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.
4D Sphere
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
That''s about all I know about string theory!
Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.
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
For a 9-dimensional object... Pffft... That would take me a day or two
Nice article there on String Theory, got some reading to do, it seems... Never quite got past Quantum Mechanics and Chaos Theory :-)
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-
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-
Last I heard, superstring theory held that there were 10 dimensions.
3 normal, 1 time, and 6 curled up in a ball.
No I did not make that up.
I know that there are RedHat Linux screensavers that project hypercubes, so you might be able to dig up their source.
Also, I think 4D to 3D is a projection, so you could do a projection matrix for it.
If a circle is reprsented in 2D by all points a certain distance from the center and a sphere is all points in 3d a certain distnace from center point, then a 4D hypersphere is all points a certain distance from the center point.
How would you think of a 4D object ?
I use time.
A 4D point is (x,y,z,t), let us say.
A 3D object in 4-space would be (x,y,z, t(o))
Where t(o) describes the duration on the 3D object.
Because: point(x,y,z,1) only exists at time unit 1.
So it only holds true for that instant.
If you manage to think as 4 axes without using time, I would really like to know how you do it. :-D
I`ve read Flatland, a book on chaos theory, part of the book Hyperspace, and some articles.
That Java applet was cool- but it hurt my head to look at.
I came, I saw, I got programmers block.
~V''''lion
3 normal, 1 time, and 6 curled up in a ball.
No I did not make that up.
I know that there are RedHat Linux screensavers that project hypercubes, so you might be able to dig up their source.
Also, I think 4D to 3D is a projection, so you could do a projection matrix for it.
If a circle is reprsented in 2D by all points a certain distance from the center and a sphere is all points in 3d a certain distnace from center point, then a 4D hypersphere is all points a certain distance from the center point.
How would you think of a 4D object ?
I use time.
A 4D point is (x,y,z,t), let us say.
A 3D object in 4-space would be (x,y,z, t(o))
Where t(o) describes the duration on the 3D object.
Because: point(x,y,z,1) only exists at time unit 1.
So it only holds true for that instant.
If you manage to think as 4 axes without using time, I would really like to know how you do it. :-D
I`ve read Flatland, a book on chaos theory, part of the book Hyperspace, and some articles.
That Java applet was cool- but it hurt my head to look at.
I came, I saw, I got programmers block.
~V''''lion
I just can''t resist. I tried, but I can''t. I just can''t accept the arguement put forth that time is not a dimension. Conceptually we view the three spatial dimensions as infinite. If they are instead finite does that mean they are not dimensions? An example of how they can be finite is travelling across the surface of a sphere away from a point. No matter what path you take if you keep moving you eventually start moving closer to that point. There is a limit to how far away from that point you can get. Although as you travel across that sphere you are moving through three dimensions you are only moving in two dimensions in uv space for lack of a better name. That space is warped into three dimensions. Now I feel the arguement that time is not a dimension is like saying that because you are stuck on the surface of that sphere that there are only two dimensions. Being physically prevented from travelling backwards through time does not mean the past does not exist. I would argue that if time is not a dimension then the spatial dimensions are not dimensions either because they are relative to time, i.e. time and space are relative.
Hmmm... I can''t say I totally agree with your statement that the fourth dimension is time. But, I also can''t agree with the statement that time is *not* the fourth dimension... Why? Because I can visualise both options
Fourth dimension as a spacial dimension:
Take the normal x and y axis: +
The z-axis usually is a diagonal line: /
Now, at a "w" axis (got to call it something): \
Combine it into a single graphic: errr... can''t find a key
Anyway, write out a "4d-cube" (jotting down all the points). Then, just as you would draw a 3d-cube, start plotting all the points, and draw the interconnecting lines... Basicly, what you will end up doing, is the same as going from a square to a 3d cube: you take the object, move it along the new axis, and then draw lines between the old points and the new points (as someone mentioned before).
But, time could indeed be the fourth dimenstion as well. A line in 3d or 4d is basicly a beginning and an end, totalling a "lifetime". A line along the x-axis, from 2 to 5, would indicate a life of 3. A line along the "t"-axis from 1902 to 1905 would be the same thing, indicating a lenght of 3 units... Hmmm... LOL! I need to think about this a bit, before I can elaborate
Fourth dimension as a spacial dimension:
Take the normal x and y axis: +
The z-axis usually is a diagonal line: /
Now, at a "w" axis (got to call it something): \
Combine it into a single graphic: errr... can''t find a key
Anyway, write out a "4d-cube" (jotting down all the points). Then, just as you would draw a 3d-cube, start plotting all the points, and draw the interconnecting lines... Basicly, what you will end up doing, is the same as going from a square to a 3d cube: you take the object, move it along the new axis, and then draw lines between the old points and the new points (as someone mentioned before).
But, time could indeed be the fourth dimenstion as well. A line in 3d or 4d is basicly a beginning and an end, totalling a "lifetime". A line along the x-axis, from 2 to 5, would indicate a life of 3. A line along the "t"-axis from 1902 to 1905 would be the same thing, indicating a lenght of 3 units... Hmmm... LOL! I need to think about this a bit, before I can elaborate
what we call 3d really is 4d, we see it at a specific point
in time.
if you want to visualize a 4d sphere, throw a tennis ball,
that''s it.
it''s like inifinity, just wait til nightfall, go outside, and
look up. it''s boring.
in other words, we already exist in 4d,
for the 5th dimension, I''d say that that''s probability,
but that''s just my opinion-(Hitchiker''s Guide to the Galaxy)
in time.
if you want to visualize a 4d sphere, throw a tennis ball,
that''s it.
it''s like inifinity, just wait til nightfall, go outside, and
look up. it''s boring.
in other words, we already exist in 4d,
for the 5th dimension, I''d say that that''s probability,
but that''s just my opinion-(Hitchiker''s Guide to the Galaxy)
*sigh*
I''m sorry I have to weigh my opinion in on this one... I really am... because this is going to be long.
