Visualizing 4-D: Is My Mind the Problem?

[Char. Limit]

I can't get my mind to do this. Two questions come from this observation.

1. Is something wrong with my mind (well, more than usual)?

2. How do you guys do this?

 

[ramb]

1. Nothing's wrong with your mind, we are born in a 3D world, we can't fathom to understand a fourth dimension physically.

BUT.

2.
My professor introduced the idea of 'temperature' for visualizing 3D in 2D space, meaning you write numbers on an xy plane for z, so translating this idea to 4D.

I use density to see it. Think of a 3 dimensional object, then how dense at some point is the 4th dimension.

 

[mikeph]

Try to think of a continuous ordered set of 3D spaces... for example, in 2D you have a square, in 3D a cube, in 4D try to think of a set of cubes each with a reference number which is a real number from 0 to 1. Then in this 4D "box" space, you can specify a point by giving the three 3D coordinates, and then a box number to specify which box you're talking about.

Where it becomes confusing is intuitively recognising that there is nothing special about the "box number" dimension, ie. (x,y,z,n), where each is a real number in [0.1]. The distance between any two points is just sqrt[(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2 + (n2-n1)^2]. For example, the diagonal of a square is sqrt(2), the diagonal of a cube is sqrt(3), and the diagonal of 4D space is just sqrt(4) = 2. That's my interpretation of higher dimension spaces (where all dimensions are measured in the same units, length).

 

[A.T.]

Googling the thread topic gives intersting stuff. Once you get it, try to visualize curved 4D!

 

[Neo_Anderson]

You can visualize the fourth dimension, but in order to do so you have to carefully read the following post I made in another thread of mine. It may seem long-winded, but you will be rewarded for your efforts. If you're not serious about visualizing the fourth dimension, then be lazy and skim over it.
Here it is:

You will observe the following if you were to venture into the fourth dimension:

• If you were to look at a complex mechanical device such as an internal combustion engine, you'd see every component of that engine, all at the same time. You'd see the outside of the engine and it's innards (pistons, valves, oil, etc) simultaneously. You'd see every internal component of that engine (including every single molecule that makes up each component if a 3D microscope is handy) simultaneously, regardless of where you are in this 4-D.
• If there is a 3-D cube inside a 3-D sphere lying in the 4-D, you'd see both the cube and the sphere simulteniously. But in our dimension, you'd just see the sphere, since the sphere is a closed surface in our dimension. Please note that you'd always see the cube inside the sphere regardless of where you are in the 4-D.
• In the fourth dimension, you'd be able to move left-right, forward-backwards, up-down, and another direction that we have never experienced before: direction Omega and direction Alpha.
• In the second dimension, a 2-D spinning top spins around a point. In our dimension, a 3-D spinning top spins around a line (also referred to as an 'axis', whether it be the x, y or z axis), and in the fourth dimension, a 4-D spinning top spins around a plane.

I have a hypothesis that says the fourth dimension can in fact be visualized by us, but will require we first consider the second dimension. It goes something like this:
A 2D organism has two eyes to perceive depth. One eye consists of a 1D collection of rods and cones ("retina"). These rods and cones are joined to form a 'line' of light-sensitive nerves.
A 3D organism (homo sapiens, for example) has a retina that forms a plane, or 2D collection of cones and rods. Again, this is called the 'retina', and it is a 2-d plane of visual nerves on the back of our eyeballs.
A 4D organism, therefore, must have a retina that is cubic (literally, 3-dimensional) in shape and depth.
We must ask ourselves: can the 2D organism--with it's 1D line of cones and rods--visualize the third dimension? By extension, it stands to reason that if he can, then we can visualize the fourth dimension. But how?
The 2D organism must have a good memory in order to visualize our dimension. He must take the following steps in order to create an accurate visualization of something 3D-ish, such as a sphere. He must:

1. Look at the far-left area of the sphere and memorize what he sees (his memory from our perspective will be an infintesmally thin slice of the sphere).
2. move his 2D eye a bit to the right and memorize what he sees.*
3. Move his 2D eye a bit more over to the right and memorize what he sees.
4. He must continue doing this--moving his eyes ever so slightly to the right, remembering what he sees, move his eyes to the right a little more and remember what he sees each time--until a complete memory of the 3D sphere is formed in his memory.

