Astronomy and navigation on a 4D hyperspheric planet

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.

Astronomy and navigation on a 4D hyperspheric planet

Postby Lucia the Phosphorus » Wed Aug 08, 2012 9:01 pm

Hello all~

I know there's another thread about it but I didn't feel like posting on an old thread... so here I am -hehe-

I've thought about how exactly day/night, seasons, stars etc. would work with four dimensions of space which isn't exactly easy to think about but I do have a few ideas though with a low level of rigor....

Let's suppose the planet rotates around two mutually perpendicular planes (x1-y1) and (x2-y2) with equal periods; in this case every point revolves around the centre at maximum celerity (magnitude of the velocity vector) as the following equations demonstrate: (here I consider the boundary of a unit 3-sphere for simplicity)
x1(t) = cos(a)*cos(t+b) -- 'a' represents latitude between the two planes
y1(t) = cos(a)*sin(t+b) -- 'b' represents time shift
x2(t) = sin(a)*cos(t+b+c) -- 'c' represents phase shift on the second plane
y2(t) = sin(a)*sin(t+b+c) -- 't' represents time
dx1/dt = -cos(a)*sin(t+b) -- cos(a) is constant with respect to 't' and d/dx(f(x+c)) = f'(x+c)
dy1/dt = cos(a)*cos(t+b) -- see above
dx2/dt = -sin(a)*sin(t+b+c) -- see above
dy2/dt = sin(a)*cos(t+b+c) -- see above
(ds/dt)^2 = (dx1/dt)^2 + (dy1/dt)^2 + (dx2/dt)^2 + (dy2/dt)^2 -- by the distance formula
(ds/dt)^2 = cos(a)^2*sin(t+b)^2 + cos(a)^2*cos(t+b)^2 + sin(a)^2*cos(t+b+c)^2 + sin(a)^2*sin(t+b+c)^2
(ds/dt)^2 = cos(a)^2*(sin(t+b)^2+cos(t+b)^2) + sin(a)^2*(cos(t+b+c)^2+sin(t+b+c)^2)
(ds/dt)^2 = cos(a)^2*1 + sin(a)^2*1 -- Pythagorean identity
(ds/dt)^2 = 1 -- Pythagorean identity again

So every point on the planet revolves around the centre in a great circle which suggests that there is no region of maximum velocity.. is that true? If so then every circle on the 3-sphere can be specified with 'a' and 'c' while 'b' specifies the point on the circle. Every point is on an unique circle and it seems that there is a one-to-one correspondence between circles on the 3-sphere and points on a 2-sphere -- it's like the 'a' value corresponds to 'latitude' of the 2-sphere and 'c' value corresponds to 'longitude' of the 2-sphere... does this make any sense? Or am I just rambling...
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Re: Astronomy and navigation on a 4D hyperspheric planet

Postby wendy » Thu Aug 09, 2012 8:41 am

A good deal of thought has gone into this, even here. Here is my take on it. It pretty much follows the geocentric idea (rotating earth, sun goes around the earth, stars fixed). You can make it more complex by moving the earth around the sun, but it makes the maths harder.

A planet in four dimensions will, under the laws of physics (specifically, tidal equalisation of energy in different modes of rotation) will tend to adopt a 'clifford rotation', where every point goes around the centre.

The effect of this, starwise and sunwise, is that all of the heavens will be visible from all parts of the planets. In our world, venturing into the northern hemisphere will reveal entirely new constellations not previously known. The sun (essentially stationary or slow-moving in the stars), will rise and set exactly 12 hours apart, so the day never lengthens or shortens according to the seasons. (Yes, seasons do occur).

Now, we look at the instruments of navigation. Longitude as in 3D, requires accurate clocks. The stars give siderial time, and the time of year gives the drift from siderial time to solar time.

The lattitude is the selection of great circle. This is most interesting, because lattitude corresponds to points on a sphere. Consider a person in 4D, you have a horizon (which gives a 3sphere). There is an 'east' pole and a 'west' pole, and the line between these gives the line of skimming. Stars that lie on the line of skimming never rise or set. The number of degrees the star rises eastwards of the skimming-line, would cumulate the same number of degrees, and set exactly antipodally, in the western half of the sky. A star rising at the east-pole, will cumulate at the zenith, and set at the west pole.

