4D planets (split from "Rings in 4d")

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4D planets (split from "Rings in 4d")

Postby Keiji » Tue Jan 25, 2011 6:49 pm

Eric B wrote:Marp and Garp? Never heard of that one! I imagine that is ana and kata changed into global circles. A second kind of lattitude or longitude?


Using the definitions from Alkaline's glossary:
  • In 3D, a planet has an equator. The direction which the planet rotates in is called east; the opposite direction is west.
  • In 4D, the analogous equator is called the solar equator - it is the one most aligned with the plane of its orbit around its star. The planet still rotates around this equator, and the terms east and west retain the same meaning.
  • There's another equator for 4D planets, perpendicular to the solar equator, which is called the polar equator. The planet rotates around this equator independently from the other one, and the direction which the planet rotates in this equator is called marp. The opposite direction is garp.
  • This leaves only one axis not yet defined - so we can define this axis to be perpendicular to both east/west and marp/garp, and we call the directions in this axis north and south. In 3D, we could define north as being 90 degrees counter-clockwise from the direction east; in 4D we could define it as the direction of the vector n = e×m where e and m are vectors pointing in the east and marp directions respectively (the cross product works here because the surface of a 4D planet is 3D).
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Re: Rings in 4d

Postby quickfur » Tue Jan 25, 2011 6:57 pm

Interesting. How confident are we that these definitions would actually prove useful, though? The impossibility of stable orbits aside, what if the planet is rotating in XY and ZW, but the plane in which it orbits its star is closest to XZ? How will we define the equator then?
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Re: Rings in 4d

Postby Keiji » Tue Jan 25, 2011 7:01 pm

I hadn't thought about that, I was just going off of Alkaline's definitions.

Presumably there is some mathematical formula to calculate the "closeness" between two planes as a real number, so the probability of it being equally close to both is infinitesimal and thus we wouldn't have to worry about such a possibility.
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Re: Rings in 4d

Postby quickfur » Tue Jan 25, 2011 7:06 pm

Well, that case I suggested isn't exactly a corner case... the solar plane is actually orthogonal to both rotation planes, even if you allow some leeway for slight misalignments. It's not completely independent of them the same way they are independent of each other -- it does overlap in 1 dimension each. However, the point is that the respective rotations are so far apart that I can't see any reasonable justification for identifying the XZ solar plane with either the XY or ZW rotation. I mean, there's an entire plane in which you can rotate the planet, and the XZ rotation will still be orthogonal to both rotation planes, so you can hardly reasonably connect it with either one.
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Re: Rings in 4d

Postby Keiji » Tue Jan 25, 2011 7:09 pm

It might not be the corner of a cube compared to its volume, but it's still an edge, or maybe a face, if you see what I mean. ;)

From any "problematic" situation, a random rotation of the planet would almost never leave it in that vital plane, hence the problem would disappear.
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Re: Rings in 4d

Postby quickfur » Tue Jan 25, 2011 7:21 pm

Keiji wrote:It might not be the corner of a cube compared to its volume, but it's still an edge, or maybe a face, if you see what I mean. ;)

Fine. But you have to agree that in such cases, the definition is no longer that useful, since both of the planet's double rotation will play approximately equal roles in various phenomena, so identifying one of them as "equator" over the other doesn't seem to be worth the trouble. You might as well regard both rotations equally. :)

In fact, I have a counterproposal: completely disregard the solar plane altogether, and consider a planet in double rotation alone. Let's assume that both rotations proceed at equal rates (it seems reasonable to assume that tidal forces or something along those lines would tend to equalize both rotations over time). Then there is a 2D toroidal sheet wrapped around the planet's surface, which experiences the greatest total velocity. This sheet corresponds with the ridge of the duocylinder. By analogy with a 3D planet's equator, which constitutes the points with greatest velocity, wouldn't it be reasonable to call this sheet the "equator"? Then the two interlocked circles where one of the rotations is not felt would be the rotational poles. They will not be stationary, but they will be the points on the planet with the least velocity.

As to how this interacts with the solar plane... you could either say that it should be regarded as something completely independent, or you could dismiss it as irrelevant since there are no stable orbits anyway. :P
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Re: Rings in 4d

Postby Keiji » Tue Jan 25, 2011 7:43 pm

Okay, firstly, I have been wondering for a while exactly how you can find a set of points T such that T is equivalent to a torus and that T is a subset of the points on the surface of a 3-sphere. I can see the argument why it should be possible (when going from one ring to the other perpendicular ring, one sweeps out a torus), but I can't visualize a torus being on the surface of a 3-sphere (if I try to, it makes the 3-sphere seem like some toroidal form, i.e. non-convex).