At the end of the mental excersize he notices that his memory--a plane of 1-D images of the sphere--is completely full.
The important thing to understand is that the 2D organism's memory is now a 2D plane consisting of many 1D memories of the 3D object! The 2D being now acesses every memory of each 1D slice of the 3D sphere lying in his 2D memory simultaneously. He thus has sucessfully visualized a 3D sphere.

By extension, we may do the same excersize, if our memories are good enough (we'd need a few harddrives of memory). We must visualize something like a hypersphere (a 4-D sphere), but only one 3D visualization at a time, as with the 2D organism and his visualization of the 3D sphere. As we build our visualization of the hypersphere, we keep a memory of all our previous visualizations of the hypersphere (which will appear as many, many memories of many, many 3-D spheres), until we have a 4D visualization of many 2D spheres stored in a 3D memory.

Because the excersize operates in the second dimension (remember, our retinas are 2D, or planar, curved sheets of cells) and the third dimension (the stored memories of the hypersphere are stored in a 3D memory), then the fourth dimension can conceivably be visualized. this ends my hypothesis.

And here's the thread that discusses the very topic: https://www.physicsforums.com/showthread.php?t=357086

 

[tot]

how about color? we use color all the time for this.
http://www.mediafire.com/file/xdtbyldidzy/4d.png

 

[tot]

You can also do it by animating a variable.
like this
https://www.youtube.com/watch?v=F3d07Xe4WPQ

 

[Char. Limit]

how about color? we use color all the time for this.
http://www.mediafire.com/file/xdtbyldidzy/4d.png

[/URL]
I don't know about that. That just looks 3d to me.

I'm going to try these methods while I take a shower.

 

[tot]

the color defines the location on another axis for every point does it not?

 

[sokrates]

You can visualize the fourth dimension, but in order to do so you have to carefully read the following post I made in another thread of mine. It may seem long-winded, but you will be rewarded for your efforts. If you're not serious about visualizing the fourth dimension, then be lazy and skim over it.
Here it is:

And here's the thread that discusses the very topic: https://www.physicsforums.com/showthread.php?t=357086

This assumes that there *is* actually an observable "spatial" 4th dimension.

What makes you think the observable nature is 4-D (spatially, as you imply) ??

Why isn't there ANY account of direction alpha and omega? Do you think we wouldn't INFER the existence of 3D if we lived in flatland?

It could be argued whether a 2D organism could ever "visualize" 3D, but "homo sapiens" visualizing 4th dimension?

Before thinking hard on how to "see" a 4th dimensional cube, maybe you should provide some justifiable remarks on the existence of it.

Your engine example is not at all warranted either. It's a poor analogy taken from the Flatland example. Is there any evidence to support these claims? (apart from simple flatland arguments)

 

[Neo_Anderson]

This assumes that there *is* actually an observable "spatial" 4th dimension.

What makes you think the observable nature is 4-D (spatially, as you imply) ??

Why isn't there ANY account of direction alpha and omega? Do you think we wouldn't INFER the existence of 3D if we lived in flatland?

It could be argued whether a 2D organism could ever "visualize" 3D, but "homo sapiens" visualizing 4th dimension?

Before thinking hard on how to "see" a 4th dimensional cube, maybe you should provide some justifiable remarks on the existence of it.

Your engine example is not at all warranted either. It's a poor analogy taken from the Flatland example. Is there any evidence to support these claims? (apart from simple flatland arguments)

Horrible.
Just horrible.

 

[mikeph]

the color defines the location on another axis for every point does it not?

No.. I think there is a difference. The colour example (or temperature in a room) is not 4D space, it is a scalar defined on a 3D space. The colour function maps R^3 -> R^1, nowhere is a 4D space used. I think a 4D space is defined by its metric, and not by a function which uses it as a domain.

 

[diazona]

Yeah, but we (well, most of us) can't actually see four-dimensional images in our minds. We have to have some way of projecting it into three dimensions. The color thing works for a limited set of surfaces embedded in 4D space.

The continuous ordered set of 3D spaces you mentioned earlier is my preferred method of doing it.

 

[tot]

if time is a dimension then we see 4d all the time don't we?
we live and breathe in 4d don't we?

 

[sokrates]

But that's a very different dimension than the usual spatial dimensions, isn't it?

You might want to check:
http://en.wikipedia.org/wiki/Minkowski_space
In contrast to:
http://en.wikipedia.org/wiki/Euclidean_space

The 4th spatial dimension is a mathematical construct. It may not even exist.