We now construct the 'lattitude 3-sphere'. Between the east-pole and west-pole, there is a 3-space which contains all of the cumulation-points. In this 3-space, we draw a 3-sphere with a diameter between the observer and the zenith point. Now any star that cumulates does so in this 3-space, and we plot its lattitude, by drawing a line from the star to the observer. Where it strike the lattitude-sphere is the 'lattitude'.

What's interesting is that every place on the 4-sphere uses the same lattitude circle, but according to where the place is, the zenith stars are different. We might, for example, have for every star, a town where this star reaches zenith.

The zodiac is the trace of where the sun moves through the year. The actual track in the sky is a circle, but not necessarily corresponding to any of the zenith-arcs that the rotating earth makes on the sky. In practice, it doesn't, and the angle between these (when they cross), is the 4D version of the tilted earth. The sun, for moving in front of stars not on the same zenith-traces, will track a cicrle (like a lattitude in 3D sphere, not necessarily the equator).

The points on the longitude circle that the sun falls directly over is the 'tropics'. The points directly opposite form the 'artics', the axis which makes these points as 'lines of lattitude' in 3D correspond to the 'Equator' and 'Polar' circles. If you take, say the northern hemisphere of the earth, and then reduce it in the manner of the lattitude-sphere above (from an observer at the centre of the earth, the reduced sphere runs diametrically from the centre of the earth to the north pole). The lattitudes are effectively doubled (so that SP = 0 degrees, NP = 90 degrees). You have at 90-23.5×2 = 43 S, the tropics, at 43 north the artics.

Most of the planet would be temperate climates, with bands of really hot areas and really cold areas. The sun skims the horizon for one day (mid-winter), on the artics (only), and is zenith only in the tropics (only).

Now. The motion of the sun on the zodiac circle is to pass through the 12 signs over a year, and therefore to move around the tropics once a year. You can then mark the tropics with the twelve signs of the zodiac (eg tropic of cancer, tropic of leo, tropic of virgo etc), as points on the 43 S line. This line of zodiacs is similar in function to when one might see a zodiac line drawn across a map of the earth (representing the zenith-lattitudes of the sun).

When you take a particular day, say 5 May, you can place the sun at a specific point on the zodiac point. We put the lattitude sphere with this point at the top. The sun is maximum at this point, and rises only as high as the the angle between the sun-point and the locale. Taken at a given localle, the sun rises at different points of the horizon, (in the shape of a circle), where it rises east-most, it corresponds to the highest rise of the sun (mid-summer). Where the sun rises west-most (mid-winter), the sun rises lowest in the sky, and the land recieves less warmth from the sun.

What this means, is while our planet only has a 'tropic of cancer' and 'tropic of capricorn', the 4-planet has tropics of all signs, and therefore there is a place at every season of the year. In 3d, we have diverse regions have N = season, S = season + 6 months, in 4D, there are season-zones like time zones. The circle having 360 degrees, has a change of season (at month level), at a distance of 30 degrees or 1800 n. miles.

You could, for example, have sports-teams who literally 'follow the sun'. Cricket for here, is something that internationally, is played here around late december to janruary etc. Once this season has moved on, you could then decamp the teams to the next part of the world, and let it go all again. The autumn teams move in for their spin, and the the winter teams follow, and so forth.
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Re: Astronomy and navigation on a 4D hyperspheric planet

Postby quickfur » Thu Aug 09, 2012 5:40 pm

In the other thread, I considered the heliocentric case where one of the planet's plane rotations coincides with (or is close to) its orbital plane. This has the interesting effect of dividing the surface of the planet into zones delineated by concentric tori, which begin as a circle in one plane and ends as an orthogonal circle in the orthogonal plane.

Since one of these planes coincides with (or is close to) the orbital plane, the sun always passes overhead at midday when you stand on the corresponding circle. So the innermost torus around the circle has a full day/night cycle. For reasons that will become clear, we may call this region the tropical region.

Once you depart from this tropical region, the sun will not pass directly overhead, but move across the sky at a shallower angle. The further you depart from the tropical region, the lower the sun will be at midday; when you reach the orthogonal circle, the sun is always on the horizon. So we may call the toroidal region around this other circle the arctic region.

If the arctic plane is perfectly aligned with the orbital plane, then the sun never rises and never sets in the arctic region; it just circles around the horizon. If the arctic plane is slightly off-alignment, though, as is the more likely case, then the sun will be above the horizon half the year, and below the horizon the other half, thus giving rise to arctic day/night lasting for 1/2 a year each. This will also cause slight variations in the length of days in the tropical region, though it will still have full day/night cycles. The regions between the arctic and tropical regions will experience considerable changes in the length of days throughout the year, thus giving rise to seasons. So they may be called the temperate regions.