Now back to the topic: Suppose we do what you suggest with the equator - then we'd have to arbitrarily define one such "rotational pole" as primary and the other as secondary, so that we could then define the 3 cartographical axes.
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Re: Rings in 4d

Postby quickfur » Tue Jan 25, 2011 7:53 pm

Keiji wrote:Okay, firstly, I have been wondering for a while exactly how you can find a set of points T such that T is equivalent to a torus and that T is a subset of the points on the surface of a 3-sphere. I can see the argument why it should be possible (when going from one ring to the other perpendicular ring, one sweeps out a torus), but I can't visualize a torus being on the surface of a 3-sphere (if I try to, it makes the 3-sphere seem like some toroidal form, i.e. non-convex).

It's actually very easy. Just take a duocylinder and relax the requirement that its two bounding 3-manifolds be flat, then inflate it like a balloon so that it becomes a 3-sphere. Its ridge is now a torus that lies on the surface of a 3-sphere.

How to visualize it, you say? Just take this:

Image

And inscribe it inside the spherical projection of a 3-sphere.

This is a perspective projection, of course. If you use a parallel projection, then it's just a cylinder inscribed in a sphere.

Now back to the topic: Suppose we do what you suggest with the equator - then we'd have to arbitrarily define one such "rotational pole" as primary and the other as secondary, so that we could then define the 3 cartographical axes.

Why do we need to designate them as primary and secondary? We could just as easily say that one pole is "north" and the other is "south". Or "blouth" and "clouth", if you prefer to avoid the connotations associated with north/south. :P
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Re: Rings in 4d

Postby Keiji » Tue Jan 25, 2011 8:20 pm

quickfur wrote:Why do we need to designate them as primary and secondary? We could just as easily say that one pole is "north" and the other is "south". Or "blouth" and "clouth", if you prefer to avoid the connotations associated with north/south. :P


Well, one such "pole" would define the east/west directions and the other would define marp/garp. They aren't really poles as they are not opposites in the same sense, so cannot be called north/south, and as for making up random names - well I arbitrarily chose the names primary and secondary! The point is, you would have to arbitrarily pick which particular "pole" used the name which comes first alphabetically.

Also, thank you for explaining the torus thing, I understand it now.
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Re: Rings in 4d

Postby quickfur » Tue Jan 25, 2011 8:37 pm

Fine, although "primary" and "secondary" implies order of importance, whereas alphabetical order has no relevance to whether one "pole" is more important to the other. :) And they actually are opposites in the sense that, if you were to start from a point on one pole and walk towards the other pole, then exactly halfway you will cross the equatorial torus. (Remember that the 3-sphere's surface is 3D, so the 2D torus divides the surface into two pieces that mirror each other. The analogy with the 3D planet's equator dividing the northern hemisphere from the southern hemisphere is quite clear, albeit somewhat twisted because of the toroidal shapes involved. :P) So we could in fact call these two great circles "north" and "south" poles, except that they no longer designate just points, but two great circles. Isn't 4D fun?
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Re: Rings in 4d

Postby Keiji » Tue Jan 25, 2011 9:42 pm

Oh, I see! Well, thinking about it: with my way, you'd define north/south as being the axis not in either ring. With your way, you'd define north/south as being the axis you move the torus along to get from one ring to the other - which is of course perpendicular to the torus. Now the vital point I hadn't noticed until now is that the axes through that torus are actually the two rings themselves. Which means that whichever method we use, the set of axes is the same! So we may as well just use your way, since it's more intuitive.

So the definitions are now:
  • A 4D planet has a single toric equator. This contains the points of greatest rotational velocity on the planet.
  • There are two circular poles, the north pole and the south pole. These circles are in the two planes in which the planet independently rotates. The points in the equator are equidistant from the north and south poles.
  • From any point not in either pole, the direction north points towards the nearest point in the north pole, and the direction south points towards the nearest point in the south pole. From a point residing in one pole or the other, north and south are not defined. It turns out that, when they are defined, north and south always point in opposite directions.
  • The direction east is the direction of rotation within the north pole. The direction west is the opposite.
  • The direction marp is the direction of rotation within the south pole. The direction garp is the opposite.

