Lets say you have an object in a 4d coordinate system.
X=left and right
Y=back and forth
Z=up and down
T=past and future
Now you can also rotate each axis: pitch, yaw, and roll.
Is there a way to rotate an object in the forth dimension, and if
so, how does the view change?
I’ll let you know last Thursday.
Neat question. What’s rotation? It’s tempting to want it to be “about an axis”, but it’s actually “in a plane”. In two dimensions, there’s only one such plane (if we thought of rotation as about an axis, it would be an axis that doesn’t exist in our coordinate system). Rotation of a vector (or point) consists of multiplying its coordinates by the matrix:
[cos(a) -sin(a)]
[sin(a) cos(a)]
Where a is the angle of rotation. Both coordinates change in a 2D rotation. In three dimensions, there are three possible planes we can rotate “in” (corresponding to the three axes we can rotate “about”). The three rotation matrices are familiar (well, maybe not, I had to look them up in my third year physics book):
[cos(a) -sin(a) 0]
[sin(a) cos(a) 0]
[ 0 0 1]
[cos(a) 0 -sin(a)]
[ 0 1 0 ]
[sin(a) 0 cos(a)]
[1 0 0 ]
[0 cos(a) -sin(a)]
[0 sin(a) cos(a)]
Corresponding to rotation in the xy plane (“about the z axis”, or “yaw”), rotation in the xz plane (“about the y axis”, or “pitch”), and rotation in the yz plane (“about the x axis”, or “roll”). In a 3D coordinate system, there are three possible rotations corresponding to which pair of coordinates we allow to change. The rotation appears to be specified by choosing an axis to not vary, but I think that’s a coincidence of three dimensions.
In a 4D coordinate system, it’s not clear how to build a rotation matrix that would represent rotating “about an axis”, but rotation “in a plane” is easy. There are six such rotation matrices, corresponding to which pairs of coordinates we choose to allow to vary:
[cos(a) -sin(a) 0 0]
[sin(a) cos(a) 0 0]
[ 0 0 1 0]
[ 0 0 0 1]
[cos(a) 0 -sin(a) 0]
[ 0 1 0 0]
[sin(a) 0 cos(a) 0]
[ 0 0 0 1]
[1 0 0 0]
[0 cos(a) -sin(a) 0]
[0 sin(a) cos(a) 0]
[0 0 0 1]
[cos(a) 0 0 -sin(a)]
[ 0 1 0 0 ]
[ 0 0 1 0 ]
[sin(a) 0 0 cos(a)]
[1 0 0 0 ]
[0 cos(a) 0 -sin(a)]
[0 0 1 0 ]
[0 sin(a) 0 cos(a)]
[1 0 0 0 ]
[0 1 0 0 ]
[0 0 cos(a) -sin(a)]
[0 0 sin(a) cos(a)]
The first three are your “yaw” (rotation in the xy plane), “pitch” (rotation in the xz plane), and “roll” (rotation in the yz plane) extended to four dimensions.
The last three correspond to rotations in the xt, yt, and zt planes. What’s that? Think of a rotation in the xt plane. It’s sort of like a translation along the x axis (sliding “forward” or “backward”) accompanied by a translation along the t axis (sliding “ahead” or “back” in time). Meanwhile, our y and z coordinates are remaining constant. But, don’t forget it’s a rotation, and not really a pair of translations. That is, if we’re farther from the “origin” of the rotation, then our coordinate changes more, as a function of a. It would look like you stayed in a fixed yz plane, and your xt position ratated about a fixed point in that plane.
I’m not sure where you got the idea from Enola, but you may be interested to hear that there’s a Robert Heinlein book called (I think) The Number of the Beast that relies on this premise. IIRC, there’s some device that allows exactly such rotations (actually, I think it was rotation of the axes, which just means you flip the signs on the sines in the rotation matrix).
(kgmm)/(s*s)
Let’s start with three dimensional space:
Rotation in Yaw ( rotation about x) changes your view as you rotate. You can go from positive y, positive z, to +y-z, to -y-z, to
-z+y. Rotation about each axis changes your “view” of the two perpendicular axes. Assuming T, time, is perpendicular to x,y,and z, then rotation about T will give you a “view” of all the “planes”.
Myself, I don’t buy that time is perpendicular to the three planes, but that’s another story.
b.
I’m not sure if this is what you’re getting at, but a rotation around an axis is very similar to moving in Special Relativity. In SR, moving at a constant velocity is equivalent to a hyperbolic rotation - replace the sin’s and cos’s in Newton meter’s matrices with sinh’s and cosh’s. But, it gets difficult - in some ways a hyperbolic rotation is the same a circular one, and in some ways it’s different. So this may or may not correspond to what you’re thinking of.