Okay, the debate started with how you would represent a 4D (hyper)sphere. The equation w^2 + x^2 + y^2 + z^2 = r^2 was suggested (or one like it). That equation, AFAIK, is correct. It yields points (w,x,y,z) which are always r away from the origin, (0,0,0,0). An anonymous poster also gave the general form, but convince yourself that this one works first.
The reason I use (w,x,y,z) instead of the more common (x,y,z,t) is that by including t you imply that time is somehow involved, when really it isn''t. So don''t think of there being time here. I''ll come back to this bit.
If you want to see a hypersphere? I think you''re out of luck. Not to mention, you''d have a hard time seeing it, because as it rotates around any axis, the projection of it (onto 3, 2, or 1 dimensions) doesn''t change. So you wouldn''t see it rotating.
Now, the hypercube is a different story, it''s actually quite cool to look at... and if you look long enough... you might start to get it (I''m going to have to go look again now, I forget). Get a pair of 3D-goggles, and go to: http://dogfeathers.com/java/hyprcube.html Weird stuff, I tell you.
Speaking of projections, here''s how you''d perform that (in case you want to do some graphics stuff with 4D objects). Take every point in your 4D object (x,y,z,q). Now, you need to perform a linear transformation from R4 to R3. So, you perform the following matrix mult:
There you go, now you''ve got a point (x,y,z) you can map into three-space. Take your renderer and go wild!
Now, the other debate seems to be over whether and how time fits into all this. In the strictest mathematical sense, it doesn''t. What I said up there holds no matter what you do. However, if you try to treat the fouth dimention as time things get really ugly, and you should probably take your questions to the likes of Prof. Stephen Hawking or Sir Roger Penrose.
One point that was made is that time is not a dimension because it breaks symmetry. Well, IIRC, there are three symmetries to our universe, Charge, Spin, and Time (I might not have those right). Now the odd thing is that if you were to switch +ve charges for -ve ones, or if you were to flip the Spin axis, our universe would evolve exactly the same, but it turns out if you flip the Time axis, things don''t work out the same for exactly the reason that the anonymous poster suggested: The 2nd law of thermodynamics. That law says that entropy increases with time. Well, if you flip time into reverse, then entropy decreases with time. That causes our universe to not evolve the way we like it to.
There are those who believe that Time actually is symmetric, it''s just that the model we have of the universe thinks it isn''t and therefore is inaccurate... but that''s getting pretty deep into theoretical physics... and I don''t know my way around that well.
Well, there''s my ramble. Hope it helps!
Eric
I''m sorry I have to weigh my opinion in on this one... I really am... because this is going to be long.
Okay, the debate started with how you would represent a 4D (hyper)sphere. The equation w^2 + x^2 + y^2 + z^2 = r^2 was suggested (or one like it). That equation, AFAIK, is correct. It yields points (w,x,y,z) which are always r away from the origin, (0,0,0,0). An anonymous poster also gave the general form, but convince yourself that this one works first.
The reason I use (w,x,y,z) instead of the more common (x,y,z,t) is that by including t you imply that time is somehow involved, when really it isn''t. So don''t think of there being time here. I''ll come back to this bit.
If you want to see a hypersphere? I think you''re out of luck. Not to mention, you''d have a hard time seeing it, because as it rotates around any axis, the projection of it (onto 3, 2, or 1 dimensions) doesn''t change. So you wouldn''t see it rotating.
Now, the hypercube is a different story, it''s actually quite cool to look at... and if you look long enough... you might start to get it (I''m going to have to go look again now, I forget). Get a pair of 3D-goggles, and go to: http://dogfeathers.com/java/hyprcube.html Weird stuff, I tell you.
Speaking of projections, here''s how you''d perform that (in case you want to do some graphics stuff with 4D objects). Take every point in your 4D object (x,y,z,q). Now, you need to perform a linear transformation from R4 to R3. So, you perform the following matrix mult:
[1 0 0 0][x] [x][0 1 0 0][y] = [y][0 0 1 0][z] [z] [q]
There you go, now you''ve got a point (x,y,z) you can map into three-space. Take your renderer and go wild!
Now, the other debate seems to be over whether and how time fits into all this. In the strictest mathematical sense, it doesn''t. What I said up there holds no matter what you do. However, if you try to treat the fouth dimention as time things get really ugly, and you should probably take your questions to the likes of Prof. Stephen Hawking or Sir Roger Penrose.
One point that was made is that time is not a dimension because it breaks symmetry. Well, IIRC, there are three symmetries to our universe, Charge, Spin, and Time (I might not have those right). Now the odd thing is that if you were to switch +ve charges for -ve ones, or if you were to flip the Spin axis, our universe would evolve exactly the same, but it turns out if you flip the Time axis, things don''t work out the same for exactly the reason that the anonymous poster suggested: The 2nd law of thermodynamics. That law says that entropy increases with time. Well, if you flip time into reverse, then entropy decreases with time. That causes our universe to not evolve the way we like it to.
There are those who believe that Time actually is symmetric, it''s just that the model we have of the universe thinks it isn''t and therefore is inaccurate... but that''s getting pretty deep into theoretical physics... and I don''t know my way around that well.
Well, there''s my ramble. Hope it helps!
Eric
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