Visualising the fourth dimension-can it be done?

[Neo_Anderson]

Visualising the fourth dimension--can it be done?

Let's start with the second dimension. Can it be visualized? I believe so. It would not! appear as an infinitely thin vertical section of 'space' surrounded by void; rather, it should be visualized as a region where all objects appear to be 'horizontal' in nature. For example, a 2-D box should be correctly visualized as a horizontal region of 'brightness' (where the 2-D light is shining upon its surface), and directly underneath, a horizontal region of shadow (where the 2-D light source is not falling on the 2-D box).
Turning left or right is meaningless--you'll still see the horizontal region of the side of the box the 2-D light falls on and it's shadowed part as if you never turned left or right.
If you make a 180-degree turn, you see objects behind you--all as the horizontal 'smear' of the 2-D object you're looking at.
A circle should appear as a horizontal smear who's height is equal to the height of the original circle, with its width extending from your left all the way to your right. The smear will have a smooth transition from bright (where the 2-D light falls upon the circle's surface) to dark (where no 2-D light falls upon it's surface).

So as you can see, visualizing the 2-D is in itself somewhat of a mental challenge. But the fourth dimension? That dimension where the corner of your room will have not three but four folds coming from it, with each fold being at a 90-degree angle to the other three. Our three dimensions have three "folds" coming out of the corner: one for the ceiling, one for one wall and the last for the other wall.

Can you visualize a 4-D region of space? If so, post your ideas here!

 

[Sorry!]

Have you ever seen an object move? I'm assuming you mean 4 spatial dimensions huh.

 

[Tobias Funke]

I can't do it. I've looked at drawings of a 4-cube, and it just looks like a small cube inside a bigger one with corresponding vertices connected. I have no idea what people are thinking when they say they can "see" it from that picture, but I may just have poor visual skills. I also can never follow those 4 or 5 sequential pictures of a particular topological shape being inverted or something like that.

 

[waht]

Although its impossible to visualize 4 spatial dimensions, it's possible to visualize how you would behave in a 3 dimensional slice embedded in 4th dimensions.

For example, a bug is traveling on a surface of a sphere which is 2 dimensional and embedded in a 3rd dimension. The bug can only perceive 2d, and can only move in two independent directions from left-right or front-back. If the bug keeps walking in any these directions, it will eventual come back to its starting point.

Now let's say you are embedded in a 3d volume slice of a 4th dimensional sphere. Now you have three independent directions to take, either from your left-right, front-back or up-down. If you go left, eventually you will come back to your starting point. If you go ahead, you will eventually come back to your starting point. And if you go up, you will eventually come back to your starting point.

Or on any other manifold, imagine you are inside a 3d cube which is embedded in 4d. You can move freely inside the cube. Now if you were to take a step through the 4th dimension, you will find yourself inside of another cube in which you can move freely in 3 dimension.

Hide your personal belongings in one cube, and then comeback to your original room. Nobody will find your stuff.

 

[Phrak]

Although its impossible to visualize 4 spatial dimensions,

How so?

 

[waht]

How so?

The difficulty lies in stacking 3d pieces of volume in a sequence like pancakes, and seeing all of them at the same time.

The projection of a terrasect (a 4d cube) onto a 3d structure only captures a certain angle of the actual terrasect. And there is many such angles. It's like looking at a projection of a 3d cube on a 2d plane, you only see a certain portion of a cube at time.

hence in a sense, people perceive in 2d, and it's already difficult to perceive 3d objects because you have to look at them from different angles.

Perhaps with sufficient mental training you can get better. I don't know the limits of human mind so perhaps it's not impossible, I take that back.

 

[Dale]

Can you visualize a 4-D region of space? If so, post your ideas here!

If you have ever caught a ball then you visualized a 4D line, and did so quickly and naturally. We live in a 4D world and your brain is very well-suited to such visualizations. You can easily use that to visualize 4D geometric objects, for instance a 4D ball is just a 3D ball that begins as a point, expands to some radius, shrinks down to a point, and disappears.

 

[Phrak]

Perhaps with sufficient mental training you can get better. I don't know the limits of human mind so perhaps it's not impossible,

I wonder, myself. I can't do but the simplest exercises. How would one even train?

