Exploring 4-Dimensional Fields: A Mind-Bending Concept

[strid]

Thougt of a thing, reading a topic about dimnesions...

if we think like this...

a point had 0 dimensions...
there are infinite many points in a line (which has 1 dimension)
there are infinite many lines in a square (which has 2 dimensions)
there are infinite many squares in a cube (which has 3 dimensions)

following this patterns, I see it reasonable to suggest:

that there are infinite many "cubes" (objects with 3 dimensions) in a 4 dimensional "object"...

so in an "field" (or what to call it) with 4 dimensions we can fit EVERYTHING that we can see.. like the sun... or maybe ven the whole univerese...

it might then be that our universe is 4 dimensional, an therefore has infinite "space" (3 dimensions)...

hope I wasnt confusing...

 

[Kerbox]

wouldnt the fourth dimension be... time? :P

 

[danne89]

Don't forget the hidden ones! You can always stuck your garbage in those...

Seriously, so you mean you can put infinity volume in a finte 4d-space? That sounds quite strange, but facsinationg.

 

[Kerbox]

I don't really think that is what he ment. I think he is confusing infinite steps with infinite size.

 

[strid]

time is not the 4th dimesion I'm talking about...
I'm talking about a 4th "spacial" dimension with the quality of fitting infinite much "3dimensional stuff"

danne89, understood my point i think...
maybe volume is the better word to use instead of space...

let me then put it like this:



in a "4 dimesnional thing" there is infinite volume
as there is infinite area in a cube...'

was this clearer

 

[Crosson]

Yeah, since the fourth dimension is time, what you say is exactly correct. Let me describe a 4-d object that contains "infinite" 3-volume: Me walking from the kitchen to my room.

If you think of this "content" (the generalization of length, area, and volume) as being made up of 3-volumes, you would need in an infinite number of them (1 three volume for each instant in time, exactly analagous to there being one square for each coordinate in the z dimension if you construct a cube from squares).

I am going to think about how a two dimensional creature could possibly use the third dimension as time.

 

[Kerbox]

If you bound the volume in all four dimensions, then the volume will be finite. I can't see how summing the area of the infinite number of cubes would make any sense.

 

[Moo Of Doom]

I don't really think that is what he ment. I think he is confusing infinite steps with infinite size.

Actually, I do believe that's what he meant. And why couldn't you? Say you have a beach full of sand you want to clear out that is full with 1000x100x10m of sand. You could (theoretically) fit all of that sand in a 1x1x1x1m 4D box (or smaller even) by filling the first layer of 1x1x1 with sand, then adding another 1x1x1 layer on top of that, since the sand is only 3D (which we are only assuming :P), it has no hyperdepth, and so, does not take up more of the hypervolume of the box than the first one (which took up 0). You can keep going until the beach is cleared and still have infinite volume to fill. (Now getting the sand back out... that's a problem :P)

EDIT:

time is not the 4th dimesion I'm talking about...
I'm talking about a 4th "spacial" dimension with the quality of fitting infinite much "3dimensional stuff"

Actually, it is many people's belief, including mine, that the "time" dimension is indeed just a spatial one. Our brains (and eyes) just take in one "slice" at a time (much like a 2D creature would need to experience a 3D object).

 

[strid]

Crosson.. time isn't the 4th dimension, as there is a finite volume in you walking to the kitchen...

ito make an easier example.. a cube with 3*3*3cm sides is placed on the table.. over a period of time it is moved 3 cm to one side... see it than as if you had one of this really old cameras where the photographing was really slow.. every area that is blurred is the volume... so the cube would, with the movemment include the area would be 54cm^3... it would be as two cubes next to each other...

But there is still a point in what you said...
if we have a point.. and move it straight.. we get a line... if we take this line and move is perpendiculat to the line we get a square... and if we move this we get a cube... so we should get some sort of 4dimensional thing if we moved a 3 dimensional thing... but the human eye can't observe it...

but we could guess of some of its qualities... whereof one is, I suggest, that it has an infinite volume...

 

[Kerbox]

Mmm... yea. So what you are saying is that a hypercube encloses infinite amount of 3d space, and a normal 3d cube encloses infinite 2d area, is that right?
Then I wonder, what possible use is it to consider the area of each of the infinite planes that makes up the 3d box?

 

[strid]

moo of doom made som good points..


about that with infinite area in a cube:

say you have a square with area 1*1cm...in a cube of 1*1*1cm, you could place infinately many such square in the cube...

 

[Kerbox]

say you have a square with area 1*1cm...in a cube of 1*1*1cm, you could place infinately many such square in the cube...

But what physical sense would it have to sum up all the areas of the squares?
What you are doing is simply extending the 2d space to a third dimension by adding an axis perpendicular to the two others.

