Visualizing the Fourth Dimension

[Thefourth]

I have been interested in a fourth spatial dimension for a while now. It's incredible to me that these basic shapes such as cubes and spheres can be continued on into a fourth dimension; a direction that doesn't even exist in our universe.

As interesting this topic is, it's been quite hard for me to really dwell deeper into this subject, without having a good idea of what this shapes really are and how they might look. I know that a tesseract is basicly two cubes connected, but I still can't quite understand. Could anyone provide a good example for me? How would a hypersphere appear? That I don't understand in the least.

[ajb]

I have been interested in a fourth spatial dimension for a while now. It's incredible to me that these basic shapes such as cubes and spheres can be continued on into a fourth dimension; a direction that doesn't even exist in our universe.

As interesting this topic is, it's been quite hard for me to really dwell deeper into this subject, without having a good idea of what this shapes really are and how they might look. I know that a tesseract is basicly two cubes connected, but I still can't quite understand. Could anyone provide a good example for me? How would a hypersphere appear? That I don't understand in the least.

 

You mean in terms of Schlegel diagrams? That is as a projection of a polytope in [math]\mathbb{R}^{d}[/math] to [math]\mathbb{R}^{d-1}[/math].

 

Spheres are not polytopes. Thus what would be their Schlegel diagrams?

 

Maybe stereographic projection is what you are looking for?

[Shadow]

In less technical terms, you basically can't visualize a four dimensional anything the same way you can visualize a 3D cube for example. It's like trying to draw a 3D cube on a piece of paper; just can't be done. What we do instead is called projection. Projection is what you do every time you draw a 3D object, be it a cube, a tetrahedron, or even more complex objects such as a house, on a piece of paper; I could explain further, but I think it'll be much easier to comprehend if you look at the image at the top of the last page ajb links to (stereographic projection). This process can be extended to higher dimensions, which is the closest you can come to "visualizing" a higher dimensional object. But it doesn't actually tell us what it'll look like, it just gets us a little closer. If you only draw the outer edges of a projected 3D cube, you get a sort of weird looking hexagon if my imagination is working correctly, but that doesn't mean a 3D cube will actually look like a hexagon. In the same way, a projection of a tesseract into 2D looks like two 3D cubes projected into 2D, but that doesn't in the slightest mean that it actually looks that way.

There are other forms of "visualizing" higher D objects, such as the Shlegel diagrams ajb mentioned or making a net of an object. There are probably more that I'm not aware of, explore the internet and see for yourself.

 

Another thing you can do is conceptualization. Even though you can't picture it in your mind, you can work with an abstract concept obeying the rules of higher dimension geometry, and potentially do lots of cool stuff in you head without actually seeing a single picture.

 

Also, and I probably should have led with this, I'm no expert in this matter. The above was merely an attempt to convey my view of the subject, and may contain untruths. If so, I apologize.

Edited October 6, 2010 by Shadow

[granpa]

4D space has an up and a down just like 3d

4d space has a forward and back just like 3d

 

4d space has (as it were) 2 rights and 2 lefts.

use time to represent the new dimension.

start at the new right and take a 3d slice of the 4d object.

then move over time to the new left.

[Shadow]

I don't think this warrants a separate topic and since it's related, I'll ask here. Can motion in 3D be represented as rotation in 4D? I was thinking about the 2D projection of a point on a rotating sphere. It would be moving in 2D. If the answer is yes, can time be understood as 4D rotation, since time is basically something that allows 3D objects to change, ie. move? I apologize if the questions are absolute rubbish, but they have been lying around in my head for a while now, and I'm curious as to what the answers are.

[michel123456]

As Granpa said, the 4th dimension exists and is called Time.

 

A 3d cube is the projection of a 4d "spacetime-cube".

What is a 4d "spacetime-cube"? It is a 3d cube travelling through time.

You can imagine that through motion, because motion is displacement through space & through time (in comparaison with standing at rest which is displacement through time only).

So, when a 3d cube is moving, you get a good approximation of the 4th dimension.

[Shadow]

I know that time is considered to be the fourth dimension, but (and here I'm not sure if it's just because of my interpretation of the texts I've read), I've always distinguished between spatial and temporal dimensions. Time is a temporal dimension, and our "3D" is made out of three spatial dimensions. Since I've found the same argument made on Wikipedia, and while I know it's not considered a reliable source, in this case I think it's safe to assume that it's a correct one. What I'm asking is if displacement in space and time, as in a temporal dimension, can be explained/represented/modeled as rotation in the fourth spatial dimension, the same way the example I gave represents displacement in 2D using rotation in 3D. The only part that contradicts this view is that you would have to have some kind of "time" that would allow rotation in 4D, if you understand what I mean, which would lead to an infinite number of dimensions each allowing motion in the one before, which is probably one of the reasons spatial and temporal dimensions are so firmly separated.