Nice visual of tesseract, by the way. Is it rotating about one of it's main axiis?

 

[Ben Niehoff]

In 4 dimensions, objects do not rotate "around axes". They rotate around planes. It looks like this tesseract is undergoing double rotation; i.e., rotation in the xy plane, and in the zw plane, simultaneously (and at different rates).

 

[Math Jeans]

In my honest opinion, I do not believe that it is possible to perceive the fourth dimension without making compromises.

By this I mean: it is physically impossible to see an actual four dimensional image in your mind because everything that makes up your mind is itself 3 dimensional. Unfortunately mental discipline can't bring the particles making up your body into a different plane of existence (although that might be an arguable point for some).

 

[Neo_Anderson]

How so?

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

1. 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.
2. 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.
3. 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 x and direction y.
4. 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. 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'.
A 4D organism, therefore, must have a retina that is cubic 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 cube. He must:

1. Look at the far-left area of the cube and memorize what he sees (his memory from our perspective will be an infintesmally thin slice of the cube).
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 cube is formed in his memory.
5. At the end of the mental excersize he notices that his memory--a plane of 1-D images of the cube--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 cube lying in his 2D memory simultaneously. He thus has sucessfully visualized a 3D cube.

By extension, we may do the same excersize, if our memories are good enough. We must visualize something like a tesseract (also known as a 'hypercube,' or a 4D cube), but only one 3D visualization at a time, as with the 2D organism and his visualization of the 3D cube. As we build our visualization of the tesseract, we keep a memory of all our previous visualizations of the tesseract, until we have a 4D visualization of many 2D images stored in a 3D memory.

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

* At this point you may protest saying," But how can the 2D organism move his eyes in 3-space? Aren't his actions restricted to the second dimension only?" But the hypothesis still stands. If you wish, you can presume that each "infintesmally-thin slice" of the 3D cube is lying in the 2D organism's plane, stacked one on top of the other. Thus, there would be many infintesmally-thin slices of the cube--each slice representing a slightly different viewing angle of the 3D cube--on top of one another in 2-space. It would be the 2D organism's job to memorize each 1D slice and store them in his 2D memory until a complete 2D image of the 3D cube is formed.

 

[Neo_Anderson]

Visualizing the fourth spatial dimension was popularized by Charles Hinton around the time of James Clerk Maxwell. Here he is:
http://en.wikipedia.org/wiki/Charles_Howard_Hinton"
And here's his paper on trying to visualize the fourth dimension. Unlike myself, he offers no hypothesis on whether or not the feat can be done, however. Nonetheless, an interesting read:
http://www.ibiblio.org/eldritch/chh/h1.html"

 

[Neo_Anderson]

This thing:
http://upload.wikimedia.org/wikipedia/commons/d/d7/8-cell.gif
is utterly meaningless to us. It doesn't do the fourth dimension justice. This video is to the fourth dimension what the drawing of a cube (all 12 lines) is to a 1D region of space. Try drawing a cube on a line and you'll see what I mean. If you do, you'll just end up with 12 equally-long lines drawn on a line--not much there to tell you it's a cube instead of 12 lines joined together on a line...
We'd have to generate a tesseract video using holography to get a better understanding of what the tesseract looks like from a 4D point of view...

By the way, Late Dr. Carl Sagan has alleged to easily and effortlessly visualize four spatial dimensions. Dr. Steven Hawking (Cambridge), OTOH, was quoted as saying, "I have a hard time visualizing the third dimension let alone the fourth..."

 

[Phrak]

In 4 dimensions, objects do not rotate "around axes". They rotate around planes. It looks like this tesseract is undergoing double rotation; i.e., rotation in the xy plane, and in the zw plane, simultaneously (and at different rates).

Excellent. (you can't get away with anything around here, can ya?) So, in general, you hold N-2 dimensional displacements constant, and rotate 2 others. This also means that one can perform N/2 independent rotations in N dimensions at the same time, where N is even.

I can't think of any notion of a "3-rotation" that is a generalization of the usual 2-rotation that isn't a composite of two 2-rotations.

