# Us humans visualising the fourth dimensional line element

1. Nov 26, 2006

### PMCIIII

Alright. So you guys already know that the first dimension is simply a line (according to our 3-space perspective).
The second dimension extends a line at 90 degrees from the first, creating a plane.
Adding yet another line 90 degrees from the first two lends a cube. You may spare no great expense visualising the corner of a 'box' to see the corner has three line elements; all at 90 degrees from each other, that lend it the property of the cube.
Now here comes the tricky part: Add another line to the corner of our cube; 90 degrees in relation to the other three lines, and we obtain the hypercube ("Tesseract"), which is the fourth-dimension analogue of our regular, ol' cube.

Now I'm asking if the human mind is capable of visualising something like this. I have a hypothesis that answers this in the affermative, but I still need help. But first, some background.

1. A 3-D sphere entering then leaving the second dimention would appear as a circle that starts from a point, grows to the max diameter of the sphere, then shrinks back into nothingness
2. Likewise, a hypersphere entering our 3-D space would appear to us as a regular ol' sphere beginning at a point, growing to the maximum diameter of the hypersphere, then decreasing in size to nothingness.
3. A 2-D organism would observe the sphere entering his universe as an object with depth and height, but no width. We may interpret this as an infinitely-thin sliver; an infinitely thin slice of a 3-D sphere.
4. As 3-D organisms, we percieve the 4-D hypersphere entering our universe as a regular ol' sphere having height, depth and width but not omega (omega is the fourth-dimensional line element. Thus, we have width, length, height and omega). The fourth-dimensional organism would interpret the sphere we see as an infinitely thin slice of his hypersphere.
Interesting properties:​
When we, as 3-D organisms, look at any surface in our universe, what we're actually looking at is second-dimentional planes. Just look at a box if you don't believe me: The box you're looking at is consisting of three planes joined in 3-space to form a cube.
Likewise, the organism in 2-space, when looking at his square, is actually looking at lines. Two of them, provided the corner of the square is seen.

It stands to reason, therefore, that any fourth-dimentional organism looking at the surface of his hypercube would actually be looking at cubes. That's right: The surface of the hypercube is the cube, just like the surface of our cube is the plane.

Now my question is this: Since we are 3-D organisms with 3-D imaginations and 3-D powers of visualisation, is it possible for us to actually visualise a hypercube? Or, does our 3-D brain prevent us from fathoming such an object? Remember now, regardless of size, our cube (think box or something) to the 4-D organism is an infinitely thin sliver of his 4-D hypercube. Think of the possibilities: I can stuff a box of cereal into a box that measures 14" by 10" by 3". But in the fourth dimension, I can actually stuff an infinite number of cereal boxes in the hypercube, even though the dimensions of the hypercube are finite (14" by 10" by 3" by X" omega)!

I have alot more to say on the subject. But do you guys think that visualising such a space is possible? I mean, Dr. Sagan once was quoted as saying he can visualise 4-space. Then Dr. Hawking turns around and says, "I have a hard time visualising the third dimension -- let alone the fourth!"

2. Nov 26, 2006

### GeoMike

I'm not an expert on this (by far), but one thing I always wondered:

If you take a 2D object (e.g. a square), you can "unfold" it to make a line. Likewise, a hollow cube can be "unfolded" to make a figure that can be completely contained on a single plane. So a 4D "cube" could be "unfolded" to make some figure that is completely described within a 3D space, right?

-GeoMike-

3. Nov 26, 2006

### 0TheSwerve0

Really quick, have you seen the movie "Cube 2: Hypercube"?

4. Nov 26, 2006

### PMCIIII

The 2-D "object" is transformed into 1-D only after it -- and everything in it -- is "unraveled" onto a linear surface: No width or height; just depth.
A 4-D tesseract, when likewise unraveled, will reveal the surfaces of that tesseract. And the surfaces of the tesseract, as we already know, are 'cubic' in nature; that is to say, there will be the three line elements of length, width and height.

Remember, when we, as 3-D organisms look at our cube, what we are seeing are planar, 2-D surfaces -- not 3-D surfaces. And what's remarkable is that we see all points of the planar surfaces simultaneously. This is remarkable, because the 4-D organism -- looking at the surface of his tesseract -- is actually seeing cubes. Not just cubes as we see them (2-D, planar surfaces), but every point on and in the cubes, all at teh same time!
When we see a cube, we only see, at most, three of its planar surfaces (look at the corner of a cardboard box). When the 4-D organism looks at our cube, however, he not only sees all six of its sides, but everything contained within that cube all at the same time!
Is it possible for us to visualise the same?

5. Nov 26, 2006

### fourier jr

with some things it's best to just get used to them, rather than try to understand them, as john von neumann said. someone could go crazy trying to visualize a vector like $$[x_1, x_2, ...., x_n]$$ or especially $$[x_1, x_2, ...]$$

Last edited: Nov 26, 2006
6. Nov 26, 2006

Mathematics is not mysticism.

7. Nov 26, 2006

### Alkatran

I wouldn't be surprised if there are people who can, to a limited degree, visualize things in 4D. However, most people have difficulty even visualizing 3D objects in their head.

