Extending lower dimensional rules to 4D

[PWrong]

I'm a bit worried about the way we simply extend basic mathematical facts into the fourth dimension.

A common process on this forum is to say:

In 1D a particular number will always be 1,
In 2D this value will be 2,
In 3D the value will be 3,
So therefore in 4D, the number must be 4.

Can we always be sure of this? There are cases where this doesn't happen. Here are some examples:

1. We don't know whether 4D gravity is inversly proportional to the distance squared or the distance cubed.

2. An orbit is always two dimensional, no matter how many dimensions are available.

3. An object can oscillate in 1 dimension, and there is such a thing as circular motion, but there is no such thing as spherical motion.

4. Sound is always two-dimensional.

5. Tetronians can pluck sheets and tie them into knots.

Most of these can probably be proven easily with enough work. But some of the more abstract problems have to be considered carefully.

We can't just say, "this happens in 2D and this happens in 3D, so this must happen in 4D". We have to understand why it happens in our universe and reduce it to a mathematical problem.

Some apparently obvious things can be extremely hard to prove. The Poincairre conjecture is the perfect example. I don't understand it myself, but it's something to do with the 4D analogue of rubber bands stretched around a sphere and a donut.

If we stop assuming things about the 4th dimension, we might find another problem like this for mathematicians to occupy themselves with.
Any comments on this? Could there be any completely unsolvable questions about the 4th dimension?

[Geosphere]

Boy, I pretty much disagree with every example you have given.

Orbits are 2D? Not if you're an electron.

No such thing as spherical motion? Fill a balloon sith soda, swirl it. Watch the bubbles.

Sound is 2D? Man, I thought this had been beaten to death in the Music thread...

And since the sheets into knots thing is without backing or definition, I can't comment.

[pat]

I'm a bit worried about the way we simply extend basic mathematical facts into the fourth dimension.

Understandable and good. For each of your points, I'm going to give my justification as to the most natural way to consider extending the points and to give justification of them.

1. We don't know whether 4D gravity is inversly proportional to the distance squared or the distance cubed.

The assumption here is that there is only a particular amount of gravity being "emitted" from a given source and that it is emitted equally in all directions. Thus, by a Gauss's Law sort of argument, spherical shells centered at the mass must have the same gravitational flux. In 4-D (assuming that all four dimensions are homogeneous), the amount of "space" those spherical shells take up increases as the cube of the radius.

If we truly think of gravity as bending and warping of space-time instead of something "emitted", then it could conceivably act in any way. It could fall off exponentially with distance or linearly or what-have-you. If we want to think of it in the sort of rubber-sheet universe often pictured in mass-market-science magazines, then it'd be pretty tough to have it be anything beyond a monotonically decreasing function of distance. But, possible, I suppose.

2. An orbit is always two dimensional, no matter how many dimensions are available.

Here, the assumption is that gravity acts as if radiating from a point, that it acts equally in all directions, and that orbits are a function of the gravity and the momentum of the body. The momentum of the body is one vector, the direction of the gravitational field is the another vector. There are no other forces. There is only one plane containing those two vectors.

3. An object can oscillate in 1 dimension, and there is such a thing as circular motion, but there is no such thing as spherical motion.

This follows mostly the previous paragraph's argument. There is the momentum of the body, the force pulling the object to the center, and any other force acting on the body. If there is no other force, then there are only two vectors... only 2-d.

4. Sound is always two-dimensional.

That doesn't make sense to me. I cannot argue for that one. 8^)

5. Tetronians can pluck sheets and tie them into knots.

I always have problems with these kind of statements. Bob can pluck stuff out Fred's safe. Emily can pluck stuff out of Bob's safe. Sure, those are true if Fred is actually entirely flat and Bob is flat to Emily... But, I usually prefer to think that if Emily is 4-D, then so are Bob and Fred. It just so happens that Fred only has a 2-D perception and range of motion, and Bob only has a 3-D perception and range of motion. Thus, it's not immediately clear whether Fred's safe is really just a bottomless and topless box to Bob or whether it's a box with top and bottom to Bob.

