However, in order for the equations to work out requires more than three dimensions for the strings to vibrate in. Some theories have as many as 11 dimensions to allow all the required degrees of freedom for the strings to produce the essential characteristics. Yet our experience is limited to only three spatial dimensions. (We aren’t counting the time dimension of relativity, just spatial dimensions like up/down, left/right, and in/out.)
Although mathematics has no problem dealing with higher dimensions and the math has been worked out for years to calculate the volume of, say, a six dimensional sphere or 6-ball. But, if there are more than three dimensions, why aren’t we directly aware of them?
One answer is that they are all rolled up in that same tiny area that these strings operate. It is called “Compactification.” Although some very famous and smart scientists such as my hero, Richard Feynman and Roger Penrose have rejected String Theory, largely due to the complete lack of experimental confirmation, others such as Stephen Hawking and my professor, Leonard Susskind, believe String Theory to be an elegant solution to the merging of quantum physics and relativity and a route to unification of the four forces of the universe including gravity which is not included in modern quantum mechanics.
So think back to your high school math classes, especially geometry. Remember that you were taught that a “point” is an object with no dimension and a “line” is an object with only one dimension. Basically a line was what you got when you moved a point. Now imagine a straight line stretched between two points. It has only one dimension: left and right, or East and West, or x and minus x. You can only move in two directions on this line. Sort of like a tight rope stretched between two poles. You can only move East or West, assuming that’s the direction of the line. You can’t move North or South without falling off the line.
From a distance the rope looks like a simple line, but you know that up close it has three dimensions. It is probably like a circle in the N/S and up/down dimensions. Now imagine that, instead of the tight rope walker, you were a tiny ant on the rope. Of course you could go East or West down the rope, but you could also turn ninety degrees and walk North or South. You would wrap around the diameter of the rope and come back to where you started. Fortunately, as an ant, you can walk upside down and make that journey.
Well that’s what Compactification is. It may appear to us in our universe of miles and inches and even tiny areas like the size of an atom or the size of a proton or neutron that the line (tight rope) is one dimensional, but at the tiny, tiny size scale of strings, like the ant, we know the line has other dimensions. Of course, my analogy sticks to three dimensions since that’s all our brains can normally imagine, you go by analogy showing how one dimension can expand to three if you look close enough or magnify it enough or you’re small enough like the ant on the tight rope. So it’s the same idea to reach 6 or 9 or 11 dimensions. We can’t sense them because they are just so small or “compact.”
Don't worry that the ant travels around the rope and comes back to the same place. In theory, if the rope was stretched clear to infinity on both ends, it, too, would just wrap around. Sort of like the surface of the earth. It is a three dimensional sphere, but we experience it as a two dimensional plane. If you travel North or South, East or West, if you go long enough or far enough you'll wrap back around to the same place you started. Most physicist believe the three dimensional universe is the same. Travel far enough in a straight line and you end up back where you started. Mind boggling … isn't it?
That's the essence of String Theory and Supersymmetry. The extra dimensions allow extra modes of vibration that we sense at our higher size as physical properties such as mass or charge or momentum. Although the math is crazy difficult, it is still doable. Getting your mind to imagine more than three dimensions … that is a much tougher issue. Maybe you don’t need to have a mental image as long as the math works. Frankly, I struggle with both.