circle.
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Re: Map of a 4D Earth

Postby PatrickPowers » Sun Aug 18, 2019 1:32 pm

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?



I found out that on a planet with isoclinic rotation this approach will not work. The direction of iso is useful. It could be found using a compass. But the other two directions are hard to distinguish. The idea that one direction is phase and the other amplitude won't fly: you can't tell them apart. Indeed, the planet itself seems lend no clue. It is necessary to refer to a heavenly body. The Sun is the obvious choice.
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Re: Map of a 4D Earth

Postby PatrickPowers » Tue Aug 20, 2019 3:13 am

Here's how it could work.

Start wi9th the direction in which an observer on the surface of the planet is moving as it rotates. It corresponds to our east-west. The sun rises in the east, in a statistically average sense. It is also the direction that a compass would show.

The ecliptic interacts with the planet in a similar way as to our Earth. (I'm not certain of this, but believe so.) That is, it has a limited extent. There are some parts of the planet where on some day the Sun will appear directly overhead, and other parts where it never does. The middle of this limited range compares with out Equator. That is, on a particular equinoctial day the Sun appears directly over a great circle. (This great circle is NOT yet an equator because a great circle doesn't have enough dimensions to partition the planet. But it's a start.) Similarly we have a sort of tropic of Cancer and of Capricorn.

A person on this great circle desires to sojourn to the tropic of Cancer by the most direct route. We could call this direction north, and the direction to the tropic of Capricorn south. If you want a change of climate, this is the direction you would go. It would arise naturally.

There remains a third direction, which is the unique direction perpendicular to both our north-south and our east-west. No need to give it a name just yet.

Is east-west usually perpendicular to north-south? I don't know yet. But this will work either way. It isn't important. Or so I now believe.

Next, a map. I've got to figure out the best way(s) to extend the equinoctial great circle to an equator that partitions the planet into hemispheres, and also facilitates a practical, useful, attractive map. The properties of the new third direction need to be explored.
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Re: Map of a 4D Earth

Postby PatrickPowers » Thu Aug 22, 2019 12:55 am

The solar thing doesn't work. That leaves the stars. That DOES work. The 4D sky is much bigger so there are many more constellations. Each is fixed above a great circle of rotation. So find three constellations above two such circles mutually perpendicular. Then the third direction doesn't have a circle of rotation: its the direction perpendicular those. As for daytime, it would be possible to use the Sun, but no easy. But everything is much closer together. Oceans are much smaller. So there is much less need for navigation. On the other hand the planet has a small radius so the horizon is a lot closer so you can't see very far.
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Re: Map of a 4D Earth

Postby PatrickPowers » Sat Aug 24, 2019 8:32 am

So how about a map. Since it is necessary to navigate by the stars, I think there would be a big emphasis on that.

Start with a great circle that you like. Maybe it is the great circle in the middle of the ecliptic. Then use that as the axis for the croissant-shaped map mentioned elsewhere. So far, it is just like a non-isoclinic rotation planet.

Now add the locations of certain stars. Each star is fixed over a great circle. So emphasize that great circle on the map, by making it a different color or whatever. With three such sightings you ought to be able to "triangulate" your position. Though it would be more accurate to call this "tetrahedronation."

What would one of those great circles look like? Go back the croissant map. Each great circle is confined to one of those cylinders that comprise the map. A great circle is a helix that goes around the cylinder exactly once.
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Re: Map of a 4D Earth

Postby wendy » Sat Aug 24, 2019 9:36 am

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.

1. Every point on the surface goes around the centre in a circle. This means that every star rises at some point on the horizon, and sets diametrically opposite.

2. The horizon of a person on a 4d earth is a 3-sphere. Here is the trick. The rising stars form a hemisphere, with an 'East pole'. As time goes by, the hemisphere turns around the EW equator like a page in a book, but as it does, it also rotates around the EW equator. In the 3d world, the sky stripes that rise are fixed to the visible pole in the sky, and spin like a fan. This is not happening in 4d. Instead, the sky rotates around the horizon as the sky progresses.

