unicole wrote:In a very basic sense, where do the theories of spacial dimensions higher than 4 come from?
This is a case of plain ole' geometry. By defining a dimension as an axis of freedom to move back and forth in, you can build these axes together into an n-dimensional universe. No matter how many dimensions you have, they all intersect at 90 degrees of each other. Moving left/right is 90 degrees to forward/backward. And, both of these directions are 90 degrees to up/down. Combining all three into one makes right/left, for/back, and up/down as a 3D space. If you attached future/past, you get a 4D space-time.
But, if keeping to space dimensions only, you end up with more and more 90 degree axes to move around in. So, for a real 4D space, we will find ana/kata to be perpendicular to all up/down, right/left, and for/back at the same time. A 5D space will have an additional axis 90 degrees to all four, and so on.
In the book Warped Passages by Lisa Randall, she describes some dimensions as being "very tiny" but what the hell does that mean? How can a spacial dimension be tiny? Aren't they all built up on each other?
I guess there could be two ways to be tiny. One type is just a small amount of thickness in that higher dimension. Like a sheet of paper, it's mostly 2D, with a very tiny amount of 3D thickness. In the same way as this, it's possible to have a mostly 3D object, with a tiny 4D thickness. Another way is the curled up, compactified dimension. This is the kind String Theory is based on. In this scenario, we have something more like a garden hose universe.
Imagine a person tightrope walking across a garden hose, suspended off the ground. The only sizable part of the hose is the length, which has only one dimension, only capable of forwards and backwards movement. Now consider what an ant would perceive. This critter is so small, it has two dimensions of freedom. Not only can an ant move lengthwise, but also around the hose, in the second dimension. Note how the second direction of left/right is self-repeating. Moving in a straight line leads back to the start. A person would not experience this hidden extra dimension because of size. Last time I heard, our universe had somewhere around 26 dimensions, as 4 extended large, and 22 tiny curled up directions, all 90 degrees to each other. Theoretically.
Theorizing that there are 10 or more spacial dimensions, we would soon stop being able to comprehend them. So at these higher dimensional levels, would you still experience the first dimension?
Well, if there's anything I've learned about wild high-D stuff, is that it's not all that incomprehensible. There are various tools at one's disposal that can potentially make high-D stuff easier to see. Well, you don't actually "see" them, but you have a very good model of it in your mind. It also depends on which shapes you're working with. Polytopes are highly complex, but toratopes (hyperdonuts) are very simple, and can be understood in very high-D. Of course, these are just geometric objects, when it comes to even more abstract spaces, then it's more difficult.
If you were 10D, you could see, perceive, feel and understand all 10 axes, even if it's the x-axis of the 1st dimension. Where, any axis could be x, it depends on orientation. Take a 10D cube for example. Holding it in your hands, you could still feel every sharp pointy 0D corner, joined by every 90 degree sharp 1D edge branching off the corners, joined by every flat 2D square panel, etc, all the way up to the 9D surface panels that your 10D hands are touching.