We can't just say, "this happens in 2D and this happens in 3D, so this must happen in 4D". We have to understand why it happens in our universe and reduce it to a mathematical problem.

Very true. It's far too easy to generalize patterns from smaller dimensions.

If we stop assuming things about the 4th dimension, we might find another problem like this for mathematicians to occupy themselves with.
Any comments on this? Could there be any completely unsolvable questions about the 4th dimension?

There are quite a few interesting things that appear only in the 4th dimension. Another example (somewhat related to the Poincare conjecture) is this. If you make two different smooth coordinate systems in the plane, then for any given region around a point, you can make a smooth mapping from one coordinate system to the other. This is called a diffeomorphism. So, for example, if I had the polar coordinates and the Euclidean coordinates of the the point whose polar coordinates are r=5 and theta=30-degrees, then there is a region around that point where both coordinate systems are smooth. And, there is an invertible, smooth (differentiable) function that maps polar coordinates to the corresponding Euclidean coordinates. We can do this for any smooth coordinate system in 2-D. All 2-D coordinate systems are diffeomorphic.

In 3-D, we can also do the same. All 3-D coordinate systems are diffeomorphic.

In 4-D, it turns out, that there are infinitely many classes of smooth coordinate systems which are not diffeomorphic to each other. There is an infinite set of smooth coordinate systems such that it's impossible to take any pair of coordinate systems in the set and make a smooth, invertible function that maps one to the other. How wacky is that?

It turns out, that for 5, 6, and 7 dimensions (IIRC), all coordinate systems (within the same dimension) are diffeomorphic. And, it turns out that in 8 dimensions, there is a set of 28 coordinate systems (and no more than 28) such that if you take any pair of coordinate systems in the set, it is impossible to make a smooth, invertible map between them. And, if you take some coordinate system not in the set, then there exists a smooth, invertible map between that coordinate system and one of the 28 in the set.

And, that's it. In higher than 8 dimensions, every coordinate system can be smoothly mapped to every other coordinate system (within the same number of dimensions).

Wacky, wacky stuff which I hope some day to understand.

[PWrong]

Orbits are 2D? Not if you're an electron.

I meant gravitational orbits.

No such thing as spherical motion? Fill a balloon sith soda, swirl it. Watch the bubbles.

I don't mean a sphere that moves, I mean an object that moves around to cover the surface of a sphere. If you spin a bucket around you, the bucket moves in a circle. That's circular motion. Spherical motion doesn't exist even in the 4th dimension.

Sound is 2D? Man, I thought this had been beaten to death in the Music thread...

I know it has. I'm not trying to bring it up again, I'm just taking examples from popular threads.

And since the sheets into knots thing is without backing or definition, I can't comment.

Tying sheets into knots is mentioned in the glossary under "sheet".



Thanks for your explanation about the gravitational flux pat. I'm learning about magnetic flux in physics, and I didn't think of applying it to gravity.

Tetronians can pluck a sheet the same way a guitarist plucks a string, as mentioned in the music thread.

I'd never admit it to my friends (they all hate maths), but that coordinate system thing looks really interesting. I think I vaguely understand it. Are Euclidean coordinates the same as Cartesian coordinates? So, using your example, I can convert the polar coordinates(5, 30) into (sqr(3)/2, y=2.5), using the function:

x=r*sin(theta)
y=r*cos(theta)

I can do a similar conversion in 3D, but not in 4D. Is this right, or are you talking about something far more complicated than year 11 trigonometry?

Either way, that's exactly the kind of problem I was talking about. In fact, I didn't realise it was such a big problem. Maybe it would be easier to understand the 5th dimension than the 4th, if we can't convert the coordinates smoothly.

[jensr2000]

I can't understand it either. From my point of view, every coordinate system, for which a unique coordinate produce a unique point - I think it's called injective - must be interchangeable with another system also being injective.