3. We take the cumulation-line, that is, the plane vertically to the EW horizon, this is a hemisphere where the stars cumulate, or reach their highest point. Each point on this hemisphere is a great circle in the sky, two stars that fall on the same great circle will cumulate in exactly the same spot. So we now have divided longidude (rotation) out and the hemisphere is 'lattitude'.

4. The lattitude-sphere is actually a sphere (3d), where the poles are the observer at one pole, and the zenith or 'straight above' on the other pole. We map the stars from (3) onto this lattitude-sphere, by drawing a straight line from the observer, to the star, and where that line crosses the lattitude-sphere, that is the 'lattitude' point. Unlike the EW sphere above, the lattitude-sphere is identical for all parts of the world. The only difference is what part is set tangential to the horizon.

5. The sun is taken as not being on the same set of clifford-parallels, and thus the path of the sun (zodiac), is a helix confined in a torus. The sun actually moves in a circle in the sky, and this circle crosses lattitude-points that itself forms a circle on the lattitude-sphere. The circle of the sun is equidistant from exactly two great-circles of the earth. These act as 'climata' poles.

6. The point in the 'south' on the lattitude-sphere at the centre of the zodiac-circle, is the "south-polar", has a tropical climate. The point opposite the south-polar is the north-polar, has a frigid climate. In 3D, the south-polar represents the equator (as is), and we have two seasonal sides. If you rotate (through the polar-axis + 4d), the northern hemisphere using the equator as a hinge, the north pole would sweep a circle that passes through the south pole six months later. The 3d sphere is like a double-end compass pointer, where the 4d circle is the full circle.

7. From south to north, the climates run as they do on earth: tropical, subtropical, temperate, subpolar, polar. Unlike earth, the sun always rises, so there is no 'land of the midnight sun'. Where it differs, is that on 3d earth, you see maps with a snakey line of the zodiac, showing its lattitude at each season: a sinewave between capricorn and cancer. In 4d, the sun is always over a tropic, and it's a helix in a tropic-torus. On the lattitude-sphere, the zodiac-circle is a line of lattitude 43°S (by 3d standards), which makes for a tilt equal to the earth's, This is 2×23.5 degrees from the south-polar on the lattitude circle. The equator of the lattitude-sphere is then marked out in months (as lines from pole-to-pole or 'longitude on lattitude). The months represent the mid-summer, in much the same way that we might mark out time-zones on the 3d earth.

8. If we head back to the EW horizon-hemisphere thing, the zodiac circle on the lattitude-sphere projects to the highest point in the sky where the sun rises to. This is not a great circle as such, but a line that might dip to the horizon etc. In any case, if you imagine standing on a 3-sphere (at any point), the sun is chugging around 43S at 1 cycle per year. Unless you are at the poles, the distance will vary from when it's on your longitude, to when it's opposite. The length of this arc, is exactly twice the cumulation or height of the noon-sun. The low sun is what makes winter, and the high sun makes it hot.

9. The coordinates on this figure is then an EW direction, (as the time of day), a NS direction (as climata), and a third direction that goes as the seasons, say AB.

The AB and EW directions are both perpendicular to the NS direction, but the angles between AB and EW vary from 0 to 180 degrees, or twice the climata.
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Re: Map of a 4D Earth

Postby mr_e_man » Sun Aug 25, 2019 11:40 pm

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?
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Re: Map of a 4D Earth

Postby PatrickPowers » Mon Aug 26, 2019 12:38 am

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?


Here on mundane 3D Earth we get gigantic whirlpools and magnetic fields due to the Coriolus effect. It is kind of amazing that this doesn't slow down the rotation much. It appears the flows are basically laminar so not much heat is dissipated. I think that 4D Earth would be the same.
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Re: Map of a 4D Earth

Postby PatrickPowers » Mon Aug 26, 2019 12:50 am

I think I get it. The plane of the ecliptic intersects the various planes of rotation of 4D Earth. Let's say that that angle is in the interval [0,90] degrees. The higher the angle, the less solar energy on average. The lower the angle, the more solar energy on average. That's climate. So there is a hot region and a cold region, each (I think) with shape of nested tori. The two regions are perpendicular to one another and roughly parallel.