Is this OK?
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Re: Rings in 4d

Postby quickfur » Tue Jan 25, 2011 10:08 pm

I like it. :)

Well, except that now I'm having second thoughts about using "north" and "south", because our 3D-centric assumptions on it causes dimensional analogy to break down. Because now north and south aren't antipodal, but orthogonal. A naïve application of dimensional analogy would expect north/south to be antipodes, and the equator a 2-sphere. This is the approach I've adopted when dissecting polytopes on my website.

However. From the POV of a native 4D dweller on a 4D planet, there is absolutely no reason why two antipodes on the planet should stand out from the rest of the surface. Assuming that all 4D planets will have some sort of residual angular momentum (whatever the 4D equivalent might be), this will over time redistribute itself into an equal double rotation, and so the structure we've described will arise. A native dweller, then, will naturally recognize the torus as the area of maximal velocity, and the two polar great circles as the lines of least velocity. From this realization, it is only natural to base the navigational directions on this structure. So in this sense, "north" and "south" are appropriate from the native's POV.
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Re: Rings in 4d

Postby Keiji » Tue Jan 25, 2011 10:27 pm

Yes, I realize that there is no dimensional analogy, but there is a conceptual analogy, which is really all that is necessary.

Also, perhaps this will give us new ideas for orbits? Notice that around some "glomic" object, two other objects can orbit at identical radiuses in perpendicular circles and never collide, whereas in 3D, they would. Perhaps such a model would grant some form of stability (at least at the macroscopic scale of planets, rather than the microscopic scale of wave functions).

If it did, this could raise some interesting issues, namely that around a star you'd get pairs of planets, mutually stabilizing each other.
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Re: Rings in 4d

Postby quickfur » Tue Jan 25, 2011 11:41 pm

Wait, so you're saying you're trying to solve the 3-body problem in 4D? My hats off to you, sir!
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Re: 4D planets (split from "Rings in 4d")

Postby Keiji » Wed Jan 26, 2011 12:34 pm

No, actually, that had occured to me during the consideration. I'm just suggesting considering two planets in mutually perpendicular orbits around a star.
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Re: 4D planets (split from "Rings in 4d")

Postby quickfur » Wed Jan 26, 2011 4:10 pm

Hmm. What effect would orbiting in mutually perpendicular orbits have on the orbits themselves? Some special features of such an orbit are that two planets orbiting at the same radius will always be equidistant to each other. (Think an n,n-duoprism here: every cell in one ring touches every other cell in the other ring.) Since the planets would attract each other gravitationally, it seems that their tendency would be to pull each other out of such an orbit, until they collide. Or am I missing something here?

Or maybe, if their orbit is fast enough, maybe they will settle into a spiralling motion instead? I'm not sure. It's hard to say without actual calculation. But since there doesn't seem to be a counteracting force to the planet's mutual attraction, it doesn't seem likely they will remain in such an orbit for long. If they don't collide, they would quickly twist their orbits out of the mutually-perpendicular system.
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Re: 4D planets (split from "Rings in 4d")

Postby wendy » Thu Jan 27, 2011 8:04 am

For what it's worth, here is my take on 4D planets.

A planet is a system with several modes of rotation, the nature of this is an energy transfer so that the modes of rotation become equal by energy. For a planet in four dimensions, this means that clifford rotation applies. All stars travel on non-intersecting hedrices or 2d space, this cuts across the sky as great circles. The lines swirl across the sky, because every line must set 180° opposite.

One can make a 'universal sphere' that lies on the ground, where one stands opposite the rising of the zenith-star rising, and sets the rising of other stars accordingly. It's kind of like a compass based on the local star-risings. What it gives you is what 'trace' or great circle you are on, which appears as a point on the sphere.

The sun moves, or rather moves in the same coordinate system that the earth and the stars are still. In practice, it's like our sun moving through the stars. There is no reason to assume that the sun follows a trace or great circle on the earth, rather the zodiac or sun-risings follow a great circle that crosses the zenith traces. That is the zodiac is a small circle on the zenith-sphere.

We can assume, for no good reason, that an inclination of 23.5 degrees applies. The effect is that as follows.

The zodiac appears as a small circle at 43°S (ie 2×23.5 from the south pole), these traces represent the tropics. The 43°N represents the polar circles.

The points on the tropics, for one day of the year, the sun is over-head. We have the tropic of cancer and capricorn. In four dimensions, you have the tropic of leo, the tropic of libra, &c.