So in this particular setup, we have the 4D analogues of the division of the earth's climate into arctic, temperate, and tropical; but with the interesting twist that the arctic region is not confined to two disjoint polar regions, but lies along a great circle (a longitudinal circle), just like the equator (but orthogonal to it).
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Re: Astronomy and navigation on a 4D hyperspheric planet

Postby quickfur » Thu Aug 09, 2012 6:08 pm

A peculiarity of spheres in even dimensions is that the hairy ball theorem does not hold, so the weather system will be considerably different from our Earth's weather system. On our 3D earth, when there is wind, there is always a point where no winds are blowing, and around it winds will blow in a circular direction -- i.e., there is always an eye of a tornado-like system. Or, IOW, there's always a storm somewhere on the planet (unless the entire planet is devoid of wind, which is very unlikely due to differential heating by the sun, etc.). In 4D, however, this is not true; it's possible for wind to be present at every point on the planet without any "eye of the storm". Weather is incredibly difficult to model, so I don't really know too much beyond this, save that any weather systems on a 4D planet must be quite different from what we are used to.

Now with regards to navigation: again, using the heliocentric model I posted above, if one is in the tropics, then the path of the sun overhead will define an east/west direction (we may adopt the convention that the sun rises in the east and sets in the west). Unlike the earth, however, this does not uniquely define the lateral directions, because the surface of the planet is 3D! The path of the sun only fixes one of the 3 pairs of directions across the planet's surface. The other two directions can be arbitrarily rotated around the path of the sun, which is bad because there is no fixed reference frame as to which of the 360° of lateral directions we want to refer to. So we need another fixed reference in order to fix the orientation of the lateral directions.

However, one cannot simply fix a direction by reference to any "pole", because there are no rotational poles here. Neither does the direction of the arctic region help at all, because if you walk across the planet in any of the 360° of lateral directions from the sun's path, you will always end up in the arctic region. Every lateral direction eventually converges on the arctic region. So you can't fix a reference direction that way, either.

And since nobody has worked out a workable version of electromagnetism in 4D yet, we know nothing about the presence/absence of magnetic poles or their analogues thereof, so we can't say if magnetic compasses exist in 4D. And even if they did, there's no guarantee they won't just correspond to the arctic region (i.e., a circle that doesn't help fix a reference direction among the 360° of lateral directions, rather than a pole with a fixed direction).

So the only way to fix a reference lateral direction is by the stars.
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Re: Astronomy and navigation on a 4D hyperspheric planet

Postby quickfur » Fri Aug 10, 2012 6:45 am

quickfur wrote:[...] So the only way to fix a reference lateral direction is by the stars.

Actually, I thought a bit more about this today, and realized that it's still possible to have a navigational system without relying on the stars, as long as you're reasonably far away from the equator (say in the temperate zones), and the planet's double rotation has a rotation plane shows a measurable difference from its orbital plane while being somewhat close to it.

The basic idea is this: in the temperate zones (or even the arctic zones in arctic summer), the sun will not be directly overhead at midday, but will only reach a certain maximum height depending on the season and your location on the planet. That is, it will not reach the zenith throughout the day; at midday the line from the observer on the ground to the sun will have a non-zero angle to the zenith. This non-zero angle thus defines a direction perpendicular to the path of the sun across the sky. This affords us a way to unambiguously fix all 3 pairs of cardinal directions on the planet:

First, we mark a point on the ground to be our reference point, and plant a vertical stick there. At sunrise, we trace a line on the ground where the shadow of the stick falls. At midday, we do the same again: the shadow will be short and will make an angle close to 90° with the sunrise line. At sunset, the shadow will be long again, and will be almost 180° with the sunrise line (it won't be exactly 180° because the sun does not pass the zenith at midday; at the equator it does, but then the short shadow has zero length and is useless in fixing a perpendicular reference direction). Midday can be determined by making temporary traces of the stick's shadow and picking the shortest one. The short shadow should then bisect the angle between the sunrise/sunset lines (or be pretty close to bisecting the angle). If we draw a line perpendicular to the short shadow, in the plane of the sunrise and sunset shadows, then it will give us the east/west directions. The short shadow itself will give us north, and its opposite will give south. These two pairs of directions therefore determine a plane that bisects the ground, so we draw another line perpendicular to this plane to fix the third pair of cardinal directions (called "marp" and "garp" by some people -- you can invent your own pair of words here).

