4-d coordinate systems

Higher-dimensional geometry (previously "Polyshapes").

4-d coordinate systems

Postby alkaline » Tue Jan 06, 2004 10:15 pm

4-d coordinate systems:

- cartesian: x,y,z,w

- cubindrical: r,θ,y,w (same as cylindrical + w, with r and θ specifying the coordinates on the x/z plane)
w = w
y = y
z = r cos θ
x = r sin θ

- spherindrical: r,θ,φ,w (same as spherical + w)
w = w
z = r cos θ
x = r sin θ cos φ
y = r sin θ sin φ

- glomar: r,θ,φ,ζ
z = r cos θ
x = r sin θ cos φ
y = r sin θ sin φ cos ζ
w = r sin θ sin φ sin ζ

whoever is knowledgeable in coordinate systems, check over my equations.
Last edited by alkaline on Wed Jan 07, 2004 3:44 pm, edited 1 time in total.
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Postby Aale de Winkel » Wed Jan 07, 2004 6:21 am

good move of items. However my "Tetra Planetairy Coordinate" thread ought also be here!

The there listed polar coordinates mimics your "glomar coordinates" so those should be oke

a mere typo I think in the spherindrical: y = r sin(θ) sin(φ) I think :wink:
further a bit irregular permutation of coordinates, I'ld say:
sphere [x,y,z] = r [cos(θ),sin(θ)cos(φ),sin(θ)sin(φ)]

adding on
-duocircle; r,θ,φ
x = r cos(θ)
y = r sin(θ)
z = r cos(φ)
w = r sin(φ)
or simpler notated: [x,y,z,w] = r [cos(θ),sin(θ),cos(φ),sin(φ)]

also listed in some other places!

btw: the glossary still list the duocylinder as a tetraspace object, in stead of the pentaspace object it really is:
[x,y,z,w, x[sub]5[/sub]] = [r cos(θ),r sin(θ),r cos(φ),r sin(φ), x[sub]5[/sub]]
.
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Postby alkaline » Wed Jan 07, 2004 3:46 pm

alright, i fixed the typo with the spherindrical coordinates.

i'll get to that duocylinder eventually :-)
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Postby Aale de Winkel » Wed Jan 07, 2004 6:53 pm

I do think there the following formulates the polar coordinate system for all hyperspheres:

hypersphere: x[sub]k[/sub]: x[sub]k[/sub] x[sup]k[/sup] = r[sup]2[/sup]
x[sub]i[/sub] = r { [sub]k=1[/sub]Π[sup]i-1[/sup] sin(θ[sub]k[/sub]) } cos(θ[sub]i[/sub]) ; i = 1..n-1
x[sub]n[/sub] = r [sub]k=1[/sub]Π[sup]n-1[/sup] sin(θ[sub]k[/sub])

so with n-1 independent θ's

perpendicular linear extrusions onto these merely adds a further coordinate, the conic extrusion gives a certain angle say 2α at the conepoint which gives a value for r = d sin(α)
Last edited by Aale de Winkel on Thu Jan 08, 2004 6:35 am, edited 1 time in total.
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Postby alkaline » Wed Jan 07, 2004 7:35 pm

this one is wrong:

x[sub]n[/sub] = r { [sub]k=1[/sub]Π[sup]n-1[/sup] sin(θ[sub]k[/sub]) } sin(θ[sub]n-1[/sub])

It creates one too many sine's (it duplicates the last one). it should be this:

x[sub]n[/sub] = r { [sub]k=1[/sub]Π[sup]n-1[/sup] sin(θ[sub]k[/sub]) }
.
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Postby alkaline » Wed Jan 07, 2004 8:18 pm

Aale de Winkel wrote:adding on

-duocircle; r,θ,φ
x = r cos(θ)
y = r sin(θ)
z = r cos(φ)
w = r sin(φ)
or simpler notated: [x,y,z,w] = r [cos(θ),sin(θ),cos(φ),sin(φ)]

This coordinate system is very strange - it describes a point in four dimensional space, but with only three coordinates. Is this really possible? It violates the principle that you need four quantities to describe a point in space of four dimensions. Maybe it needs two radii:

coordinates: r,θ,ρ,φ
x = r cos(θ)
y = r sin(θ)
z = ρ cos(φ)
w = ρ sin(φ)