The points at the sun-pole, is a great circle, which during the day, the sun rises to a maximum height of 66.5 (90-23.5), every day of the year. It just reaches a different point in the sky every day. Now, imagine pulling this circle towards the horizon. One As one pulls it towards a point, the circle has a maximum and minimum, ie at 50°s, the circle comes within 3degrees of the zenith, and at winter, only 43-degrees into the sky.

For points in the temporate range, the circle of risings lies between the zenith and the horizon, still of 47° diam. For example, at the middle on the zenith-sphere, the sun rises as high as 45+23.5 = 68.5, and as low as 45-23.5 = 21.5.

In points in the polar region, the sun sometimes hangs on the horizon, and sometimes rises higher, but never higher than 47 degrees, and often very much lower. At the poles, it never rises more than 23.5 degrees.

Back at our zenith-sphere, we now put the zodiac, with the months &c, like you sometimes see on earth-zodiacs. Anything that has the same longitude as say Leo, would have their summer in Leo. Where the longitude is that of Taurus, the summer falls then. It's just like our N=cancer, S=capricorn, but you have a full circle, not just the opposites.

We then have
  • point on zenith-circle = time of the day
  • zenith-longitude = time of the year
  • zenith-latitude = climate (the less the sun rises, the colder it be)

On the ground, you now have time-zones (as before), and season-zones [great circles falling on a longitude on the zenith-sphere], and climate-stripes [latitude on the zenith sphere]. The climate-lines run perpendicular to the timezone and season zones, but the other two run at angles equal to distance from the climate-line (latitude on the sphere) to the nearest pole.

How much space is in the tropics and poles. This is easy to enumerate, since this is the versine of the zenith-latitude (climate).

  • Polar and Equatorial = 15.9%
  • Temperate = 68.2%

None the same, the entire polar region is connected, so polar bears and penguins can live side by side.
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Re: 4D planets (split from "Rings in 4d")

Postby anderscolingustafson » Sun Mar 13, 2011 3:40 pm

What about if there was a tidally locked 4d world? How would the directions be defined for a planet like that? For that matter how would one define the directions of any tidally locked world?

One couldn't measure East and West along the Solar Equator for a Tidally Locked world regardless of it's number of dimensions as a tidally locked world would have no Solar Equator. For a tidally locked world that was 3d one could define one direction as going from the light side to the dark side of the planet and the other as going from the dark side to the light side of the planet. For instance North could be the direction going from the light side of the planet to the dark side and south could be the direction going from the dark side of the planet to the light side. For 3d one could then define the remaining two directions as those around the ring that would be in between the light and the dark side so with one being East and the other West. For a 4d tidally world this would make it easy to define the North South axis but then it would make defining the East West, and Marp Garp axis problematic as even if East West were to go along a ring with East going one way and West the other and then Marp was to point toward one point and Garp the other and both axis were to be on the imaginary sphere that was between the light and dark side of the planet were the ring and two points would be that would define East West, Marp and Garp would still be arbitrary as so were these directions would be would be arbitrary as there would be no way of defining the place of East West, and of Marp Garp for a tidally locked 4d world.
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Re: 4D planets (split from "Rings in 4d")

Postby anderscolingustafson » Tue May 24, 2011 12:58 am

For a 4d planet would we define North as the direction pointing from the solar equator to the polar equator or the direction pointing from the polar equator to the soler equator?
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Re: 4D planets (split from "Rings in 4d")

Postby wendy » Wed May 25, 2011 9:19 am

Tidally locked planets still rotate, but just show the same face to another body, like the moon.

Really, you can't expect the rotations of things to follow 3d just because 3d does. In 2d, the whole planet spins - everywhere is the equator.

When you look at four dimensions, and some fairly basic laws (that where several modes of rotation exist, the distribution of energy will happen to equalise the energy), then the clifford rotation will happen.

And what's wrong with this?

In two-dimensions, the effect of rotation is to add time zones.

In three dimensions you add a single spinner, that sets the hemispheres 6 months apart. (It's kind of like the spinners that were dice-replacements).

In four dimensions, you get a full circle of season-zones, like time zones. That is, there's a zone corresponding to +3 months, and another to +4 months. Where-ever you are, some zone will be +2 months ahead, and some -2 months behind etc.

It's only those folk that think that four dimensions is a kind of 3d, that run to problems.
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Re: 4D planets (split from "Rings in 4d")

Postby Halfbaker » Sat Jun 04, 2011 4:18 am

Wait, but wouldn't a hyperplanet rotating in two different directions just be rotating on it's side? You can add any two vectors regardless of dimension, so...