Assuming polar coordinates can be described as:
x = r cos(theta)
y = r sin(theta)
and
r = sqrt(x^2+y^2)
theta = arctan2(x,y)

So where is the problem to translate that into 4D (glomar coordinates):
x = r sin(psi) sin(phi) cos(theta)
y = r sin(psi) sin(phi) sin(theta)
z = r sin(psi) cos(phi)
w = r cos(psi)
and
r = sqrt(x^2+y^2+z^2+w^2)
psi = arccos(w / sqrt(x^2+y^2+z^2+w^2))
phi = arccos(z / sqrt(x^2+y^2+z^2))
theta = arcsin(y / sqrt(x^2+y^2))

So what's this 'diffeomorphic' in relation with coordinate systems?

[Aale de Winkel]

this mimics the stuff already handled in the Geometry 4-d coordinate system thread. It is the first time I saw the term diffeomorphic, for me every coordinate system are isomorphic to one another, no matter what dimension.
A coordinte system is a mapping M from one object onto an other so the isomorphisms between maps can be described as M[sub]1[/sub]M[sub]2[/sub][sup]-1[/sup].
.

[pat]

I'd never admit it to my friends (they all hate maths), but that coordinate system thing looks really interesting. I think I vaguely understand it. Are Euclidean coordinates the same as Cartesian coordinates? So, using your example, I can convert the polar coordinates(5, 30) into (sqr(3)/2, y=2.5), using the function:

x=r*sin(theta)
y=r*cos(theta)

I can do a similar conversion in 3D, but not in 4D. Is this right, or are you talking about something far more complicated than year 11 trigonometry?

You can still convert polar coordinates to Euclidean coordinates via a smooth, invertible map. But, there are other coordinate systems (infinitely many different classes of them) in 4-D that are self-consistent but cannot be smoothly mapped to Euclidean coordinates. It is far more complicated than year 11 trigonometry, and I don't really know all of the details myself. But, I know enough to explain more of it than just the hint above. It would take more time than I have at the moment to explain in more detail. But, if there's interest, I'll start another thread tomorrow on it. In the meantime, you may want to check out Exotic R4 on Mathworld.

[pat]

So where is the problem to translate that into 4D (glomar coordinates):

As I just mentioned in a reply to someone else, there is not a problem with glomar coordinates. The problem is that there are ways to make coordinate systems which are, within themselves, smooth and consistent, but which cannot be smoothly mapped to Euclidean (or glomar coordinates).

A coordinte system is a mapping M from one object onto an other so the isomorphisms between maps can be described as M[sub]1[/sub]M[sub]2[/sub][sup]-1[/sup].

Yes, if your only concern is that for every point in your space, each of your coordinate systems has one and only one set of coordinates for that point, then every coordinate system is isomorphic. Isomorphic is different from diffeomorphic. I'll get into it more in a separate thread (tomorrow).

For now, consider this really, really ugly possible change of coordinates. Given a point (x,y) in the plane, send that point to:

• M(x,y) = ( x, y ) if both x and y are rational
• M(x,y) = ( x, y+1 ) if x is irrational and y is rational
• M(x,y) = ( x+1, y ) if x is rational and y is irrational
• M(x,y) = ( x+1, y+1 ) if x is irrational and y is irrational

These coordinates would be isomorphic to regular, Euclidean coordinates. We can easily map back and forth between them. These coordinates, however, aren't smooth. And, the mapping from Euclidean coordinates to these coordinates (and the inverse mapping) is not smooth. From a topological standpoint, the fact that these new coordinates are not smooth makes them uninteresting. But, in 4-D, there exist smooth coordinate systems which cannot be mapped smoothly to (and from) Euclidean coordinates. They can still be mapped to (and from) Euclidean coordinates... just not smoothly. I think that's interesting especially since in 3-D and 2-D, every smooth coordinate system can be mapped smoothly to (and from) Euclidean coordinates.

If you're not concerned about smoothness, then a coordinate system is a coordinate system is a coordinate system. But, if you don't have smoothness, then it's tough (or impossible) to formulate physics involving differential equations. And, it'd be nice to know that if someone else also formulated physics equations but didn't use Euclidean coordinates, that you'd still be able to translate their results back to Euclidean coordinates.

[Aale de Winkel]

Yes, if your only concern is that for every point in your space, each of your coordinate systems has one and only one set of coordinates for that point, then every coordinate system is isomorphic. Isomorphic is different from diffeomorphic. I'll get into it more in a separate thread (tomorrow).