Maybe a good name for this coordinate system would be "duopolar coordinates", because it is two independent polar coordinate systems.
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Postby Aale de Winkel » Wed Jan 07, 2004 9:01 pm

Could well be that you need two radii, you then have a bigger circle "duoing" a smaller circle. Thus far I thought that the circles where of the same size,but this could well be let loose. Simular of course for the other duospheres.
In the case of what I on my own page called by the generic term "n-p-multisphere" one thus have p radii and angles.

http://home.wanadoo.nl/aaledewinkel/Enc ... hapes.html
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Postby alkaline » Thu Jan 08, 2004 12:39 am

i hope you didn't miss the fact that i posted two messages in a row (see above)
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Postby Aale de Winkel » Thu Jan 08, 2004 6:39 am

alkaline wrote:this one is wrong:

x[sub]n[/sub] = r { [sub]k=1[/sub]Π[sup]n-1[/sup] sin(θ[sub]k[/sub]) } sin(θ[sub]n-1[/sub])

It creates one too many sine's (it duplicates the last one). it should be this:

x[sub]n[/sub] = r { [sub]k=1[/sub]Π[sup]n-1[/sup] sin(θ[sub]k[/sub]) }
.


Yes, I missed this posting when reading from screen yesterday afternoon.
You are correct here, I took the liberty of editing this out.

Further as for the duocircle also in the hypersphere formulae each coordinate can have a different r, which turns the formulae into formulae for the hyperellipse.
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Postby alkaline » Thu Jan 08, 2004 4:20 pm

i good way to prevent missing new posts from a thread is to look at the page icons at the top of the post, to the left of "Posted: 08 Jan 2004 ...". The new posts will be in brown, old ones in white. Just scroll up to the first brown one then read down from there.

As for as duocircle coordinates goes, a single r for all of the coordinates limits the coordinate system to only describing points on a sphere; it isn't a generalized coordinate system. In contrast, the coordinates of the spherical coordinate system can describe points anywhere in realmspace. The glomar coordinate system is also a general system. Both spherical and glomar coordinate systems could describe ellipses/hyperellipses, but the duocircle system with a single radius couldn't.
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Postby Aale de Winkel » Thu Jan 08, 2004 6:25 pm

alkaline wrote:i good way to prevent missing new posts from a thread is to look at the page icons at the top of the post, to the left of "Posted: 08 Jan 2004 ...". The new posts will be in brown, old ones in white. Just scroll up to the first brown one then read down from there.


I haven't the faintest what this is about, that post I mised was simply due to the late hour. Also I don't see anything to the left of the "post..." headings(!?) Am I going blind :?: :lol:

alkaline wrote:As for as duocircle coordinates goes, a single r for all of the coordinates limits the coordinate system to only describing points on a sphere; it isn't a generalized coordinate system. In contrast, the coordinates of the spherical coordinate system can describe points anywhere in realmspace. The glomar coordinate system is also a general system. Both spherical and glomar coordinate systems could describe ellipses/hyperellipses, but the duocircle system with a single radius couldn't.


I don't know what you mean by "generalized coordinate system". If you mean that a duoellipse is impossible I don't know what else one should call the points:
[ R[sub]1[/sub] cos(θ[sub]1[/sub]) , R[sub]2[/sub] sin(θ[sub]1[/sub]) , R[sub]3[/sub] cos(θ[sub]2[/sub]) , R[sub]4[/sub] sin(θ[sub]2[/sub]) ]

so having 6 parameters to this "duo-ellipse"
.
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Postby Geosphere » Thu Jan 08, 2004 6:41 pm

Right above the post:

Posted: Thu Jan 08, 2004 10:25 am Post subject:

And theres a little amber or white page showing read or not just to the left.
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Postby alkaline » Thu Jan 08, 2004 9:54 pm

Aale de Winkel wrote:I don't know what you mean by "generalized coordinate system". If you mean that a duoellipse is impossible I don't know what else one should call the points:
[ R[sub]1[/sub] cos(θ[sub]1[/sub]) , R[sub]2[/sub] sin(θ[sub]1[/sub]) , R[sub]3[/sub] cos(θ[sub]2[/sub]) , R[sub]4[/sub] sin(θ[sub]2[/sub]) ]
.
so having 6 parameters to this "duo-ellipse"

By generalized coordinate system i mean that it can describe any point in four-dimensional space, with no constraints.