The device installed into the spaceship in The Number of The Beast allowed for rotation and translation about six axes, three spatial and three temporal, the upshot of which was to permit travel through alternate realities with differences ranging from the subtle (no letter “j” in the English alphabet but everything else the same) to the drastic (an inverted universe where there were balls of vacuum interspersed throughout a medium of solid matter) to the fantastic (L. Frank Baum’s Oz as an actual planet). The adventurers also pulled some interesting tricks via spatial-temporal manipulation like instantly teleporting 100 feet into the air. It’s an OK read with some very interesting ideas, but not his best work IMHO, and IIRC, it was his last book.
Edwin A. Abbot sort of goes into this in his book Flatland, which IIRC was written in the Victorian era but was way ahead of its time. It’s available for free at this link. Dionys Burger wrote a sequel called Sphereland where he delves further into the issue with more modern notions of space-time.
“Space-like” dimensions (like x, y and z) and “time-like” dimensions (like time) are probably not interchangeable in that way
Newton,
There’s also a pretty interesting novel by Christopher Priest entitled The Inverted World.
If you replace the trig functions in Newton meter’s matrices (can I just call you Joule?) with their hyperbolic equivalents, wherever they involve time, and shuffle a few minus signs, you get the generator matrices of the Lorentz group, which is the set of all “rotations” in 3+1 dimensional space (i.e., 3 spatial dimensions and one temporal). You might have more luck searching for that term.
The problem is that people try to visualize spaces which are completely unnatural to… well, vision. If you want to ROTATE about the ‘T’ axis, then you are only changing position, XYZ.
If you want to rotate about a spatial axis, then it’s likely that you’ll end up changing time. In vector manipulation, the number of coordinates doesn’t make a difference, as long as you don’t try to picture it in your head.
The identity matrices which (???) put up earlier for up to 4 dimensions are the right track. You pick two coordinates, take the plane which contains them (a unique plane) and take the angle.
You could alternatively REFLECT across a plane. Just don’t try to picture it when it’s 18 dimenional!
Another good tip is to avoid confusing geometric questions with metaphysical questions, maaaan. 
I’m going to take a stab at this, sans matrices. And I won’t even have to say “quaternions”. Oops.
Reduce to 3 dimensions, 2 spatial and 1 temporal. Consider a point that travels, in 2 space, along the x axis from -inf. to +inf. Define time t=0 to be when it crosses the y-axis. Note that this point’s path in 3 space is a line in the x,t plane.
Now rotate around the x-axis until it is a line in the x,y plane.
What an observer would see in the 2-d world, is that at time t=0, all of sudden a line would appear diagonal to the x,y axes, and just as suddenly disappear.
If we consider a 2-d object larger than a point, like a circle, what would be seen is the outer edges of two parallel lines. I don’t know what the interior would be like. Think like the trail of a sparkler on long exposure film. Again, there-not there in an instant. (All of a sudden I got a yen to listen to Allman Brothers, Eat a Peach.)
[/end of X axis case]
If you rotate a multiple of 90 degrees in the y axis, you can get different combinations of the point moving forward/backwards in time/space.
Rotation about the t axis is boring.
Rotations not a multiple of 90 …
…are just like when coming out of an Improbabilty Drive jump. Any hallucinations are your own problem.
Ooh, I forgot this one:
Assume the point follows the path of the plot for the sine curve. Wobbling around the x axis.
Rotate it just right, and you’ll see appearing out of nowhere a bunch of evenly spaced points. These points split in two, move away from each other and merge with the neighboring ones, and then vanish.
As you can imagine, I saved a lot of money not having to buy drugs in the '60s.
Nope, I’m way off – he had four original fiction works after TNOTB until his death in 1988.
If I have a point at (1, 0) and I want to rotate it by 45 degrees, then the point gets to (0.707, 0.707). I know this because I’m used to working with angles in that manner, and I know that, generally speaking, rotating in a circle around the y-z plane means moving around a circle with radius r. The translation then means that a point is at (r cos a, r sin a), where ‘a’ is the angle of rotation. I’m unclear on how I would get this result from Newton Meter’s post as I haven’t really worked with matricies since, oh, algebra two.
Can someone elaborate a bit?
Err, probably the x-y plane.
Oh, I see.
[cosa -sina][r] = [cosa*r +(-sina*0)]=[rcosa]
[sina cosa][0] = [sina*r +( cosa*0)]=[rsina]
Nifty. This totally, finally explains why you use matrix algebra in 3D graphics.
For those who don’t know matrix math, like me, I found this page which made some actual sense:
http://www.sosmath.com/matrix/matrix.html
I can’t help with the answer to your question, but I’d like to warn you to be very careful when you’re rotating your axes, wouldn’t want to slice a finger off now would we?
Would this be a hyperzombie thread?