Ah,I see "diffeomorphism" is a term when onehas multiple choises for the "source" of the mapping in order to make a one to one mapping.
This is of course the case for the polar coordinates of a sphere oneside open rectangle, or glome oneside open beam. one might choose the actual range one uses for the mapping.
I now faguely remember something, but is has been more then 20 years since I studied these kind of thing, so that thread of yours might be a real nice refreshor course for me (subthread of Geometry I think)
For your ugly coordinate changes, I've seen weirder mappings involving all kind of functions including chronecker delta's and theata functions, so nothing amases me in respect to mappng functions.

[RQ]

well, pwrong, all motion is in one dimension, even if gravity pulled you down, your path is a parabola, and still a curved one dimensional line.

[PWrong]

well, pwrong, all motion is in one dimension, even if gravity pulled you down, your path is a parabola, and still a curved one dimensional line.

hmmm, good point. I was talking about the way an orbit is always circular or elliptic, but the path the object travels is actually a line. It makes perfect sense actually. All motion is in one dimension because time is only one dimension. There's that evil word again. Sorry for bringing it up.

[BClaw]

5. Tetronians can pluck sheets and tie them into knots.

I could very easily be wrong, but I thought I had heard that knots of any kind were a 3-d phenomena that could not exist in any other dimension?

[pat]

I could very easily be wrong, but I thought I had heard that knots of any kind were a 3-d phenomena that could not exist in any other dimension?

I've heard that sort of thing before, too. I believe that the real version is this. If you have a length of string (which is really a sliced open version of the shell of a 2-D sphere S¹) in 3-D, you can tie it in knots and then fuse the ends together. If this string were then taken beyond 3-D, one could untie the knots without unfusing the ends of the string.

In 4-D, one can take a (shell of a 3-d) sphere (S²) with a slice in it, tie it into a knot and fuse the slice back together. It is not impossible to untie this knot in 4-d. If this knot were taken beyond 4-d, one could untie the knots without unfusing the slice.

This pattern continues up through the dimensions where you can tie the shell of an n-1-dimensional sphere (S^n-2)into a knot in n-dimensions and it's no longer a knot in n+1 dimensions.

I think.

[Watters]

well, pwrong, all motion is in one dimension, even if gravity pulled you down, your path is a parabola, and still a curved one dimensional line.

The path of a porabola is defined in both x and y functions. it would be imposible to define a porabola with only an x function (it would just be a distance).

[Watters]

Sry i screwed up. the top line is the quote. THis part is mien:

The path of a porabola is defined in both x and y functions. it would be imposible to define a porabola with only an x function (it would just be a distance).

[PWrong]

If you have a length of string (which is really a sliced open version of the shell of a 2-D sphere S¹) in 3-D, you can tie it in knots and then fuse the ends together.

Do you mean a string is really the outside of a 1-D sphere? A circle is the set of points on the edge of a disk, which is actually a 1D line. I know it's confusing, but isn't the usual term a 1-sphere, for this reason?

[pat]

If you have a length of string (which is really a sliced open version of the shell of a 2-D sphere S¹) in 3-D, you can tie it in knots and then fuse the ends together.

Do you mean a string is really the outside of a 1-D sphere? A circle is the set of points on the edge of a disk, which is actually a 1D line. I know it's confusing, but isn't the usual term a 1-sphere, for this reason?

Yes, I should have been more clear. The convention, from what I've seen, on these pages to call a circle a 2-dimensional sphere. The mathematical convention is to call a circle a 1-sphere denoted S¹. I was using bad terminology in hopes of being better understood. I should know better.

Yes... what I should have said here was:

If you have a length of string (which is really a sliced open version of a circle S¹) in 3-D, you can tie it in knots and then fuse the ends back together. If this string were then taken beyond 3-D, one could untie the knots without unfusing the ends of the string.

One can take a spherical shell (S²) with a slice in it, move it into 4-D, tie it into a knot and fuse the slice back together. It is impossible to untie this knot in 4-d. If this knot were taken beyond 4-d, one could untie the knots without unfusing the slice.