East/west therefore follows the apparent motion of the sun across the sky; south is the most direct direction to the equatorial zone (the path perpendicular to the equatorial circle) and north will eventually intersect with the arctic ring. Walking east/west obviously will remain in the same "latitude" of the temperate zone, as will walking marp/garp. Walking north will get you closer to the arctic zones, and walking south will get you closer to the tropical zone.

The major problem with this system is that if you're too close to the equator, the short shadow is zero length, and therefore does not help in fixing the cardinal directions. So if you live in the tropical zones, you'd have to navigate by the stars alone.

So one may surmise that temperate zone dwellers on our 4D planet will develop both solar and sidereal navigation, whereas equator dwellers will not develop solar navigation (save perhaps to determine east/west -- but that is of limited utility in navigation since the lateral directions can't be determined). Arctic zone dwellers will likely also rely mainly on sidereal navigation, since during arctic winter, the sun isn't visible at all, so one can't use it to determine directions. During arctic day, one can observe the direction of its maximum height by using the shadows method as described above, except that some modification of the technique may be necessary as the shadows will be very long. (Perhaps the use of a very short stick will be of utility.)
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Re: Astronomy and navigation on a 4D hyperspheric planet

Postby wendy » Wed Aug 15, 2012 11:23 am

Unfortunately, you can't use the noon-day sun as a position-marker, on earth or on 4-earth. Like 3d, the sun can cumulate at different heights. The higher the sun goes, the hotter it be, sort of thing.
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Re: Astronomy and navigation on a 4D hyperspheric planet

Postby quickfur » Wed Aug 15, 2012 3:14 pm

wendy wrote:Unfortunately, you can't use the noon-day sun as a position-marker, on earth or on 4-earth. Like 3d, the sun can cumulate at different heights. The higher the sun goes, the hotter it be, sort of thing.

Right, the height of the sun at midday varies over the course of the seasons, so it can't be used to determine position. However, if one is outside of the tropics, one can use it to determine directions. It doesn't work in the tropics because the direction of the short shadow will flip depending on which half of the year it is, and it's zero length when the sun passes directly overhead. But outside the tropics, closer to the "arctic great circle", the relative orientations of the dawn/dusk shadows and the short shadow will not change, even though the length of the shadows will change. So one can get a crude set of cardinal directions from it.

Navigating by the stars is still a far superior solution, in any case.
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Re: Astronomy and navigation on a 4D hyperspheric planet

Postby quickfur » Tue Aug 21, 2012 4:04 am

I worked out a little bit of 4D astronomy, and realized that, in general, the celestial equator and the ecliptic would not only be an angle relative to each other, but will not intersect. So does that mean that there is no 4D equivalent of summer/winter solstice? Or do we just pick the points when the sun is closest to the celestial equator? And what if the celestial equator and the ecliptic happen to be two fibres of the Hopf fibration, in which case the sun will always be equidistant from the celestial equator?
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Re: Astronomy and navigation on a 4D hyperspheric planet

Postby PatrickPowers » Mon Mar 25, 2019 1:31 pm

quickfur wrote:I worked out a little bit of 4D astronomy, and realized that, in general, the celestial equator and the ecliptic would not only be an angle relative to each other, but will not intersect. So does that mean that there is no 4D equivalent of summer/winter solstice? Or do we just pick the points when the sun is closest to the celestial equator? And what if the celestial equator and the ecliptic happen to be two fibres of the Hopf fibration, in which case the sun will always be equidistant from the celestial equator?


If conditions are chosen uniformly at random, then I suspect that in about half of cases the ecliptic doesn't intersect the celestial equator. It's possible for the sun to be over the same latitude all year long. Or it can oscillate between any two latitudes, each cycle taking half a year.

Each plane of rotation would have four solstices a year, these being when the sun is minimum and maximum angle to that plane. The dates of solstices of the two planes have no simple connection to one another.

The equinoxes could be the days of minimal angle with the equator. These would be the days in which the longitudinal seasons are at their strongest. There would also be days when the angle with the equator is maximal. That could be an equatorial solstice. It might not be the same day as a polar solstice.

The longitudinal seasons don't circle the planet. They oscillate along the equator, the amplitude of this oscillation being about twice the sum of the two obliquities of the ecliptic. The period is half a year.
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