 

[strid]

But what physical sense would it have to sum up all the areas of the squares?
What you are doing is simply extending the 2d space to a third dimension by adding an axis perpendicular to the two others.

it shows us that the same way, we could (COULD) have a 4dimensional "box" where we can just stock up infinite many 3dimensional objects...

the exmaple with sand was quite good...

 

[Kerbox]

it shows us that the same way, we could (COULD) have a 4dimensional "box" where we can just stock up infinite many 3dimensional objects...

the exmaple with sand was quite good...

Try stocking up infinity many planes first then, and then we can start talking about practical applications :rofl:

 

[Moo Of Doom]

But what physical sense would it have to sum up all the areas of the squares?
What you are doing is simply extending the 2d space to a third dimension by adding an axis perpendicular to the two others.

Well, imagine if Carl was a 2D being, and he lived in a 2D universe. He has calculated that his universe will be destroyed by a 3D object on a collision course with his plane. The only way to save his universe is to move it away. Being a brilliant scientist, he has discovered a way to create a 3D box. Now in order to save his universe, he just cuts it up into sections that will fit in the box (i. e. have the same dimensions as the base of the box). He then places each section in the box (hopefully in an ordered fashion). Since they have l*w*0 volume, he can fit as many as he needs in the box. He then moves his universe-in-a-box (don't ask me how) to a new location, and unpacks it and puts it back together. Yay, his universe is saved!

:P

 

[strid]

you might not have noted but mathematics is not neccesarily practical :)
try to show me the value of imaginary numbers then, which is a well-accepted thing... :)

I'm not saying anything about "doing" anything.. just pointing out the interesting relaionship between the dimensions that hints that a 4dimensional object has infinite volume...

 

[Moo Of Doom]

Actually, imaginary numbers have infinite (:P) application to specific types of engineering and various other fields of science. (But I forgot which, exactly >.<)

 

[Kerbox]

Dealing with alternating (sinousoidal) currents in electrical systems for one

And the practical aspect I brought up since in the original post, there was talk about fitting universes and suns into finite hypercubes and such...

 

[danne89]

I didn't completely understand that time interpretation for a few posts back. Can you develop it a little.

This whole discusion reminds me of a joke: "How can you visulise 4th dimension objects" "First visulise nth dimenstion and then reduce to 4th" :) Perphaps a little too old...

If I understand popular string theory right, it exist many hidden dimenstions of 4+. If you can unhide those things, what would happen. Now I maybe quite OT, but it's a fascination discusion indeed.

 

[Icebreaker]

I don't understand the original post. Are you talking about the suggestion that the universe is a sort of higher dimensional sphere? As in, if you go straight in any direction, you'll return to the same spot?

For example, if the universe was a plane, but "in actuality" is the surface of a sphere.

 

[mathwonk]

i am ignorant. just try to educate me, i dare you.

 

[strid]

I don't understand the original post. Are you talking about the suggestion that the universe is a sort of higher dimensional sphere? As in, if you go straight in any direction, you'll return to the same spot?

For example, if the universe was a plane, but "in actuality" is the surface of a sphere.

No... I'm pointing on the fact that there can fit infinite many n-dimensional objects in a (n+1)-dimnesional object...

As there are infinite many points in a line...infinte many lines in a square adn etc...

so is possible (probable?) that the universe is a 4+ dimensional object that can contain infinite much 3-dimensional stuff...

 

[Icebreaker]

No... I'm pointing on the fact that there can fit infinite many n-dimensional objects in a (n+1)-dimnesional object...

Not necessarily. Depends on what you mean by "fit".

so is possible (probable?) that the universe is a 4+ dimensional object that can contain infinite much 3-dimensional stuff...

That's more of a question for physicists. As I understand it, superstring theory says there are 10 spatial dimensions.

 

[strid]

I've hard to imagine how physicst can caonclude that there 1re 10 spatial deminsions...how would they possibly be able to do that.. sounds fascinating and inrtesting...anyone tha can explain breifly?

 

[HallsofIvy]

I've hard to imagine how physicst can caonclude that there 1re 10 spatial deminsions...how would they possibly be able to do that.. sounds fascinating and inrtesting...anyone tha can explain breifly?

A least one variety of string theory (and, apparently, the one most accepted) requires 11 dimensions (10 spatial and one temporal) in order to make the fundamental values come out right. The idea is that all but 3 of them are "rolled up" tight. Imagine a sheet of paper (2 dimensions) rolled up in one direction to form a cylinder of very very small radius. Looks like a one dimensional line, doesn't it?

 

[mathwonk]

i recall that some of the tightly rolled dimensions are those of a calabi - yau complex threefold, but this only accounts for 6 real dimensions. what is the 7th?

