Extra Dimensions in String Theory explained

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You can define any point in space relative to some fixed coordinate system using three distances. This makes it a three-dimensional world. If you follow Einstein and wish to employ the mathematical convenience of talking about 'space-time' then you need to add one time measurement. This makes three or four dimensions depending on what you are trying to measure. You can choose to measure space with things other than distances - but no matter what, you always need just three numbers...so for example, you can measure every point in space using two angles and one distance ('spherical polar coordinates') or one angle and two distances ('cylindrical polar'). In space/time, you always need four numbers. The exact formulation doesn't matter - the dimensionality of space (or space/time) doesn't change depending on how you measure it.

The extra dimensions that string theory predicts are claimed to be 'very small'. That's an odd thing to say because the three dimensions we know and love seem to be vast. Understanding what "a small dimension" means is tricky - we have to take it in small steps:

  • Suppose for a moment that we were observing some two-dimensional creatures - living on the surface of a flat piece of paper. In our present world view, the paper is flat and infinitely large. There is no 'up/down' dimension for them because they are 2D creatures - they only have left/right and forwards/backwards.
  • But suppose one of those two spatial dimension (let's pick the left/right dimension) was 'small' - just a 10 miles across say. The universe can't have 'edges' - it has to 'wrap around'. By this, I mean that moving in the left/right dimension for exactly 10 miles would take you all the way around that dimension and back to where you started - for a 2D creature this would be a bit strange - but for us 3D creatures watching them, it would be like they were living on the surface of an infinitely long cylinder of paper that's just one mile in diameter. They could move as far as they wanted along the length of the cylinder - but if they moved a long distance in the other direction, they'd go all around the cylinder and back to the start. Because their 2D light beams are stuck in the 2D surface, if they looked off to the left or right using a pair of decent binoculars, they'd be able to see themselves 10 miles away.
  • In a three dimensional universe like ours, if our up/down dimension was only 10 miles across then you'd be able to travel as far as you wanted left/right or forwards/backwards - but if you moved upwards by 10 miles (or downwards by the same amount), you'd be back where you started. Also, if you were out in space and looked up using a pair of binoculars, you'd be able to see your own feet, just 10 miles away. Looking left or right or forwards or backwards - and everything looks kinda normal.
  • Now - imagine that third dimension isn't 10 miles across - but just one millimeter across. We would be almost like 2D beings - almost all of our existance would be in two dimensions since nothing in the universe could be more than a millimeter in height - and moving up or down would have almost no effect on your life. That third dimension exists - but it's hardly any use at all. We would have to be almost perfectly flat creatures - it would be ALMOST a 2D world...but not quite.
  • Now imagine that instead of the up/down dimension being a millimeter across, it's much MUCH smaller than the diameter of an atom...in that case we'd have no way to know that there even was a third dimension - it would seem exactly like being in a flat, 2D world since any motion at all in the 3rd dimension would have no effect and no object could be as tall as even an atom...atoms themselves would have to be almost exactly 2D objects. We wouldn't even know that the up/down direction existed at all. It the third dimension were that small, we might as well be living in a 2D world for all that it would matter to us.
  • OK - so back to a normal 3D world. What would happen if there were a 4th dimension? Well - we can't see it, measure it...it's not in any way detectable...so we might jump to the conclusion that there isn't one. But if the 4th dimension existed but was very small (much less than the diameter of an atom) - then it could very well be there but we'd be totally unaware of it...unable to detect it. It would SEEM like we were living in a 3D world.

The string theorists claim that there are DOZENS of extra dimensions beyond the three we can normally experience - but all but the first three are so small that we can't tell that they are there - even with the most sophisticated equipment we have. I've heard these extra dimensions described as being 'rolled up'. They might very well be correct - but we have no way to know.