With a system of six parameters like you give, it allows a single point to be described by an infinite number of coordinates (even with θ[sub]k[/sub] limited from 0 to π). For the system to make sense, it must be true that either (R[sub]1[/sub] = R[sub]2[/sub] and R[sub]3[/sub] = R[sub]4[/sub]) or (R[sub]1[/sub] = R[sub]4[/sub] and R[sub]2[/sub] = R[sub]3[/sub]).

Look at a single plane, for example - what does [ R[sub]1[/sub] cos(θ[sub]1[/sub]) , R[sub]2[/sub] sin(θ[sub]1[/sub]) ] describe?
.
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Postby Aale de Winkel » Fri Jan 09, 2004 6:38 am

alkaline wrote:Look at a single plane, for example - what does [ R[sub]1[/sub] cos(θ[sub]1[/sub]) , R[sub]2[/sub] sin(θ[sub]1[/sub]) ] describe?
.


R[sub]1[/sub] and R[sub]2[/sub] are parameters of the ellipse at hand, I thought these where called the semi-major and minor radii, but according to: http://mathworld.wolfram.com/Ellipse.html this is something else :?:
whatever, the true and only variable is here θ[sub]1[/sub] [ R[sub]1[/sub] cos(θ[sub]1[/sub]) , R[sub]2[/sub] sin(θ[sub]1[/sub]) ] describes here thus a 1 dimensional line curved in 2 dimensional space :!: :lol:

And so the duoellipse has 4 parameters and two variables, so the duoellipse thus is isomorphic to IR[sup]2[/sup] embedded in IR[sup]4[/sup] in some curved manner. The duocircle has merely R[sub]1[/sub] = R[sub]2[/sub] and R[sub]3[/sub] = R[sub]4[/sub] 8)

With this in mind the hypersphere has 1 parameter (r) and n-1 variables (the θ's), and thus is a mapping from IR[sup]n-1[/sup] onto IR[sup]n[/sup]. described by the formula some postings back.

Note: it has been a while since I studied ellipses, looking through the reference above I reckon that the parametrisation [ R[sub]1[/sub] cos(θ[sub]1[/sub]) , R[sub]2[/sub] sin(θ[sub]1[/sub]) ] is off a bit. In the regular equation: (x/a)[sup]2[/sup] + (y/b)[sup]2[/sup] = 1; 2a is the major axis length and 2b the minor axis length. I'll look into this more carfully shortly, but anyway it'll be a mapping from IR into IR[sup]2[/sup] with two parameters and one variable, the range of the variable might be restricted to traverse the ellipse only once, this is not necesairy though! :roll:
clicking further from the refference above toward the elliptic cylinder shows the formula I suggested are right :?: . Why the reference above is so foggy I don't know :twisted: , also the elliptic cone page is of interest here :!:

A more carefull reading of the refference shows the parametrization of the ellipse as the equation pair 46 and 47, but filling them in into the equation shows that fulfillment is immediate with the identification R[sub]1[/sub] = a and R[sub]2[/sub] = b :idea:
.
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I think there are two threads going on here...

Postby pat » Fri Feb 27, 2004 5:02 pm

<em>First, sorry that I'm coming so late to this thread.</em>

There seem to be two goals expressed in this thread. One seeks to specify viable systems of coordinates for 4-space given the following constraint: Every location in R<sup>4</sup> can be specified in one and only one way.

The other thread seeks to parameterize particular surfaces in 4-space.

I'm going to focus mostly on the viable coordinate systems and get back to the parameterizations later. There seem to be five obvious, viable classes of coordinate systems: cartesian (x,y,z,w), cubindrical (r,θ,z,w), spherindrical (r,θ,φ,w), glomar (r,θ,φ,ψ), and duocircular (r,θ,ρ,ω).

I say viable classes because clearly if (a,b,c,d) is a viable coordinate system, then (f<sub>1</sub>(a),f<sub>2</sub>(b),f<sub>3</sub>(c),f<sub>4</sub>(d)) is also viable if the f are one-to-one and onto. One could also rotate, scale, or skew any of these coordinate systems and still come up with a viable coordinate system. But, these are basically the orthonormal ones.

To achieve the goal of uniqueness, one will have to restrict the ranges of the anglular coordinates. For the cubindrical: -π < θ < &#960. For spherindrical: -π < θ < π and -π/2 < φ < π/2. For glomar: -π < θ < π and -π/2 < φ < π/2 and -π/2 < ψ < π/2. For duocircular: -π < θ < π and -π < ω < π.

