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# 4D Sphere

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A 2D circle is defined as a shape where every point on it''s perimeter is equidistant from the center of the circle in 2 dimensions. A 3D sphere is the same thing, only in all 3 dimensions. So a 4D sphere (hypersphere, I guess?) would be...?

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Would be a 4D sphere : x^2 + y^2 + z^2 + t^2 = R^2

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Fruny, could you show me a graphical presentation of that?

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A sphere which exists in both space and time, which is very convenient if you have a look into relativity, because as you are most likely aware space and time are actually one thing, which Einstein referred to as space-time.

R2 in the equation above would only give the magnitude of one ray (vector).

Here is a theory on 4D for you.
If you are trying to think of an 3d object and how time is relevant to it, think of the object being like earth. Earth, I believe would be an example of a 4D sphere. The reason why I have chosen a planet is because change is very notable over time, and if you took earth at the same place and rotation in space (lets just say) in 1500AD and 2001AD, the face of the earth would have undergone massive changes, which affected the object. This massive change however is not a necessary feature, (I am only using it as a way of introducing the concept of time in relation to the sphere), and hence if I reference it at different points in time the sphere will have different properties, possibility even to a point where it is only an instance of its former state.

Well, that is my mixed up definition of a 4D sphere it is probably incorrect, but if it is good stimulus for your own thoughts, well that is enough.

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I'm not sure exactly how hyperspheres work, but I'm pretty sure to get a consistent one in real spacetime you have to multiply the t-values by i. So...

x2 + y2 + z2 - t2 = r2

Can someone correct me on that? I'm sure it must be more complicated. It's something to do with real time being all lumpy and imaginary time being homogeneous, or something

Edited by - Dracoliche on November 18, 2001 5:47:16 AM

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well here x,y,z,t is just a notation convention. It has nothing to do with the metric tensor ( ds^2 = dx^2 + dy^2 + dz^2 - c^2 dt^2 )

As for a graphical representation of the 4D sphere, check google ...

Or you can see a unit quaternion as a ''4D'' sphere : the vector (i,j,k) part as the radius vector, and the scalar as rotation around that vector ... et voilà, a 4D sphere.

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Anon said:
Fruny, could you show me a graphical presentation of that?

I think there''s a theorem that says its impossible to visualise anything in 4D with a 2D representation. Even in the 3D world we only see things in 2 dimensions, but our brain converts 3D to 2D (having two eyes helps a bit as well).
If you make one of the dimensions a time dimension you could show something changing in time, but there you are still making 3 into 2 for the brain to interpret.

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Representations of higher dimensional objects are general referenced as "shadows" cast on a lower dimension. So, just as an object in 3d space casts a 2d shadow, an object in 4d space casts a 3d shadow. Without having done the work, but just as a conjecture, I think it would be interesting to take a 4d object, cast it''s shadow on to 3d and then cast the shadow of that shadow onto 2d. Of course, since I''m posting as anonymous, I''m kind of a shadow here myself so what I think only counts in 2d, eh? :-)

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Hello guys!

Very interesting discussion about math theory you have here . Well, anyone asked for a graphical presentation of a 4D object. Well, it''s possible (looking somewhat weird, but it exists!).
For better understanding let''s analyze the coordinate systems we are used to (cartesian coordinates!):

2D (R²)... every single dot in this set of numbers can be accessed by 2 axes (x and y value).

3D (R³)... every single dot in this set of numbers can be accessed by 3 axes (x, y and z value).

4D ... so what you guess, eh? exactly, we need 4 axes. btw, you can even draw a 17D-polygon (for example).
I can''t figure out what this should help, but in theory it''s possible.

Funny to think it all over, eh?

Indeterminatus

--si tacuisses, philosophus mansisses--

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Hello guys!

Very interesting discussion about math theory you have here . Well, anyone asked for a graphical presentation of a 4D object. Well, it''s possible (looking somewhat weird, but it exists!).
For better understanding let''s analyze the coordinate systems we are used to (cartesian coordinates!):

2D (R²)... every single dot in this set of numbers can be accessed by 2 axes (x and y value).

