Science

# Let’s talk about the 4th dimension

on October 6, 2013
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Here at the lab we have all kinds of meetings with other scholars from around the world. One of our lab assistants posted this question to a few our guests.

How is this a representation of the forth dimension? What is that image supposed to tell me about what the forth dimension looks like?

The animation makes it more confusing IMO.

Think of this way:

A cube is made up of two squares connected by their vertices with lines that form right angles to their edges. Both squares are 2-d, the only way to form cube is to have a new dimension through which they can connect, we call it depth.

With a tesseract, imagine two 3-d cubes that are connected by lines at their vertices. In order to be a tesseract, these lines must be at a “right angle” to the other three edges, but this is obviously impossible in 3 dimensions. We have to make a new dimension to extend these new edges through to connect the vertices.

If we travel straight up or down (Y-axis) from a horizontal line (X-axis), it forms a right angle in two dimensions. If we travel straight backwards or forwards (Z-axis) it forms a right angle in the 3rd dimension.

If we travel straight along a new axis, it will form a right angle in the fourth dimension. I like to think of it as “inside/outside” but all of these are just ways to conceptualize a dimension you can’t visualize.

I think about the fourth dimension very often, and think what it would be like. If it really does exist, why can’t we use three dimensions to make something LOOK four dimensional? I mean we can draw in 2d and make it look 3d. Why not 3d -> 4d somehow?

Things drawn in 2D that look 3D only do so because it tricks our brain. It uses things like forced perspective and guiding lines to give us the idea that we’re looking at a static 3D image. There is no way to tell the difference between a picture of an object and a picture of a sufficiently detailed cardboard cutout of said object even though one is 3D and one is 2D. Since our brains don’t know what the fourth dimension would look like, there are no “tricks” in 3D to represent it, thus it ends up simply being another 3D object to us. There is no way to tell the difference between a 3D object and a 3D representation of a 4D object, and if there is no way to perceive any difference, there is no difference.
Wow that makes sense.

Here’s another way to realize why it looks weird.

First, imagine you are looking at the shadow of just a regular old 3D cube projected onto a 2D piece of paper. (That is, just a regular drawing of a cube.)

There are 3 different axes you can spin this cube around: X, Y, Z.

If you spin it around the Z axis (that is, rotate it around a flagpole sticking straight up out of the paper), then the shadow doesn’t look weird at all to the 2-dimensional people “living” in the flat paper. It just looks like an unchanging shape rotating around like a merry-go-round or carousel strictly in 2D.

But if you rotate the 3D cube in any other way, then the 2D shadow on the paper changes shape and does all sorts of strange things incomprehensible to the people looking at the shadow inside their flat 2D world. It only makes sense to us, because we’re really imagining the 3D cube in our heads while we’re looking at the 2D shadow.

Similarly, there are ways that you could rotate this 4-dimensional “tesseract” so that its “shadow” in our 3D world would just look like some object rotating in a normal way that we understand. But, there are other ways to rotate it in the 4th dimension! And if you rotate it around one of these other directions, then its shadow in our 3D universe will look really weird like this image.

If you were living as a two dimensional being, for instance on the surface of a paper, you’d experience three dimensional objects only in that plane, so if someone pushed a pencil through the paper, you’d see a dot at first, which would then expand into a hexagon (six corners), because that is what the cross section of a pencil looks like and because you can only see the part that is in your plane of existence, the paper.

When you live in a three-dimensional space, like we are, you can only experience four-dimensional objects through their representation in your space. You’d actually experience a 4-d object as a 3-d shape floating in mid air and changing appearance as it moves through your space.

That was an incredible explanation, thank you. I didn’t understand some previous posts about the shadows, but you have really helped me to visualize what’s happening here. I just gained a dimension.