Do dimensions apply in a universe without a center and shape?

[Blossom Morphine]

How do the dimension, at least the first three we can observe, fit in the universe? We can monitor up/down, left/right, forward/backward on our planets, yes? But then how does this fit in with the idea that the universe has no center and shape, no true directions. Does this mean that the first three only pertain to planetary shapes, or is there something I am missing? Perhaps a great deal I am missing. Hopefully someone will be kind enough to answer and clear this up.



Thank you
Blossom Morphine

 

[selfAdjoint]

How do the dimension, at least the first three we can observe, fit in the universe? We can monitor up/down, left/right, forward/backward on our planets, yes? But then how does this fit in with the idea that the universe has no center and shape, no true directions. Does this mean that the first three only pertain to planetary shapes, or is there something I am missing? Perhaps a great deal I am missing. Hopefully someone will be kind enough to answer and clear this up.

Rather than stqar4t right away with the univesrse, let's just go out in a free falling environment like the space station or an orbiting shuttle. We are familiar from pictures that the people in these environments are as comfortable "upside down" as they are "rightside up". They have no definite pointers to indicate length, width, or up and down. They can be oriented any way according to the hardware around them, and define a coordinate system that works for them. This is the general rule wherever you don't have a detectible gravity direction to orient you.

So the general situation is that you don't have a single coordinate system of length, width, and height, but rather always a family of equivalent coordinate systems, any of which can be rotated into any of the others by a three dimensional rotation^*; an "equivalence class under SO(3)", to use mathspeak.

All the physics theories and cosmology theories start from there.

^* The question of left and right, and "handedness" is more subtle. The group SO(3) takes care of this by including reflections along with rotations.