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I bet when most people think of branes, they picture a dimensional sandwich, or dimensional strata because thats what most illustrations seem to depict.
Probably because thats the easiest way to depect it.

But I think that this is an over simplification. Picture basically a lucite cube. In this cube there are floating pennies that are ultra thin. Now imagine that as you look at each side of the cube, any time you look foward, all the pennies appear facing you, and the color green. Now you look at the top, and the penny you were looking at has disappeared, but nearly in the same spot, facing up at you is the face of a red penny. If you look at the front of the cube again, but tip it down just slightly to see the depth of the pennies, the front is still green, but now you can see blue edges extending backward and any penny you view continually will also be fully blue. If you focused on the third penny back initially, it would have a green front, but every penny in front of it, and the back of the penny on to infinity would be blue.

I picture each penny as the position of the smallest measurable matter in existance (regardless of our ability to measure it or not). This smallest point of matter would look to us as a single dot. But it is not. It is 129601 dots. A central dot, and a dot for every 3d (or 4d if you count diagonal as a dimension) position 1 degree off of center. Although each of the dots have the illusion of occupying the same space, they in fact don't exactly.. They are all out of phase with each other. We images x and y co-ordinates as intersecting, but that is our brains simplifying the illusion. In point of fact, both are seperate dimensions and never actually meet. Think of a small light bulb traveling through a colored glass tube. If the tubes overlap at any point, the light will hit the point of the overlap and appear to be in both tubes. But it is in fact, only in one tube. Either the front tube or the back tube. Either way, the light appears in the same position, but is not actually in the same position for both tubes.

A more direct way to picture this is to say that both tubes DO actually merge and intersect..BUT if you were in one of the tubes, and your perceptions were limited to that tube..anything in the tube would be visible to you and solid. But you could walk through the intersecting tube and never even know it..Even if your perceptions allowed you to see what was in the tube, it would be out of phase with you, you'd try to grab something from that tube, it would be like nothing was actually there.. or it would be like trying to grab a ghost.

The reason the math still works and movement still works is a variation on the Heisenberg Uncertainty Principle. Basically, we can never know exactly where something is in relation to something else, but we can give an educated and accurate enough guess to be close to being right most of the time.

Now, if you agree with everything or most of what I've said to this point, then adopting diagonal as a 4th dimension is not that far of a stretch. Because if you always had to go on the x and y axes to move diagonally, EVERY diagonal line should look like a step latter pattern at the smallest possible size of matter. In order to follow the shortest distance between two points is a straight line rule, diagonal has to actually exist as a seperate dimension. Forget that we can make it work visually and mathematically. To me, that is just a very logically told lie of omission. I believe the universe tries to be as efficient in running as possible, and as such, I believe each dimesion was meant to be a true representation of a straight line going in a specific direction. I further believe that in time, if we learn true diagonal propulsion, we will learn that in right triangles, the actual value of a hypotenuese will become less important than the conceptual value. In other words, rather than thinking of the solution of pythagoreans theorum as a^2+b^2=c^2.. a=1, b=1 c= square root of two, it will be a=1, b=1, c=1 because the actual translation will be that you traversed the distance of the width of the cube and height of the cube in order to get to the opposite corner of the cube. The value of c might be factually variable, but it won't matter, because it will probably be thought of in terms of impulse.. And providing impulse to travel one over and one up allows you enough impulse to move one square..from corner to corner.