One of the more popular displays of a tesseract:
http://www.mathematik.com/4DCube/4DCubePovray.html

8. Nov 26, 2006

### PMCIIII

The only thing the 2-D organism needs to do is visualise his planar universe on its side. As it presently stands, he cannot do that. Instead, he is left with the constraint of viewing his universe with depth and height, but no length. That means he sees his square as having two sides: One side is height and the other depth.
Now how can he visualise his planar universe from its side, much in the same respect we see text on a sheet of paper?
Now how can we visualise a cube our 3-D universe from its side? If we can visualise a cube on its side, we'd see all six sides of this cube and everything within that cube. Kinda like looking at a 2-D universe square and seeing that it has four sides and stuff inside.
How does the 2-D organism view all four sides of his square and everything in it all at the same time? He can't. But we can. His square to him is just two lines and the only way he can check the contents of his square is if he breaks into it and looks inside.

We see a drawing on a sheet of paper. We see, actually, all parts of the said drawing, including all four sides of the paper it was drawn on. The 2-D organism, on the other hand, does not see any drawing on the paper. In fact, the only thing he sees is the two edges of the paper (he knows depth and height, but not width). If he wants to see the drawing on the paper, he'd have to break through the boundaries of the paper and take each infinitely-thin slices of the paper and comitt all of them to memory to generate a "whole".

9. Nov 27, 2006

### Volumunox

As a 3D artist i would say no its not possible for us to visualize 3D or 4D shapes.

i've been sitting with 3D for years, and yet i only see 2D.

Not even our mind can perceive a 3D object of any sort.
the mind works like our eyes, only being capable of seeing what our eyes see.

We know it got more corners, but we do not have the input to perceive it.

It is much like the 6th Sense, as we do not see or feel the object comming our way, only sense it.

much like we don't know whats on the other side of that cube, but we know its a cube.

EDIT: Found this on Wikipedia about perception

"Many cognitive psychologists hold that, as we move about in the world, we create a model of how the world works. That is, we sense the objective world, but our sensations map to percepts, and these percepts are provisional, in the same sense that scientific hypotheses are provisional (cf. in the scientific method). As we acquire new information, our percepts shift. Abraham Pais' biography refers to the 'esemplastic' nature of imagination. In the case of visual perception, some people can actually see the percept shift in their mind's eye. Others who are not picture thinkers, may not necessarily perceive the 'shape-shifting' as their world changes. The 'esemplastic' nature has been shown by experiment: an ambiguous image has multiple interpretations on the perceptual level.

Just as one object can give rise to multiple percepts, so an object may fail to give rise to any percept at all: if the percept has no grounding in a person's experience, the person may literally not perceive it."

-
Volumunox

Last edited: Nov 27, 2006
10. Nov 28, 2006

### PMCIIII

Charles Hinton was the foremost researcher in 4-D, in the respect that he claims to have visualised this space.
He suggested a cube that revolves around Omega (length, width, height and omega -- the fourth dimentional line element) and provides systems for (attempting) imagining this 4-D line element. Reviwing his work, I observe it to be crude. The better approach would be the attempt at visualising the cubic surface of the 4-D hyupercube. I have a hypothesis that shows that this is indeed possible. It's not axiomatic because the hypothesis is rooted in
-- gotta go!

11. Nov 28, 2006

### Anttech

Volumunox:

If you are correct regarding Humans not being able to visualise 3D shapes, how come Humans can rotate 3 dimensional objects in their head and then understand what they would look like at different points on the Y and X axis?

....Spacial analysis, (Im rather good at this actually, and it helps me in my work)

12. Nov 28, 2006

### Jimmy Snyder

If the issue is spacial dimensions, I don't feel that I am able to visualize four of them.

However, it is easy to substitute a non-spacial dimension for the fourth one. Time is often suggested for this purpose. Be careful not to confuse this with spacetime. Here time is being used as a way of visualizing the fourth spacial dimension. So a four dimensional cube would look like a 3 dimensional cube that pops into existance for a period of time and then disappears. In the book "Flatland", a 4 dimensional sphere was described as a figure that appears as a dot, grows into a small sphere, continues to grow until it reaches a maximum radius, then shrinks back to a dot which then disappears.

Another way to visualize the fourth dimension is as color. Imagine a monochrome cylinder which is to be turned into a Klein bottle. This cannot be done in 3 dimensions. However, we will consider that two points do not coincide unless their colors are the same. At the points which need to intersect in 3-space, change the color of those points so that they don't intersect in 3+1 space. That is, their three spacial coordinates coincide, but their fourth coordinate, color, does not.

13. Nov 28, 2006

### Volumunox

If you have to rotate the object in your mind... its still 2D. ;)

14. Nov 28, 2006

### Volumunox

Even if i had a 3D screen it would be the same as watching the object in real life.

what i see in the real life is plain.

i can perceive the depth but i can not see the object as a whole.

as i understood 4D being, they see the object as whole, the all seeing if you may.

I can look at my cup, but never know whats behind it, i might aswell look at a picture of 2D.
and how would i know it wasn't.

the only thing that makes us slightly able to percieve the depth, is when we move.

only when we move can our brain start to perceive that it has depth, but it still dosn't see it as whole.

The depth could just as easily have been an illusion

my mind works with pictures, i can picture a box in a box, as the 4D is represented, but i can still only see the boxes i do not have a 360 degree view of the two boxes.

-
Volumunox

15. Nov 28, 2006

### Anttech

In this context there is no difference between perceiving something and 'seeing' it. Due to the nature of light, and our own metaphysical state, it is impossible to perceive a 3d object from all angles at the same time. However that does not mean we cannot understand, and visualise a 3d object

16. Nov 28, 2006

### PMCIIII

You ask the wrong question to the worong person.
Anttech, you have no respect. Try and come up with an idea for a change...

17. Nov 29, 2006

### Moonbear

Staff Emeritus
This has gone on long enough. The question simply can't be answered and requires far too much speculation to try. Thread locked.