3D (R³)... every single dot in this set of numbers can be accessed by 3 axes (x, y and z value).

4D ... so what you guess, eh? exactly, we need 4 axes. btw, you can even draw a 17D-polygon (for example).
I can''t figure out what this should help, but in theory it''s possible.

Funny to think it all over, eh?

Indeterminatus

--si tacuisses, philosophus mansisses--

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We will just need a new axis Shouldn't be too hard... All the points are, in reality, on the x/y axis, just shifted according to the Z-value and the angle of view...

Something I thought about recently... In many 3d-modelers, you got 4 views of your object: 3d, and three 2d screens...

Just thinking now... Can't you simplify a 4d object into three 3d screens, and those into three * three = twelve (I can count! Jippie!) screens?

[EDIT]

Hmmm... The irony... I state i can count, yet I screw up... Just noticed it... Hmmm, okay, make it nine then

Edited by - ronin_54 on November 18, 2001 12:16:33 PM

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I am not familiar with the theories behind processing 3 dimensional data into a 2 dimensional picture, but I wonder: could that theory be adapted to display a 4 dimensional picture? Or has this already been done?

rk

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Might take a few more images I think...

Splitting a 3d image into 3 2d images, requires you to just remove one coordinate from it, so you get a x/y, z/x and y/z image.

For 4d to 3d, that would become: x/y/z, x/y/''u'', x/z/''u'', y/z/''u''. Giving you 4 3d-images...

Hey, my 12 might be right after all!

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This discussion is getting very interesting. I've always found 4D objects to be really confusing, but doing searches for 'hypercube' and 'hypersphere' brought up these pages which really cleared things up for me:
http://www.geom.umn.edu/docs/outreach/4-cube/
http://www.cyburban.com/~mrf/hierarchy(1).html
Google is such a wonderful tool.

That last site has some very relevant information for this thread. Basically what it says (under the heading: "Myth #2: The hypersphere can't be visualized easily by those of us confined to three-dimensional thinking.") is that the surface of a 3D sphere can be represented by two 2D circles, one for each hemi-sphere. In 4D, the 'surface' volume of a hypersphere can be represented by two 3D spheres, one for each 'hemi-hypersphere'.

Still, this stuff makes my 3D brains hurt.

EDIT: typos

Edited by - Scarab0 on November 19, 2001 12:07:42 PM

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Very interesting discussion!

Anyone ever hear of a tesseract? I first learned about these reading the old Madeleine L'Engle novel "A Wrinkle in Time" when I was young. A tesseract is a 4-dimensional cube, and there are graphical representations of it. I found a few sites of interest:

http://www.geom.umn.edu/docs/outreach/4-cube/

The next one has a very interesting interactive tutorial graph. The plot is always projected into the 2D screen, of course, but the shape you see does change as it rotates. And it allows you do view "cross sections" of the tesseract.

http://www.cut-the-knot.com/ctk/Tesseract.html

There's also a page on 4D geometry at Mathworld.wolfram.com:

http://mathworld.wolfram.com/4-DimensionalGeometry.html

Also, anyone here read "Flatland" (or the more recent "Flatterland")?

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

Edited by - grhodes_at_work on November 19, 2001 1:38:12 PM

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Not 12 to go from 4d to 2d, only 6 (4 choose 2): xy, xz, xu, yz, yu, zu

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Since x^2+y^2+z^2=r^2 is a spherical surface then wouldn''t w^2+x^2+y^2+z^2=r^2 be a spherical solid, i.e. a set of spherical surfaces where the radius is r^2-w^s? So couldn''t you just animate the sphere as a shrinks or expands, i.e. time as the fourth dimension? It would be like passing a plane through a sphere. It seems somewhat metaphysical to say what a 4D object would look like if we could see 4 dimensions, i.e. 3d on a 2d screen is intended to look like a 3d object, but what does a 4d object "look" like? It seems you could say it "looks" like anything you please since you can''t be proven wrong.