 

[Moo Of Doom]

i recall that some of the tightly rolled dimensions are those of a calabi - yau complex threefold, but this only accounts for 6 real dimensions. what is the 7th?

Here's a quote from The Elegant Universe that should answer that (from the notes at the back of the book):

With the discovery of M-theory and the recognition of an eleventh dimension, string theorists have begun studying ways of curling up all seven extra dimensions in a manner that puts them all on more or less equal footing. The possible choices for such seven-dimensional manifolds are known as Joyce manifolds, after Domenic Joyce of Oxford University, who is credited with finding the first techniques for their mathematical construction.

 

[mathwonk]

it answers it in the sense of giving it a name, but not in the sense of explaining what that name means.

(To me, a calabi yau threefold is a complex three dimensional manifold with trivial canonical line bundle, i.e. with a never vanishing alternating three - form on its tangent bundle.)

As such, it is a 3 dimensional analog of an elliptic curve. For instance, the hypersurface of complex projective 4 space defined by a general homogeneous polynomial of fifth degree is a Calabi - Yau.

 

[Moo Of Doom]

Sorry, but I don't know more about Joyce manifolds than you do. You can always Google it. Or maybe someone else on here happens to be able to explain them...

 

[selfAdjoint]

Sorry, but I don't know more about Joyce manifolds than you do. You can always Google it. Or maybe someone else on here happens to be able to explain them...

Try this: http://www.maths.ox.ac.uk/~joyce/mrrev.html. Joyce wrote a book.

 

[mathwonk]

thanks, that was very helpful, except i do not know which group G2 refers to.

i.e. apparently a joyce manifold is a compact riemannian manifold with holonomy group G2.

 

[meckano]

Hi, sorry for the lengthy post, but it is something I have been working towards for a few years.
I liked the 2D being trying to see 3D as time.
- there are many ideas I liked, so here I post. Again, sorry for it's length.
-- There is, at the end, a quantitative look at the 4th and well, all dimensions.

simplify calculations for Circular: Vector(1D), Area(2D), Volume(3D)...
In my search to find the the flip side of Pi, a never ending number,
I found the following:

Formulae for circles in any Dimension:

diameter ^ (dimension being worked in: 1D, 2D, 3D...) *

( π )
( _______________________________________ )
( 2 * (dimension being worked in: 1D, 2D, 3D...) )

The birth of the second part of the above equation is explained in
the following example.

2D, Area:
A circle with a diameter of 4 has an area of... π * r^2 = 12.5664
I like 1, so
I found how many circles of diameter 1 were needed to equal the area
of that 4d circle.
The answer is 16: 12.5664 / 0.7854
a) 16 is also the square of the diameter.
b) 0.7854 = area of circle with a diameter of 1.
c) 0.7854 happens to be π / 4 or
π / (2 * (dimension in question, in this case 2D)) =
π / 2 * 2 = π / 4

3D, Volume:
I did the same check with 3D: (4/3) * π * r^3 and
I found that it takes d^3 circles with a diameter of 1.
so again
4^3 = 64,
Volume = 64 * 0.5256(Volume for dia. of 1) = @33.5103
π / 2 * 3 = 0.5256

1D, Vector:
Please note this is supposed to be simple, it is 1D.
- and it relates to circles.
If you travel straight for 4 feet, how many circular feet have you traveled?
4^1 * (π / 2 * 1) = 4 * (π / 2) = 2π
What does this mean?
well we know that the circumference is π * d,
and we don't need to go in a complete circle,
we only need half the circle, hence C = 4π, 4π / 2 = 2π.

The 1D example is used to prove the validity of the formula.

so
1D gives us the circular or arc vector in feet
2D gives us the area in square feet
3D gives us the volume in cubic feet
4D gives us the ?(warp of)? spacetime in ?(quad feet)?
- is it the radius of, or area of, or volume of... who knows.
- but 4D may give radius of,
- then 5D area of,
- and 6d the volume of, warping. (still thinking about what the 4D, 5D... is really telling me.)

If the 4D is tested on a circle with a d of 4, against a circle with a d of 1,
it would take 256 for the circle with a d of 1 to fill or do the same as
the circle with a d of 4.
- assuming the 4th dimension is spacetime,
as it is accepted that space and time are together the 4th dimension,
then what is going on that it takes 256 of the circle with a d of 1
to do the same as the circle with a d of 4?