 

The more I think about it, the more convinced I am that the answer is no, you cannot model time as rotation in the fourth spatial dimension. But I just want to be sure, or to be more exact would like to know if (one of) the reason(s) for this is the one I gave at the end of the previous paragraph. And regardless, all this is assuming that motion in 3D can indeed be modeled as rotation in 4D, which I'm still not sure is true.

Edited October 8, 2010 by Shadow

[michel123456]

I am not a reliable source either. And I cannot answer your question, partly because I don't understand it clearly.

 

What I know is that space & time are not "firmly separated".

At the contrary, time & space are interconnected.

Let me give an example: you don't need to "invent" 3d space first, and then add time as the 4th dimension. You can imagine a zero dimension element (a point), and insert time as the 1st dimension. You will get a 1D manifold represented by a line. In this case you can reversely state that a line is not a spatial element, but a space-time element. Which is a quite accurate description of our real world, because Relativity states that you always need a certain amount of time in order to move along a line.

[Shadow]

My apologies, "firmly separated" was a poor choice of words; "distinguished between" is what I meant, meaning you can't treat a temporal dimension the same way you would treat a spatial one.

 

To explain my question in a better way; imagine a glass sphere with a single point painted somewhere on it. Now, imagine shining a bright light through the sphere as you rotate it (around one of its axes of symmetry, like the Earth for example) and looking at the shadow it casts on a wall or piece of paper (or in other words, project it into 2D). You would see a single point that's moving* back and forth along a segment (provided that the axis of rotation doesn't change). This means that motion on a segment in 2D can be modeled as rotation around a fixed axis in 3D, yes? Here I'm taking a leap of faith, and assuming that any motion in 2D can be modeled in this way by rotating a different "shape" and/or changing the size of the shape and/or the axis of rotation. What I'm asking is, can the same thing be done with 4D and 3D, ie. could we model motion along a segment in 3D using rotation in 4D?

 

*This is assuming that the point is not positioned such that a plane tangent to the sphere in that point is also normal to the plane we project onto.

Edited October 9, 2010 by Shadow

[khaled]

I think the fourth dimension is time, representation of a 3D object in the 4D space

 

would be either by using an Animation,

 

or by showing multiple copies of the object in 3D space where every copy has a Time tag,

[imatfaal]

I know it's not exactly on topic, but I assume that the site of the 4-d rubix cube (that's simulation of 4 spatial dimensions) has been posted here in the past. Even so here it is again - Magic Cube . Its very silly and very clever at the same time.

[michel123456]

I think the fourth dimension is time, representation of a 3D object in the 4D space

 

would be either by using an Animation,

 

or by showing multiple copies of the object in 3D space where every copy has a Time tag,

 

Correct.

[Bignose]

I think the fourth dimension is time, representation of a 3D object in the 4D space

 

But, a 4th dimension doesn't have to be time.

 

Here is a good example from my studies:

 

[math]\frac{\partial \dot{V} f}{\partial v} + \nabla \cdot (\mathbf{c} f) = h [/math]

 

where f is a particle distribution in space and in particle volume, v

 

The first term represents a particle growth, something like a species in a super saturated solution coming out of solution and causing a crystal to grow larger

 

The second term represents a convection flux through space (c is a velocity here). Could also include a diffusion term.

 

The right hand side, h, represents all the collision-based occurances. This will be collisions that could be causing the crystals to break apart, or possibly agglomerate together. Or both! Its exact form isn't important to this discussion.

 

The important thing to note is that what we have here is a distribution in space and particle volume. 4 dimensions. 3 spatial and 1 "internal" describing the particle volume.

 

Another choice could be to use a very similar equation to describe a distribution of cells in a bioreactor. This would have 3 spatial dimensions and an internal dimension describing the cell's age. Again, 4 dimensions.

 

Time can be added to each of these equations above, leading to a total of 5.

 

One can imagine many different situations where in more than just 3 or 4 dimensions are needed to mathematically describe a situation.

 

Consider a particle distribution in space (3 dimensions), distributed in velocity (3 more dimension), and particle size, and varying with time. That's 8 dimensions.