As the sun "moves" the hottest and coldest planes "move." In effect these are Villarceau circles moving around and among the nested tori. (4D tori can be constructed entirely of Villarceau circles.)

Anyway, thank you Wendy. While working through this I found out that I'd had a wrong idea about my 4D clock.
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Re: Map of a 4D Earth

Postby PatrickPowers » Mon Aug 26, 2019 12:42 pm

Here on 3D Earth the same system can be used for climate and navigation. On an isoclinal 4D Earth it would be better to separate the two. There would arise a system based on heat and the sun that would be useful for deciding where to live, what crops to grow, and so forth. We can call that the Sun system. One chooses a great circle that is more or less the hottest on the planet. It is NOT a rotational great circle, but you can walk around it and experience the heat and so forth. The great circle that is perpendicular to that great circle is the coldest on the planet. Then generate the equatorial torus of all points with the same minimum distance to each circle. The "poles" have just about no seasonal change, the equator has extreme seasons. The three directions are then are hot-cold, maximum seasonal change (mutaseason), and the third direction is isothermal. A number of maps discussed elsewhere will work.

Now for the second system. It's used to navigate by the stars using a compass. It's based around the east-west direction, the direction in which a point on the surface is moving as the planet rotates. That's what a compass will tell you: east-west. The other two perpendicular directions are arbitrary, likely based on prominent stars. Here's the map system that I think might go. Start with an torus shaped exactly like the equatorial torus in the Sun-based system. It can go whereever, but must be comprised completely of great circles of rotation. Topologists call this a "square flat torus", because it can be flattened out into a square with no distortion. But we are not going to use that square. The trouble with the square is that the circles of rotation are diagonals. That's inconvenient. We want them to go left to right, corresponding to east-west. So we transform that square. The result is a 1x2 rectangle.

Next, choose some arbitrary point on the rectangle. A corner seems like a good choice. Choose some arbitrary direction on the surface of the planet that is perpendicular to the torus. Move some tiny increment in that direction. Construct a new "square flat torus" that is parallel to the first is the same size and shape and contains that point. Convert to a 2x1. Lay that on top of the first. Continue until you have done 90 degrees worth. You have a map of the planet that is 1x2x4. That's quite close to the shape of a Swedish brick, so we will call it the "Swedish brick map." Each great circle is a left-to-right line entirely confined to a horizontal cross section of the brick. Each star corresponds to a such a line, so each prominent star gets a line. By observing the stars I believe you might be able to "triangulate" (tetrahedronate?) your position. Given a chronometer, you can get your longitude too.

Two systems, the sun system and the star system.
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Re: Map of a 4D Earth

Postby wendy » Tue Aug 27, 2019 8:55 am

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.

The motion of the sun in the sky does not have to fall to the clifford-parallels of an isoclinal rotation, but such a line will always strike the same angle against the isoclines. There is an isocline that is equidistant to the motion of the sun in the direct sense, and one in the reverse sense. Since the nature of the systems is for rotations to be in the same sense, we suppose that the motion of the sun-zenith is in the same sense too.

The tilt of the 3d earth is replaced by the notion the sun is always above a torus, at twice the tilt. In 3d, the sun-zenith is going up and down between the tropics once a year, and travelling around the world once a day. This is like a compressed helix. Some maps of the world draw just a single sine-wave, to show the lattitude of the sun at any given point of the year. In four dimensions, the sun-zenith is always on a torus that is 47 degrees from the south polar ring. The star zenith follows on any torus, a line from 0+x,0 to 360, 360+x, around the torus, where the edges of the torus are labeled 0 to 360 in both directions. The addition gives a value modulo 360, so that the torus is stripped diagonally with the star-tracks.

The sun moves through the stars in a circle, from 0,0 to 360, 361, = 0,1 and then as the same throughout the year. So the sun will be overhead at x, x+j, where x is the time, and j the part of the year.