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Was boared, so went from square to cube to 4d-object... Looks great! And mathematicly correct

Am going to show it to my math teacher tomorrow, and will post it on-line as soon as I can get my webhosting back on-line... grmbl...

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Here''s sort of a visual representation of a hypersphere...

And the POV-Ray code I used:

global_settings {  ambient_light color 3}camera {  location <0,0,-3>  look_at <0,0,0>}// main lightlight_source {  <-10,10,-10>  color rgb 1}// under lightlight_source {  <0,-10,0>  color rgb 0.5}sphere {  <0,0,0>  sin(clock*pi)    pigment {    color rgb <1,0,0>  }}

With the .ini settings:

Initial_Frame = 1
Final_Frame = 9
Initial_Clock = 0.0
Final_Clock = 1.0
Subset_Start_Frame = 2
Subset_End_Frame = 8

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That''s just an animation. I figured out a way to draw an n-dimensional object on a 2 dimensional plane

But am at school right now, so can''t upload it...

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Just if you wanted to know, time is not a dimension. Wheras the three spatial dimensions have centain symetries, time can only flow in one way, or it would counteract a fundamental law of the universe (2nd law of thermodynamics). It''s called something like symetry breaking. Multiple veiws would be an excellent way to represent larger dimensions. My maths course states that a circle and a sphere are loci of all points in 2 / 3 dimensions respectively that are a set distance from a commen centre, and from that all other conditions derive. Extending this into 4 dimensions, the same applies, i.e, a point P lies on the four dimensional sphere if || p - o || = r^2, where r is a constant, p is the position vector of P and o is the position vector of the origin. If p = (Xp,Yp,Zp,Wp) and o = (Xo,Yo,Zo,Wo) then this is represented algebraically by:
(Xp - Xo)^2 + (Yp - Yo)^2 + (Zp - Zo)^2 + (Wp - Wo)^2 = r^2
Taking the instance of o =(0,0,0,0) you have the form Dracoliche suggested. Ignore the fact that distance has no ''intuitive'' reasoning behind it at dimensions higher than 3, pythagourus rule still works (or rather is defined to calculate distance) and ignore also the fact that 4D cannot be easily spatially represented. Multiple dimensions are commenly used to solve problems, from economics, to phase space in which the universe (understood classically) is represented by something like 6*10^80 dimensions (six per molecule). Hope that helps.

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quote:
Original post by Anonymous Poster
Just if you wanted to know, time is not a dimension. Wheras the three spatial dimensions have centain symetries, time can only flow in one way, or it would counteract a fundamental law of the universe (2nd law of thermodynamics).

Good point, and quite correct. But modern cosmology''s string theory is based on the presense of not just 3, but 9 spatial dimensions in addition to the arrow of time. So all this talk of 4-dimensional geometries isn''t necessarily inconsistent with reality. (Maybe we should talk about visualizing 9-dimensional hyperspheres? Ha! )

http://www.astronomytoday.com/cosmology/superstrings.html

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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Well... Using my method, it *is* possible to draw an n-dimensional object... Though the time it takes rises exponentialy

For a 9-dimensional object... Pffft... That would take me a day or two

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Nice article there on String Theory, got some reading to do, it seems... Never quite got past Quantum Mechanics and Chaos Theory :-)

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I must say, a cool topic...

However if you want a graphical presentation of a 4d object in 3d think in this way:
Say we have a 2D world; The shadows from a 3D sphere outside would be projected as a cirkle. However, let''s say a 3d person was walking on this sphere. His shadow would also be projected on the (trancluent) cirkle in the 2d world. But those 2d-persons looking at the shadow wouldn''t be able to tell which side of the sphere he was. ( The question is, would they see the shadow at all?=)) However again, now think about the same situation but add a dimension. If someone projected a shadow of 4d cube to us, it would look like a cube, but every line would be a cube itself...
Then, it is just up to your imgaination to think about a 5 dimensional sphere would look like

Then some people say a 3d cube with a time attribute is also 4d, put i see the time as something unabsolute and abstract..

-(E)-we-