 

An additional dimension is needed whenever the dimensions that have been used already cannot be used to describe the situation. A particle at spatial position (0,0,0) can have volume 1 or volume 2 or volume 4.8125876..... Therefore, we need an additional dimension to describe the situation, and the "4th" dimension becomes volume.

 

I wrote the above just because I think that it is important not to get hung up on phrases like "4th" dimension. Depending on the mathematics needed to describe the situation, many, many dimensions may be needed.

[michel123456]

That is about the definition of the term "dimension". In your example, colour can be a dimension as well.

[khaled]

That is about the definition of the term "dimension". In your example, colour can be a dimension as well.

 

1D is a line, 2D is a plane, 3D is an object ...

 

and the 4th Dimension can be one of many things:

- Time

- Color

- Direction

- ...

 

If you are interested in physics you can consider every point (x,y,z) as an Atom

and the 4th dimension can consist many things related to Atoms in physics,

The structure can be applied in computer program to study physics of Particles ...

 

Also you can consider every point (x,y,z) as an Electron, and with giving the

required constraints for quantum mechanics, you can work on quantum

mechanics in visual ...

 

Best of luck,

[alpha2cen]

Alphabet can be a dimension , too.

[khaled]

Alphabet can be a dimension , too.

 

possible, considering textual contexts and books as a space graph of alphabets .. and since letters come as words,

we can consider words in a context as a vector spaces, where relative contexts are interpolated ...

[Svetoslav Pavlov]

Try the hypersphere as Triangle !

 

More at http://circlesinair.wordpress.com

[steevey]

(X^2+Y^2+Z^2+T^2)= a cone (with base and slant height extending indefinitely). That's FOUR dimensions AND a shape you can visually see. You have your 3 dimensions, and then the space is getting larger as time goes on. Of course, this would mean that your seeing all the time that it's been going through, which usually doesn't happen.

[alpha2cen]

Sound is can be a dimension, too.

New concept.

Edited February 8, 2011 by alpha2cen

[dalgoma]

4 D CUBES<BR style="mso-special-character: line-break"><BR style="mso-special-character: line-break">

 

To construct a cube, we join 2 squares together with 4 lines each the same length as the side of the square to each of the vertices.

 

If we put one cube on top of another, we can join the 8 vertices with lines of the same length as the side of the cube. Is this a 4D cube? If not why not!

 

 

[the tree]

(X^2+Y^2+Z^2+T^2)= a cone (with base and slant height extending indefinitely). That's FOUR dimensions AND a shape you can visually see. You have your 3 dimensions, and then the space is getting larger as time goes on. Of course, this would mean that your seeing all the time that it's been going through, which usually doesn't happen.

If you think about it, throwing in the time factor to make an extra projection is really common. We've all seen those animations of MRI slides where we are looking at a 3d object, projected onto a 2d screen but with the 3rd dimension added back in with the aid of calling it time.

 

Sound is can be a dimension, too.

I'm fairly sure any useful data relating to sound would come in at least two dimensions.

 

To construct a cube, we join 2 squares together with 4 lines each the same length as the side of the square to each of the vertices.If we put one cube on top of another, we can join the 8 vertices with lines of the same length as the side of the cube. Is this a 4D cube? If not why not!

No, it's a projection of a 4d cube onto 3d space. No simple shape is going to have more dimensions that the space it occupies. All you've made there is a picture.

[michel123456]

Take a cube, slide it, or rotate it, and you get the 4th dimension through motion.

 

If you take the cube and let it fall down, you are in fact using twice the time dimension because a free falling body is accelerating (m/s^2). It could be considered as a 4 dimensional event taking place in time: in other words maybe a 5 dimensional event.

[Sisyphus]

Take a cube, slide it, or rotate it, and you get the 4th dimension through motion.

 

If you take the cube and let it fall down, you are in fact using twice the time dimension because a free falling body is accelerating (m/s^2). It could be considered as a 4 dimensional event taking place in time: in other words maybe a 5 dimensional event.

 

No more than the curve graphed by y = x^2 is three dimensional. Which is to say, it isn't.

[michel123456]

No more than the curve graphed by y = x^2 is three dimensional. Which is to say, it isn't.

 

Interesting comment.

The curve is one dimensional.

The graph is 2 dimensional.

And if you put units on X (say meters), then Y are square meters: 2 dimensional. I guess the growing graph of increasing surfaces can be represented 3D (a pyramid).

Edited February 8, 2011 by michel123456