What does it mean for extra dimensions to be very small?

[MattRob]

Okay, this one is a bit advanced, but I was wondering...
If space really has more than 3 dimensions, then what would it be like to rotate into higher dimensions?
For ex; In a standard 3-D coordinate system, if I rotate an object by it's Y axis, then it's cross-section changes for the X and Z axes. Naming the higher dimension "W", what happens if I rotate a 3-D object by the W axis? Does the object change "cross-section" in the XYZ axes?

Say I have a 2-D plane of X-width and Y-height, but has no Z-dimensional value, then what is it's cross section edge-on? Is it zero, infinite, or undefined?

If I have a 3-D cube of X-width, Y-height, and Z depth, but it exists in a 3< -Dimensional coordinate system, then could I potentially rotate it so that it has zero cross-section in 3-D space?

Thanks in advance to anyone brave enough. It's a question I doubt I could get answered anywhere else...

 

[JaredJames]

Space has a fourth dimension. It's called time.

For your "w" example, you have to define what "w" is. Otherwise, how can you rotate in it?

 

[Bill_K]

what happens if I rotate a 3-D object by the W axis?

MattRob, Weird thing about rotation in higher dimensions: two of the dimensions change, while all the others do not. So in four dimensions if you rotate in the X-Y plane, the W and Z axes remain unchanged. Thus in a sense the 'axis' of the rotation is not a single line, it's the entire W-Z plane. Numerically the rotation would be

X' = X cos θ + Y sin θ
Y' = Y cos θ - X sin θ
Z' = Z
W' = W

 

[coelho]

Ok... let's suppose you have a finite 2-d plane (a sheet of paper for example) resting in the x-y plane. At this particular moment, the sheet is completely "in" the plane (the z-coordinate of all its points is zero).

If you rotate it around the z axis, it remains completely in the plane. But if you rotate it around the x-axis or the y axis, only a line of it remains in the plane, the line given by the set of points of the sheet whose z coordinate is zero. The rest of the sheet is "out" of the plane, because the z coordinates of its points are different of zero.

Note that it happens even with infinitesimally small rotations around the x or y axis. The slightest rotation is enough to transform instantly the part of the sheet that is "in" the plane from a sheet to a line.

The same would happen to a cube in a 3-d space. If you rotated it around some axis (or rather around some plane as Bill_K said) the cube would remain entirely "in" the 3-d space, even if its shape (or at least its direction) were altered. But if you rotated it around some another axis (or rather around another plane), it (or rather the part of it "in" our 3d space) would instantly became a plane, which would be the set of points of the cube whose w coordinate is zero after the rotation.

 

[MattRob]

Ok... let's suppose you have a finite 2-d plane (a sheet of paper for example) resting in the x-y plane. At this particular moment, the sheet is completely "in" the plane (the z-coordinate of all its points is zero).

If you rotate it around the z axis, it remains completely in the plane. But if you rotate it around the x-axis or the y axis, only a line of it remains in the plane, the line given by the set of points of the sheet whose z coordinate is zero. The rest of the sheet is "out" of the plane, because the z coordinates of its points are different of zero.

Note that it happens even with infinitesimally small rotations around the x or y axis. The slightest rotation is enough to transform instantly the part of the sheet that is "in" the plane from a sheet to a line.

The same would happen to a cube in a 3-d space. If you rotated it around some axis (or rather around some plane as Bill_K said) the cube would remain entirely "in" the 3-d space, even if its shape (or at least its direction) were altered. But if you rotated it around some another axis (or rather around another plane), it (or rather the part of it "in" our 3d space) would instantly became a plane, which would be the set of points of the cube whose w coordinate is zero after the rotation.

That's exactly what I was looking for, thanks!

In the same way that in viewing an X-Y plane in a X-Z or Y-Z cross-section would make it "vanish", the cross-sectional area becomes zero on those planes (aka coordinate grids) could you rotate a 3-D object into higher dimensions so that it's XYZ volume is zero on the XYZ coordinate grid?

Now, when I hear about certain theories that say there are more than 4 dimensions (counting time as a dimension), are those higher dimensions spatial? I.e., could you somehow "rotate" into the higher 11 dimensions in string theory, unlike the dimension of time?
Or are those dimensions entirely different from time and spatial ones?

 

[DrGreg]

Now, when I hear about certain theories that say there are more than 4 dimensions (counting time as a dimension), are those higher dimensions spatial? I.e., could you somehow "rotate" into the higher 11 dimensions in string theory, unlike the dimension of time?

It is postulated that the extra dimensions, if they exist, would be very small. What does that mean? Here's an analogy.

Consider a very long but very thin hollow tube. From a distance it looks one-dimensional, but up close it is actually a two-dimensional surface. (I'm thinking of the tube surface, not the interior.) That's an example of how an apparently 1D object could be, on a very small scale, really 2D.

So our apparently 4D universe could be, on a very small scale, really 11D.