 

[waht]

Excellent. (you can't get away with anything around here, can ya?)

that's the fun of it,

A general rotation in four dimensions has only one fixed point, the centre of rotation, and no axis of rotation. Instead the rotation has two mutually orthogonal planes of rotation, each of which is fixed in the sense that points in each plane stay within the planes. The rotation has two angles of rotation, one for each plane of rotation, through which points in the planes rotate. If these are ω1 and ω2 then all points not in the planes rotate through an angle between ω1 and ω2.

If ω1 = ω2 the rotation is a double rotation and all points rotate through the same angle so any two orthogonal planes can be taken as the planes of rotation. If one of ω1 and ω2 is zero, one plane is fixed and the rotation is simple. If both ω1 and ω2 are zero the rotation is the identity rotation.

http://en.wikipedia.org/wiki/Rotation_(mathematics)

 

[Phrak]

If ω1 = ω2 the rotation is a double rotation and all points rotate through the same angle so any two orthogonal planes can be taken as the planes of rotation.

http://en.wikipedia.org/wiki/Rotation_(mathematics)

Interesting. I'd like to see a proof of this. This seems to say that if you have two orthogonal rotations about the planes (x, y) and (z, w), called them omega_xy and omega_zw respectively, then rotate your coordinate system (whatever that means) to (x', y', z', w'), then there is a new parameter omega_x'y' = omega_z'w' that describes rotations about the planes (x',y') and (z',w').

 

[droogie01]

http://www.youtube.com/watch?v=JkxieS-6WuA"

possibly relevant?

definitely interesting

 

[Office_Shredder]

Take R⁴.

Project onto R³ along the 1st dimension. Your z axis is the 4th dimension

 

[sir_manning]

Take R⁴.

Project onto R³ along the 1st dimension. Your z axis is the 4th dimension

That's how I do it. Imagine our usual 3d world (ignoring time) inside a box. Then whatever moves inside the box is 3 dimensions, and however the box moves itself is higher dimensions - so if the box moves along a line, you have 4 dimensions. Then every 3 dimensions after that, put the whole system in a new box.

 

[Codexus]

Playing 4 dimensional tic-tac-toe is a fun way to practice one's ability to "see" in 4 dimensions. We used to do that between classes when I was in engineering school. I think we used a 4x4x4x4 hypercube. Anyway that was fun but it won't really help visualizing more complex shapes.

 

[xxChrisxx]

Mind = Blown.

 

[Sorry!]

Man, I'm trying to visualize what this looks like even with the diagrams and some extra google research and I just can't do it. I have no idea what any of how it should look. I think I pulled my brain.

 

[leroyjenkens]

Man, I'm trying to visualize what this looks like even with the diagrams and some extra google research and I just can't do it. I have no idea what any of how it should look. I think I pulled my brain.

It's hard to visualize something so different from anything you've ever seen before. It's like trying to visualize a color that doesn't exist. Try inventing a new color in your brain and visualizing it.

 

[Galteeth]

I met a girl who claimed she could do it. She was trying to explain it to me. I couldn't do it.
She was really good at math, but also insane.

She carved euler's identity into her back, amongst other things.

 

[Loren Booda]

I either visualize concentric spheres expanding outward (to represent spacetime's isotropic expansion), or a three dimensional space where every point has an associated scalar value, totaling four dimensions.

Rather than a scalar, a two or three dimensional space could map on the latter 3-geometry, making a 5-D or 6-D space visual.

 

[Neo_Anderson]

It's hard to visualize something so different from anything you've ever seen before. It's like trying to visualize a color that doesn't exist. Try inventing a new color in your brain and visualizing it.

About 800 Mg of Mescaline extracted from the San pedro cactus makes this a piece of cake!

I saw a void-like black color that was among the brightest color sources I had ever seen, and really queer bronze-like hue that was at the same time aluminum-grayish. Extremely strange! :tongue2:
www.erowid.com actually published my mescaline experience on their website.

 

[bassplayer142]

http://www.flowerfire.com/ferrar/java/hypercuber/HyperCuber.html

Here is a cool applet where you can rotate 4D.

 

[Galteeth]

Also, here is a 5d rubix cube puzzle you can download and play: http://www.gravitation3d.com/magiccube5d/