Also, how would seasons and time zones work for a pentonian planet?
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Re: 4D planets (split from "Rings in 4d")

Postby quickfur » Sat Jun 04, 2011 5:27 am

In 1D, there is no rotation. In 2D, there's really only 1 rotation (in the 2D plane itself). In 3D two (simple) rotations combined equals a third simple rotation.

However, in 4D and above, this is not necessarily the case. In 4D and 5D, it is possible to simultaneously rotate in two mutually-orthogonal planes, such that the two rotations have two independent rates of rotation. These rotations do not combine into a third, simple rotation. In 6D, it's possible to have 3 simultaneous, independent rotations. In general, in N dimensions, an object can rotate in N/2 independent planes simultaneously. In all cases, these planes cannot be just any arbitrary planes; they must intersect each other in no more than a point (otherwise they will combine into a simpler rotation).

So here's another example of why generalizing from lower dimensions without careful analysis can lead to wrong conclusions. :)
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Re: 4D planets (split from "Rings in 4d")

Postby Keiji » Mon Jun 06, 2011 8:47 pm

quickfur wrote:However, in 4D and above, this is not necessarily the case. In 4D and 5D, it is possible to simultaneously rotate in two mutually-orthogonal planes, such that the two rotations have two independent rates of rotation. These rotations do not combine into a third, simple rotation.


Does this mean a 4D orientation could not be represented by four numbers, unlike how all 3D orientations can be represented by three numbers?
Or can all possible 4D orientations be normalized into xy and zw rotations?
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Re: 4D planets (split from "Rings in 4d")

Postby quickfur » Mon Jun 06, 2011 9:23 pm

Keiji wrote:
quickfur wrote:However, in 4D and above, this is not necessarily the case. In 4D and 5D, it is possible to simultaneously rotate in two mutually-orthogonal planes, such that the two rotations have two independent rates of rotation. These rotations do not combine into a third, simple rotation.


Does this mean a 4D orientation could not be represented by four numbers, unlike how all 3D orientations can be represented by three numbers?
Or can all possible 4D orientations be normalized into xy and zw rotations?

Hmm... I know that 3 angles are sufficient to uniquely specify a direction in 4D (analogous to how 2 angles specify a direction in 3D), but as for orientation, I think it's a little more complicated. So we need a little analysis here:

To uniquely determine an orientation of an object, we can fix a particular point of the object as the "forward-pointing" part. This forward pointing part can then point in any direction in N space, which means (N-1) angles are required here. Furthermore, after the object is pointed in a particular direction, it can still freely rotate in the (N-1)-hyperplane perpendicular to this direction. The number of angles needed to fully specify its orientation in this (N-1)-hyperplane is the number of angles required to specify an orientation in (N-1)-space.

So in 1D, zero angles are needed to specify an orientation (there is only one possible orientation; there is not enough dimensions to rotate an object so that its front and back are exchanged). In 2D, exactly 1 angle is needed. In 3D, it takes 2 angles to specify a direction, and another angle to specify its orientation in the perpendicular 2D plane, so that makes 3 angles. In 4D, it takes 3 angles to specify a direction, and 3 angles to specify its orientation in the 3-hyperplane perpendicular to this direction, so you need 6 angles to fully specify an orientation. In the same vein, in 5D it takes 4 angles to specify a direction, and 6 angles to specify its orientation in the 4-hyperplane perpendicular to this direction, so a total of 10 angles is required to fully specify an orientation.

In general, if we denote the number of angles needed to represent an orientation by Q(n), then we have:

Q(1) = 0
Q(n+1) = (n-1) + Q(n)

Solving this recurrence, we get Q(n) = n*(n-1)/2, which is the sequence of triangular numbers. A rather neat coincidence, don't you think? :) So the first few values are:

Q(1) = 0
Q(2) = 1
Q(3) = 3
Q(4) = 6
Q(5) = 10
Q(6) = 15
Q(7) = 21
Q(8) = 28
Q(9) = 36
Q(10) = 45

As you can see, the fact that in 3D exactly 3 angles are needed to specify an orientation is just another of those coincidences in 3D (along with the number of principal rotations). The number of angles is not equal to the number of dimensions, in general. :)
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Re: 4D planets (split from "Rings in 4d")

Postby quickfur » Mon Jun 06, 2011 9:25 pm

P.S. It just occurred to me that this sequence is exactly the same as the number of principal rotations in N dimensions. In other words, to fully specify an orientation you need to specify a rotation in each of the possible principal rotations in N space. Which makes sense, intuitively.
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