If we take a glome, and unpack it, torus by torus, and open each torus out, we start with the south-polar arrow, and progress through a range of thin rectangles, to squares, to a series of thin rectangles to the north polar line. The motion of the sun and stars stay in the same layer. The over-all assembly is the intersection of two perpendicular cylinders of equal diameter, and the central axis of each is tangential to the surface of the other.

If we now rotate the north polar to match the south polar, each of the layers will also be turned. Because the diagonals of the layers are always equal to a great circle, the resulting figure will become a lattitude-sphere times a longitude circle (which is opened up). The East-West axis can be found from the motion of the stars, and the triangulation of the stars in the sky can be done with a device which when the axies are placed east-west, and the star is placed in the sight, the current angle and elevation will lead directly to the cumulation of the star.

The lattitude-sphere is as described, derived from a sphere drawn in the plane between the E and W parts of the sky. The diameter is from the zenith to the observer, the stars map onto this by a ray drawn from the star's cumulation to the observer. The zenith sphere would rotate by 90 degrees in each direction, representing the motion of the star at its rise to set.

One would first need to know E/W axis. The instrument is a paged sphere that freely spins in any direction in a plane. One would set the axis to be perpendicular to the EW, and then turn the paged sphere to align with a visible star. As the page turns, a scale at the base turns also to indicate the angle added to or subtracted by the angle of the page. From a table of lattitudes, one finds that the star (Sirius), is on a particular ring. If the sphere is rotated to this point, and clipped, it will still turn on the axis around the clip, and a second reading can be taken, for a second star. When this is also clipped, the base of the sphere is standing on the correct lattitude for that point. The longitude requires, as in 3d, an accurate clock.

The map that shows latitude and longitude correctly, would be a spherical sphere. This is a 4d figure, but to unfold it onto a 3d page, one would have to find a suitable projection of a sphere to 3d.
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Re: Map of a 4D Earth

Postby PatrickPowers » Wed Aug 28, 2019 2:24 am

Right, I'm using "map" a colloquial sense, as something flat you can fold or roll for storage. There are many ways to map 4D to 3D, the question is which would be most useful for a particular purpose. On 3D we need only one map, in 4D I think there have to be two.

Here on 3D Earth the Mecator is most useful for navigation, but that is largely unneeded now. Equal area maps are finding favor.
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Re: Map of a 4D Earth

Postby mr_e_man » Wed Aug 28, 2019 9:10 pm

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.


It's true that nothing is solid on astronomical scales.

In general, angular velocity only makes sense for solid objects, or fluids with a fixed shape. But angular momentum makes sense for any system, regardless of its composition, and is always conserved (ignoring General Relativity). Two particles moving in opposite directions along parallel lines have non-zero angular momentum (proportional to the distance between the lines), even if they're not spinning individually.

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.
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Re: Map of a 4D Earth

Postby mr_e_man » Wed Aug 28, 2019 11:13 pm

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
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Re: Map of a 4D Earth

Postby PatrickPowers » Thu Aug 29, 2019 2:51 am

Wendy is arguing that stress forces like torsion would slow the faster rotation, like the tidally locked moon. While there might be such forces, they would be constant. To get heat you have to have a varying force, right?
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Re: Map of a 4D Earth

Postby PatrickPowers » Thu Aug 29, 2019 5:47 am

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


The gnomic has the virtue of simplicity and works in any number of dimensions but can only cover a fraction of the planet.

With the spherical you are mapping the 3sphere to a flat 3 ball. Wild! Which is the radial dimension? And are those typos in your notation formula?
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Re: Map of a 4D Earth

Postby mr_e_man » Fri Aug 30, 2019 5:13 am

What typos? Are you thinking that cos(b+c) should be cos a cos(b+c)? I factored it out.

The domain of the "Hopf coordinates" is

0 < a < pi/2,
0 < b + c < 2pi,
0 < b - c < 2pi,

or alternatively

0 < a < pi/2,
0 < b < 2pi,
0 < c < pi.

We can put (a,b,c) as Cartesian coordinates on a 3D box-shaped map. The coordinate system is degenerate at a=0 and a=pi/2. The wrap-around behaviour is

(a, b, c) ~ (a, b+pi, c+pi) ~ (a, b+2pi, c).

