What is the idea behind multiple dimensions?

In another post, you said space is flat like a piece of paper. The thing that really gets me is the concept of different dimensions. What are these dimensions, where to they exist and what is the theory of what exists within them? Just thinking about it twists my mind!
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It’s definitely a mind bending thing! Have the mindbendingest .gif I can find. Let’s first clarify what the geometry of space actually means; what do we mean when we say that space is flat? 

Looking around ourselves, we can see that we live a universe of 3 dimensional objects. If you happen to be inside right now, we can make a basic grid out of the walls and the floor. We’ve got up/down, right/left, and front/back as our dimensions to move around in. We also move through time, since it takes us some time to change our position. However, if we’re just interested in describing an object that isn’t moving or changing, we can stick with 3 dimensions. 3 dimensional distortions in space are almost always hard to illustrate, so usually we have to simplify down to a 2 dimensional plane. If space is intrinsically flat, this means that our 2 dimensional plane has no bends or warping to it. If, for instance, space were so curved that it was shaped like a cylinder, I could walk in a straight line around the cylinder and wind up exactly where I started. We’re pretty sure that our universe doesn’t work like this; if I walk for a long time in a straight line, I’m just really far away from where I started.

The theory that suggests we should be in an 11 dimensional world (or more!) is string theory, and string theory really didn’t start out with the extra dimensions in mind. It was attempting to explain the difference between the way that quantum mechanics describes the world, and the way that general relativity describes the world. Quantum mechanics works phenomenally well to explain extremely small objects: the way that single electrons behave, for instance. General relativity, by contrast, does extraordinarily well at explaining large scale objects: the way that the solar system holds together, or the way that galaxies interact. But even though both of them are excellent descriptions of the universe at their own scales, if you try and scale up quantum mechanics, or scale down general relativity, you get complete nonsense. String theory was trying to tie these two together by coming up with a mathematical language that both the tiny and the large could speak, and in doing so found that they needed a pile of extra dimensions to make it work.

However, we still observe that we live in a 3D universe (+time), so if there are extra dimensions as the math of string theory suggests, they can’t interfere with the 4 that we observe. The solution was that perhaps the extra dimensions are curled up like a little MC Escher house at every location in our 3D space (see the .gif above). These would be so tightly curled up that for the purposes of any observation we’d make, the extra dimensions would appear to be effectively nonexistent. This allows an 11 dimensional space to appear 4 dimensional just because the other seven are wound very tightly and very small. Because they’re so small, there shouldn’t be much of anything hanging out in that dimension, but the idea is that the ‘strings’ that string theory proposes can live in (or at least wiggle in) those dimensions.

In case this wasn’t mind-bending enough already, the dominant flavor of string theory (called M-theory) also predicts that this 11 dimensional universe could be thought of as many-dimensional sheet (or membrane), and there could be a large number of other universes on other membranes with totally different laws of physics. However, the existence of these extra membranes should not affect our universe’s membrane, except possibly to help explain why gravity is a relatively weak force of nature. If every membrane has different set of physical constants, some of the universes they contain would be very exotic to our eyes. The existence of the objects within our universe is finely balanced on the constants of physics we have in our universe.

However, I should emphasize a very critical point. So far, string theory is just a mathematical possibility. All these extra dimensions and membranes are the physical interpretations of the equations that string theorists have developed. So far, they are completely untested and untestable, since the sizes of the objects uniquely predicted by string theory are enormously smaller than the smallest objects we can currently detect. String theory may turn out to be false, in which case these mathematically mandatory extra dimensions that string theory needed to have in order to function may no longer be necessary in a new theory of how the universe works.

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