Now, as for specifying ellipsoids and such, these are most easily specified as scalings of the glomar coordinate outputs. One could construct an ellipsoidal coordinate system with glomar coordinates (r,θ,φ,ψ) through the mapping:
  • x = α r cos θ
  • y = β r sin θ cos φ
  • z = γ r sin θ sin φ cos ψ
  • w = δ r sin θ sin φ sin ψ

where α,β,γ,δ are all greater than zero.
Last edited by pat on Tue Mar 16, 2004 4:34 pm, edited 2 times in total.
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Animation of glome showing glomar coordinates...

Postby pat » Tue Mar 16, 2004 7:47 am

The following is an animation of a rotating glome. Given the glomar coordinate system:

x = cos b
y = sin b cos g
z = sin b sin g cos r
w = sin b sin g sin r

The red color is banded according to angle r, the green color is banded according to angle g, and the blue color is banded according to color b.

Glomar-Coordinate Animation
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Postby Aale de Winkel » Tue Mar 16, 2004 11:18 am

glad to see you here.

The hyperpolar.avi somehow bolts
Windows Media Player for Windows XP
version 8.00.00.4477

to be precise, the other avi's downloaded from this forum work fine. so might there be something wrong with this avi?

To the post before this one, other are many ways to specify location in any dimension. They are all diffeomorphic iff not isomorphic, as you also stated in your formula with f's.

To force the diffeomorphism from IR[sup]3[/sup] to the IR[sup]4[/sup] ellipsoids into an isomorphism, any choise of one end open beam (or perhaps better said bi-cube (π,π,2π)) is ok.
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Postby pat » Tue Mar 16, 2004 4:23 pm

Aale de Winkel wrote:The hyperpolar.avi somehow bolts
Windows Media Player for Windows XP
version 8.00.00.4477


Hmmm... I re-encoded it with some slightly different settings and uploaded it again to the same spot. Unfortunately, I don't have a Windows XP box on which to test. And, Windows Media Player version 9.0 for MacOSX doesn't even try to play AVI files.
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Postby Aale de Winkel » Wed Mar 17, 2004 5:53 am

Doesn't work either, Media Player can't find a suitable decompressor. Peculiar since the hypercube.avi's played without any problem.
Can you direct me to a suitable player / explorer plugin to play your avi's
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Postby thigle » Sat Aug 06, 2005 11:54 pm

sorry if the following sounds dumb, i am not a mathematician, just meta.thema.tician :oops:

in Miyazaki Koji's 'Adventure in Multidimensional Space', he states that a possible construction of the projection of 4d coordinates into 3-space is as follows: edges of 2 diamond lattices, painted red & blue, having all their lattice points in common, become projections of positive(red) and negative(blue) coordinate axes in 4-space into 3-space. does this correspond to projection of any of the coordinate systems mentioned already in this thread to 3-space ?

btw, it somehow doesn't click for me completely. if there are 4 axes in 4d, each having 2 directions to it, there should be 8 rays from each lattice point. or not ? however, as you can see when you juxtapose 2 diamond lattices, half of the points are just single-colored. ?

pat: I watched hyperpolar.avi it really is great. howevever, i don't get it satisfyingly enough. why do these families of confocal conics appear, in convex/flat/concave pulse ? I just know that there is some connection between glome and family of nested toris, which in section resemble ('topologicaly') to family of hyperbolas in confocal conics. but actually, i don't know how does one get family of nested toris from glome. what (if any) role does infinity & 0 play in this ? also, what is the representational method here ? the smallest slices on the sides are from the most distant 3-spaces and the middle slice is from the space into which it is projected, ?
also, you say that ellipsoids are most easily specified as 'scalings of the glomar coordinate outputs'. what are these ? confocal ellipses and hyperbola ? so what is the relation of quadrics and glome ?
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Postby Marek14 » Sun Aug 07, 2005 7:16 am

If every rotatope can be turned in a coordinate system, how about some of mine? :)

The system based on Dome in 3D would have coordinates x,y, and theta. Points in xy plane would be described easily, and theta would work the same as in spherical system.

For duocylindrical system, of course that there must be two different radii. Imagine it this way: first two coordinates tell you where you are in xy plane. The other two coordinates tell you where you are in zw plane. They are independent.

