This is a Sideband to the previous post, The 4th Dimension. It’s for those who want to know more about the rotation discussed in that post, specifically with regard to axes involved with rotation versus axes about which rotation occurs.
The latter, rotation about (or around) an axis, is what we usually mean when we refer to a rotation axis. A key characteristic of such an axis is that coordinate values on that axis don’t change during rotation. Rotating about (or on or around) the Y axis means that the Y coordinate values never change.
In contrast, an axis involved with rotation changes its associated coordinate values according to the angle of rotation. The difference is starkly apparent when we look at rotation matrices.
I’m not going to assume any special familiarity with transformation matrices. (If you’re curious, see the Matrix Magic post.) This is mostly going to be about what those matrices look like.
Here you can just take me on faith on these two points:
¶ If we take the cosine and sine of an angle, we get X and Y values that plot a circle (with a radius of one). If we reverse it, use sine for X and cosine for Y, we get the same circle, except it’s rotated 90 degrees.
(Not that we could tell visually unless we marked, for instance, the points where the angle was zero. In the first example, we’d mark point [1,0]; in the second, we’d mark point [0,1].)
All you need to know is we view the vertical columns of the matrix as pointers that (literally) point to the transformation (rotation, in this case). For now, from left to right, we’ll call them X, Y, Z, & W.
(Assuming we need all four. If we need two, it’s X & Y. If three, X, Y, & Z.)
That’s all we really need to notice something important about rotation matrices.
Let’s take a look at this one:
This describes rotation about the vertical (Y) axis. The two involved axes are X and Z. We can tell this just by looking at the matrix.
The middle vertical column, the Y column, from top to bottom, reads [0,1,0]. This means all Y coordinate values are preserved in the transformation.
In other words, no Y values change. That is the definition of an axis of rotation.
The left and right columns, X and Z, both have a cosine–sine pair. And note that one is reversed from the other (don’t worry about the minus sign for now).
This means:  both X and Z coordinate values change;  they change in a circular way;  X and Z are 90° from each other (because of the reversed sine–cosine).
All together, that means X and Z coordinates rotate around the Y axis.
(An additional detail: The middle zero in the X and Z pointers — their Y value — tells us that they leave the Y value alone.)
Here’s another 3D rotation matrix:
This one, as should be apparent, rotates about the X axis and involves the Y & Z axes.
The left column, the X pointer, has the fixed value [1,0,0], which means the X coordinates won’t change.
The Y and Z pointers have the cosine–sine pairs that involve rotation. Notice how they have the same basic form, just in other parts of the matrix.
(The zeros in their X values indicate they don’t change X coordinates during rotation.)
Here’s the final 3D rotation matrix:
You could almost guess this has to be the Z axis, and — as the matrix makes clear — you’d be right.
It’s the same deal as the first two, just rearranged a bit.
That repeated pattern of cosine-sine stands alone if we look at a 2D (proper) rotation matrix:
Notice how it looks exactly like the upper left corner of the Z axis rotation matrix. That’s because they are essentially the same thing.
When we rotate, say, a piece of paper, all the points move; all the points rotate about the center. Thus, the matrix also changes everything. No ones or zeros.
This is what does the actual work of rotation (hence the name involved axes).
If you got this far, 4D is easy, almost trivial.
- In 2D, we have the one proper rotation (shown above).
- In 3D, we have three proper rotations (also shown above).
- In 4D, there are six proper rotations!
In the same way the 2D (Z axis) rotation is included in the three 3D rotations, the three 3D rotations are included in the six 4D rotations.
So the first three will look familiar. Here they are in the same order as presented above.
That’s a rotation about the Y axis (and the W axis). That is, Y and W coordinates don’t change. The involved axes, exactly as above, are X & Z.
That’s a rotation about the X axis (and the W axis), with Y & Z involved.
And that’s a rotation about the Z axis (and W), with X & Y involved.
These three rotations of a 4D object would just rotate it “normally” from our 3D perspective. A rotating tesseract would look like a rotating cube.
There is an interesting question about whether one could physically rotate a 4D object from only three dimensions. It requires moving fourth dimension points.
Could Flatlanders move a 3D cube that had intruded into Flatland? An interesting question. (What if it was attached somewhere in 3D Land?)
From a 4D perspective, there are three other proper rotations (all involve the W axis):
This is the rotation discussed in detail in the previous post. It’s the first rotation mode of the tesseract in the video.
This is the second tesseract rotation mode. You can see the Y axis is involved, and so the motion is vertical.
And this is the third tesseract rotation mode. This involves the Z axis, so the motion is front-to-back.
If you use one of the last three transforms to rotate a 3D cube, you get an improper rotation of the cube.
That is, you end up reflecting or mirroring the cube and reversing the order of its points. This happens because the 3D cube is rotated through the fourth dimension.
With all that in mind (and the previous post), this should now make some degree of sense:
You should now be able to see and have some understanding of the three rotation modes the tesseract makes.
(And, if not, that’s okay. After all, you’re only three-dimensional!)
I’ll leave off today (and probably for some time to come) with another work in progress that demonstrates a cube being improperly rotated.
That is to say, it’s being rotated through the fourth dimension, which turns it “backwards” until it’s rotated back again:
As the camera descends, the cube is spinning properly on its Y axis. After pointing out the order of the corners, the video shows rotation into the W dimension as first horizontal movement (of varying degree), then vertical movement, and finally a combination of both.
It also shows what the rotation looks like if the W dimension isn’t rendered. Then the cube appears to shrink to a 2D plane before growing back to full size, but reversed. The full rotation reverses it back again.
(As I said, a work in progress. 😀 )
And on that note (or set of notes), I’m done talking about the fourth physical dimension for a while. And no more transformation matrices, I promise!
I am planning to talk a lot about phase space (which has as many dimensions as necessary, but they’re all virtual). It continues a topic I started when this blog first began, and it’s (I think) one of the best tricks since sliced bread.
I think it’s a really useful tool for looking at life!
Stay 3D, my friends!