π = Pi (to be clear :)
If comparing a circle with a d of 1 with that of one with a d of 4 then:
in 1D we need 4 to go the same distance: π/2 vs. 2π ... (4^1=4)
in 2D we need 16 to cover the same area: π/4 vs. 4π ... (4^2=16)
in 3D we need 64 to fill the same volume: π/6 vs. 64π/6 ... (4^3=64)
in 4D we need 256 to xxxx? the same xxxx?: π/8 vs. 256π/8 ... (4^4=256)

Teachers will not like this but:
2D, area:
Take any diameter, square it, multiply by π/4 = area
3D, volume:
Take any diameter, cube it, multiply by π/6 = volume

 

[meckano]

There is a link from another thread that explains the beach sand going into the sand box.
It is under the title of topology. http://en.wikipedia.org/wiki/Topology

I'm referring to the line that reads:
However, it is not possible to deform a sphere into a circle by a bicontinuous one-to-one transformation.

If you take a square foot and stand in the middle of one side, looking toward the opposite side, there is an infinite number of steps. But,
if you set a step-size, to keep things 'real', set by the smallest known particle/wave/energy-level, there is now a finite number of steps.
It would then be possible to transform a 2D representation into a bunch of lines.
- good for getting a reference frame from which to 'see' what is going on. (something I do a lot of.)

Now back to the sand. To start:
1 layer of sand(2D + 3D estimate based on 2D diameter) 2 ft^2 into a 3D box volume of 2ft^3.
- I won't use 1 for dimension lengths. Based on my formula for circles, a diameter of 1 is the only one that gets smaller as we prgress into the next dimension.
-- re: 1D gives circular vector π/2 ft, 2D gives area π/4 ft^2, 3D gives volume π/6 ft^3...
(makes me think of water, being the only one to expand below 4C.)
CHANGE: We know there is a guesstimated volume fill limit based on the dimensions of the 2D sand, so:
2 ft^1 * 2 ft^2 = 2(ft^1 * ft^2) = volume of 2 ft^3
ADDITION: volume / volume of sand grains = how many grains.

Now in the 3D to 4D example,
there must be a limit, but what?
Mathematically it is, if dealing with circles and trusting my formula:
2 ft^1, so:
2 ft^1 * 2 ft^3 = ?spacetime? of 2 ft^4 of sand

But that number will only hold true if I can get a relation between all dimensions.
CHANGE: - hmm found something here to help get the 4th dimension wording. It's in the wording.

This is here to help in understanding why I no longer deal with the size of the grains of sand in 4D:
0D: Can not divide by zero. Not yet part of the 'real' world. See quantum physics?
We first had to calculate the physical dimensions of the mass-x grains of sand we would work with.
- quantum physics I presume.(based on strong, weak, whatever forces.)
-- I will think of that as quantum dimensions for now.(Amplitude, Spin...)

1D: Used to calculate the boundary lines/vectors of physical space to fit siz-x grains of sand for diameter y ft.
CHANGE: ---- not yet a tangiable dimension. remember sand has Second dimension.
2D: Used to calculate the boundary area of physical space to fit size-x grains of sand for diameter y ft^2.
---- Not yet a tangiable dimension. remember sand has Third dimension.
3D: Used to calculate the boundary volume of physical space to fit size-x grains of sand for diameter y ft^3.
---- Now a tangiable dimension. remember sand occupies 4D spacetime, or
------ is spacetime the start of something larger? One that covers 2, 3, or more dimensions?

a reach here?
4D: Used to calculate the boundary lines/vectors of spacetime that is affected by mass-x sand * grains in diameter y ft^3.
5D: Used to calculate the boundary area of spacetime that is affected by mass-x sand * grains in diameter y ft^3.
6D: Used to calculate the boundary volume of spacetime that is affected by mass-x sand * grains in diameter y ft^3.

The following is babbling, but I left it in.
after writing all that,
I'm not sure now if the 4th dimension is looking at a larger or smaller scale of the sand.
- it may be looking at the quantum properties of circle with diameter y.
-- so much to think about.
is spacetime the 4th dimension? it can't be the 1st? or just another?
OR
is the 4th dimension, like I say, just on a smaller physical/energy scale but felt farther out, like gravity?
Can't see it, but it's effects are surely seen in real time on large physical items.

 

[Doodle Bob]

thanks, that was very helpful, except i do not know which group G2 refers to.

i.e. apparently a joyce manifold is a compact riemannian manifold with holonomy group G2.

G_2 is the automorphism group of the octonian algebra. it is also one of the exceptional Lie groups that Calabi (or was it Cartan?) found could be the holonomy group of a compact, simply-connected Riemannian manifold.


For specifics, including the Dynkin diagram of its Lie algebra, go here:
http://en.wikipedia.org/wiki/G2_(mathematics)

 

[meckano]

I realize I have taken Pi*r^2 and 4/3*Pi*r^3 and found them to be dimension based.
1D: (2^1*Pi*r)/(2*1)
2D: (2^2*Pi*r^2)/(2*2)
3D: (2^3*Pi*r^3)/(2*3)

and started working on
4D: (2^4*Pi*r^4)/(2*4)
- glad most of the work is done for me.