I was puzzling over something today and perhaps someone can help to agree with, or more probably, dispel the notion...
I was looking at a cube directly at one of its square faces by itself with the top-left edge going back away from me.
I imagined that, if the cube were a 3D slice of a tesseract, that the retreating Z edge line was actually a square; even though I could only see a line of it.
I imagined that retreating square rotating around between the z and w axes and realised it would get longer in our view as it rotated to being diagonally in line with us at 45°.
It would return to its original length at 90° with this cycle repeating on and on...
I'm guessing that this would also change the appearance of the front face which would have to elongate and shrink as well.
I'm not sure exactly what else would happen.
So as a side question: Would the front face just keep rotating around itself also or would the opposite face come emerging through?
I should contemplate it but this other question is puzzling me more first...
I realise that the W-axis is going off in an invisible direction (just as the X-axis goes off in an invisible direction for the 2Der).
The W-axis is perpendicular to the front face just as the Z-axis is perpendicular to the front face.
There are actually a full 360° of perpendicular directions to the front face and not just the W and Z axes.
So this lead me to wonder if I could make the tesseract's Z line actually offset some into the W direction while making the tesseract's W line offset an equal amount towards the Z line.
My question is: In 4D are there actually a whole 90° of varieties of tesseracts of same size that don't actually occupy the same identical space as each other?
If that is not correct can you help me understand why please?
The closest 2D-3D analogy I can give is that the x-line can be extruded either left or right.
In both instances the result will be cube's that occupy the exact same space but they'll just have opposite side colour's.
However in 3D-4D I'm wondering if two 90° axes can have a variety of perpendicular orientation's to two other 90° axes for a tesseract build?
And if that is so then does this support the idea of non-identical equally dimensioned tesseracts in 4D?