BTW, I'm not exactly sure what you mean by hyperellipses here. It can be cartesian product of two ellipses in perpendicular planes, or perhaps a hyperellipsoid, generalized glome?
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Postby wendy » Sun Aug 07, 2005 11:55 pm

The only rototopes likely to yield a coordinate-system are those expressable in terms of the rss and max products. A coordinate system has as many variables as the dimension, while rototopes are defined in terms of N(N-1)/2 free variables.
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Postby PWrong » Wed Aug 10, 2005 9:34 am

Isn't a coordinate system essentially just a change of variables?
For instance, in 2D,
x = f(u,v) and y = g(u,v)
where f and g are one-to-one and have continuous partial derivatives.

In most cases, we use rotatope coordinates because it's convenient, but we could really use any kind of transformation.

Apparently Laplace's Equation can be solved in 13 different coordinate systems, and that's just in 3D.

Then again, I haven't actually learnt this stuff, so I could have misunderstood. My textbook just happens to cover a lot of 2nd year calculus, so I tend to look ahead to the interesting bits instead of studying.
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Postby wendy » Wed Aug 10, 2005 11:24 pm

Laplace's operator is a real thing, and not a fait of coordinate-system. It has a solution in any coordinate system.

While one can in 4d devise many coordinate systems, one might look at what is practical. I tend to be a bit dis-satisfied with many of the current proposals, since this is not how i see space moving in 4d.

On the other hand, i have not figured out things like the shape of winter (ie what areas are in winter) in 4d. If the shape of '4-pm' is any thing to go by, this is simply a half glomohedrix, (ie the hemi-3-sphere), which is rotated around the world.

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Postby PWrong » Thu Aug 11, 2005 12:28 pm

Laplace's operator is a real thing, and not a fait of coordinate-system. It has a solution in any coordinate system.


I think what it means is there are 13 systems where it can be solved by separation of variables. Some solutions are easier to find in one coordinate system than another.

On the other hand, i have not figured out things like the shape of winter (ie what areas are in winter) in 4d. If the shape of '4-pm' is any thing to go by, this is simply a half glomohedrix, (ie the hemi-3-sphere), which is rotated around the world.


Sorry, you've lost me. :? The shape of 4-pm? Winter happens because sunlight is focused on a larger area. It's the same reason that antarctica is cold. The reason it changes is that the axis of the earth's rotation is not quite perpendicular to the plane of orbit. Other planets have completely different "seasons".

You could probably work out the seasons for a 4D planet, but it would be quite complicated, and you'd have to assume that stable orbits exist, which unfortunately they don't.
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Postby wendy » Thu Aug 11, 2005 11:22 pm

The argument around "stable orbits in 4d not existing" contains a number of assumptions, and also errors.

In the presence of a simple radiant field (like gravity or electricity), a circular orbit is indeed stable. But any variation to it is not, and in the long run, any natural system is likely to produce a non-stable orbit.

However, the nature of 4-space is that it may not necessarily be dominated by a simple radiant field. It may be some kind of polarised thing that radiates perpendicular to some kind of dipole affair, which would give rise to stable orbits from a radiant field.

The other thing is that it is better to look at it in a more historical context, such as "discover what is really needed to make it happen", rather than to presume some fundemental laws up front. One must understand that such laws are derived and describe observed variations of an existing system, so if one hopes to discover fundemental laws for 4d, one starts off with nature and works that way.

One can derive in 4d, the seasons etc, from watching the stars and the sun at a point on the surface. Because all points rotate around the centre of the earth, the days are always 12 hours long, and the nights the other 12 hours. However, there are seasons.

In 3d, seasons arise from the tilt of the earth, which leads to daylight varying to all values. In 4d, the nature of clifford-rotation makes all days the same, and tilt as we know it does not exist. Seasons derive exclusively from the elevation of the sun in the sky. When it is low, the land is less-warmed.

In our world, we have room only for two season-points: when it is 1 June, it is the first of Winter, but in the north, it is 1 Summer. In a 4d world, the same 1 June can fall on different days through-out the season-year. For example, it can be 11-Autum, or 62-Spring. There is always somewhere that is summer, somewhere that is autumn, somewhere that is spring, somewhere that is winter.