[elpenmaster]

if motion were in 2-d, then time would be 2-d. then a particle can be in two or more places at once.

[PWrong]

Isn't that exactly what quantum mechanics is all about?

[Aale de Winkel]

Quantum mechanics is about movement at sub atomic level, Heisenberg uncertainty relations and stuff ΔpΔx ~ h for a particle one can't know both impuls and location, this is essentially what's making everthing expressed as probability that it is (that's what Einstein made proclaiming "Gott würfelt nicht")
There is a change of this is such and so.

2 dimensional time is a nice concept to think about, however I haven't the faintest idea what temporal left nor right would be, current time knows only past and future. and we are currently stuck on the present which moves into the future due to the nature of time.

A particle that exist at two places at the same time, I heard such before, but I assume it was in the realm of sci-fi stories, not real physics. all quantum mechanics says that a certain particle has certain probability to be at a given position.

Interference patterns (often calculated in quantum mechanics classes) exists due to the multitude of particles (often photons) that bombards the slithed wall given the interference pattern. A singular photon would simply travel some singular path.

[Watters]

2 dimensional time is a nice concept to think about, however I haven't the faintest idea what temporal left nor right would be, current time knows only past and future. and we are currently stuck on the present which moves into the future due to the nature of time.

There is this theroy (it is the new generaly belived idea) that in 3d space we can move forward or back in sapce at any time but only go forward in time. The Theroy is that at least one referance frame always has to move forward, So if some one enters a blackhole (and for some reason doens't get crushed) they would be able to move forward and back in time, but only forward in space. This is asmuing that there is no 4th dimension and time and the 3d space are the only two referance frames that can be switched

[PWrong]

I've read about particles being in two places at once in scifi, but also in science books and reliable sources on the internet. I thought one of the most important things about quantum theory is that impossible things happen all the time.

The double slit experiment shows that interference patterns exist even with only one photon. It doesn't make any sense, but it still happens, supposedly because the one photon is travelling every possible path and it can interfere with itself.

[Watters]

Ya....i heard that too......they are now doing reserch on new "quantum mechanics computers". As every eon proble knows a computer works on a systems or 1's and 0's (on or off). these quantum mechanics computers are using photons that simultaniously are on and off at teh same time. (Paradox, but it works) thoguht it was pretty cool. They had a guest lecture on it at my university but i had an exam on that day...choked.

[RQ]

Yes, I do know that a parabola is defined by both x and y, but the path non the less is not an area traveled by objects, but a line, no matter how much you curve or twist it.

[PWrong]

I've heard of them too. I want a quantum computer

[Watters]

Yes, I do know that a parabola is defined by both x and y, but the path non the less is not an area traveled by objects, but a line, no matter how much you curve or twist it.

It is not a line. a line is y=xm+b. A parabola is not that. (basic math). The function of a porabola can't not be a line. the very functions defining the objects are diferent and there fore the objects are diferent. you can't say that just because you can straighten out the porabola means that it is a line. This is math not crapy logic science.

[elpenmaster]

it would be impossible to define a parabola in only one dimension, but you could define its length in only one direction. motion as a length is always 1-d, even if its shape isnt

[RQ]

I mean the parabola's shape is not a line, but a curved one, as in non Euclidean space. That's what I mean.

[Watters]

You are defiing a dimension that is repersented by a line, there is a big diferance between a line and.....lets say time. time can be repersented by a line and then curved, but a actual line (a object) can't be bent else it is not a line.

[RQ]

I don't think there is any practicality in changing computers' hardware. For one the cost, but besides that electricity is instantaneous (that is to connected places as in a cable or inside chips) and light isn't. I don't mean to say that the speed of electricity is infinity, it is 123 meters/second, way slower than light, but think of talking to your a person in another state. When you talk, there isn't sound from you and then a pause, until the electricity gets to the other receiver. The electrons push a line of electrons and even though it's not the same electron that makes the sound over the phone, it is the same wave pattern, and makes the other person hear it instantaneously, whereas in light it would take less time. even though the time it takes is so much less, why bother making new computers when we already have these?