PatrickPowers wrote:With the spherical you are mapping the 3sphere to a flat 3 ball. Wild!


That reminds me: Any sphere Sn+1 can be parametrized by a ball Bn and a circle S1, and the map is volume-preserving. For n=1, this is the cylindrical coordinate system (z,θ), which is area-preserving. (See 3Blue1Brown's video, particularly 1:05.)

u = u1e1 + u2e2 + ... + unen

u2 = u12 + u22 + ... + un2 < 1

(That "less than" should be "less than or equal", but "<=" could be confused with an arrow.)

x = u + sqrt(1 - u2) (en+1cosθ + en+2sinθ)

So u is a point in the ball, θ determines a point on the circle, and x is a point on the sphere. This is degenerate when u2=1 (varying θ doesn't change x), but we may include those points in the map for completeness. To prove that it's volume-preserving, you can show that the magnitude of the wedge product of the partial derivatives of x (which is a generalization of the Jacobian determinant) is 1.

For n=2,

x = u1e1 + u2e2 + sqrt(1 - u12 - u22) (e3cosθ + e4sinθ)

we get a parametrization of the 3-sphere by a disk times a circle, which is topologically a solid torus. We can cut the circle, making it a line segment, so the resulting map is a solid cylinder.
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Re: Map of a 4D Earth

Postby PatrickPowers » Sat Aug 31, 2019 8:48 am

This is all very helpful. I've always heard of the Hopf as a mapping from S3 to S2, but it can be mapped to any shape that works.

My fave definition is that the mapping is the dividend of two complex numbers, z1/z2. That is SO cool. I always wondered what use complex division was. You can use it to define a 4D torus! Who knew?

What really helped me was this animation. https://www.youtube.com/watch?v=AKotMPGFJYk. I see now what I did wrong with the "Swedish brick map." I suspected something was funky. What I was doing was decomposing the 3-sphere into tori of equal size, tilted with respect to one another. In 4D tori flatten out very nicely. I thought they were parallel. From the vid it can be seen that they aren't disjoint. Their intersection is two great circles. Not only that, with that construction ALL of the tori share those two great circles in common. So the natural mapping is to two cylinders, not a brick. The axis of each cylinder is a great circle, straightened out into a line. Each of those tori then maps to two infinitesimal wedges radiating from that line.

The end result is two cylinders. It differs from 3D Earth in that instead of two coordinate singularities at the poles there is one at the axis of each cylinder. This map would emphasize circles and radial straight lines.

I don't know how the map is distorted, aside from being weirdly divided into two, but have the feeling that is is minimal. But like the pinball wizard, there's got to be a twist.

This is vaguely related to a model for 4D dice: use two tetrahedra to model a hypercube. Elsewhere.
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Re: Map of a 4D Earth

Postby Foxbat_25 » Sat Aug 31, 2019 3:10 pm

Oh, that video was a gold mine for newbies like me, thanks for posting it... I'm starting to understand what you guys are discussing in the forums :lol: on one hand, my brain is begging me to take a break from thinking as I'm both learning on this forum and writing loan repayment programs for villas for sale in Nice for my job, but I'm paid to do the latter and the former is too fascinating for me to give up right now... I guess said brain will have to soldier on for a while!
Last edited by Foxbat_25 on Wed Sep 11, 2019 7:22 am, edited 1 time in total.
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Re: Map of a 4D Earth

Postby mr_e_man » Tue Sep 03, 2019 9:53 pm

I'm still thinking about this.

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.


No, the bivectors aren't proportional on average, because the inertia tensor is not a scalar; one plane could be scaled more than the other. But the two planes of angular velocity individually should be parallel to those of angular momentum, on average.

And the Euler "force" would also tend to change the shape of the earth...

But this is greatly simplified if we assume that gravity is much stronger than these "inertial forces"; the earth would be almost perfectly spherical, and the inertia tensor would be almost a scalar, and the angular velocity would be almost exactly proportional to the angular momentum. My main point remains the same.
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