There exists then a shape of places that are at 11-Autumn. The places that are at 4 pm make a shape too. Exactly what shapes these are, i still do not know, but i think they're hemi-glomohedrices [half-3spheres], with the rim [equator] at some great circle.
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Postby thigle » Fri Aug 12, 2005 7:04 am

i thought that minimal something you need to stand still in spacetime(="stable orbit in 4d"?) is a spinor. or isn't it?
or do you mean that centre of rotation have to be outside of rotated object for considering it an orbit? (spinning being just a special case when rotation axes pass through the rotated object, which thus rotates around it's own centre of inertia... ?

also, no-one seems to reply to my previous post in this thread (i know, i sneaked into this rather precise and quite technical discussion with a bit of metaphoric stuff, sorry), especially the 'double diamond-lattice' coordinates is interesting to me, as well as the relation between glome/torii/and sequence of parametric conics(quadrics) (i think it does have a relation to potentially best coordinate system for 4d). i am just an amateur, though.
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Postby wendy » Fri Aug 12, 2005 10:02 am

i read your double-diamond post, but i did not understand it. A lattice has positive and negative coordinates, and you can't really hive off the negatives to somewhere else :S

i am not as mathematical as it might sound, just have a good feel for the subject.
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Postby pat » Mon Aug 15, 2005 2:12 pm

wendy wrote:However, the nature of 4-space is that it may not necessarily be dominated by a simple radiant field. It may be some kind of polarised thing that radiates perpendicular to some kind of dipole affair, which would give rise to stable orbits from a radiant field.


The more I try to question this statement, the more tangled I become in the fact that I'm only guessing what you mean by "dipole".

Here's the thing though. In our 3-D world, we do very, very well with the assumption that all of a body's gravitation field eminates from its center-of-gravity. To extend this into 4-D, our options are that the center-of-gravity is still a point or a line (or line segment) perpendicular to our original 3-D. Assuming the center-of-gravity is a line is tantamount to saying that space favors (or frowns upon) this fourth dimension... it singles it out from the other three. Of course, that singling could be an artifact in the trying to extend from 3-D to 4-D.

If instead, we say that gravity is somehow polarized in 4-D so that it radiates in circles (inverse-linear) in perpendicular planes, there's still a problem with all but circular orbits. If we say that it is somehow polarized in 4-D so that it radiates in spheres (inverse-square) in one 3-space and constantly (inverse-constant) in the perpendicular direction, then all of the 3-D orbits still work... but... this sort of thing doesn't make physical sense....

Imagine if gravity in 3-D worked like this.... it radiates in circles (inverse-linear) in the equatorial plane of the orbitted body and it radiates in lines (inverse-constant) in the other direction. Now, picture a body trying to orbit but not in the equatorial plane. Does the "inverse-constant" direction affect it at all? If it doesn't, then it's neglible... it can only affect something right over the pole. If it does affect it, then how can that work at all? If it's going to apply any force to a body, it has to be a radial force.... it doesn't have tendrils out along its equatorial plane.

Of course, I suppose, this brings me back to "dipole affair". Are you suggesting positive and negative gravity? where at some angles it pulls the object toward and at some angles pushes it away? My feeling is that that could only allow monopoles to orbit. But, I haven't tried to find stable solutions where dipoles orbit dipoles.
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Postby wendy » Mon Aug 15, 2005 10:52 pm

The notion of a radiant field is that the source radiates a flux, and the relative strength at some distance is proportional to the flux density. This applies to both a scalar field (eg light / heat) and a vector field (eg gravity, electricity).

The calculations of a two-body problem in four dimensions, do not allow us to either concentrate the mass at the centre, nor allow for elliptical orbits. A circular orbit is stable, but this stability is critical: that is, it is not near some other stable thing.

If on the other hand, we can restrict the effective gravity to a chorix, we can effectively create a radiant field that over the order of several AU, makes for an inverse square law. For this end, we need to look for kinds of star that do not have a spheric symmetry.

In our own universe, we do find examples of stars with cylindrical symmetry: the ones with an accretian disk. Even our own sun and its planets, the stars etc are all examples of this.

If we posit that the radiant field emits through for example, a chorix, and this chorix is somehow restrained to a thick chorix, then we could have stable orbits as we know them in our world, without having to invoke an inverse square law over a cubic volume.

One way of defining such a plane would be like a dipole: the ends of a line perpendicular to the plane in question.
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