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Simple. These dimensions exist just like the three that you are familiar with. But, we just can't observe them as they are insignificant on a macroscopic scale.

Get yourself down to the quantum scale, and they become a whole lot more apparent.

The interesting thing is, to view space in 11 dimensions you would still only need two eyes, but 10-dimensional retinae.

The trick to visualising it is to think of 3-dimensional objects passing through 2-dimensional space, and scale up from there. If you imagine that you are a 2-d being, living in a 2-d universe, you would see everyhing as a line. You could walk around a square and know it was a square, but you could not observe the whole thing at once.

Then, if a 3-d being from a 3-d universe were to pass a sphere in front of you, you would see a circle in 2-d. If he were to push that sphere all the way through, you would see a small circle appear from nowhere, grow bigger, then start shrinking, before finally disappearing.

If a 4-d being were to push a 4-d hypersphere through our universe, you would observe a 3-d sphere appear and then disappear. There would be a 4th dimension that you could not see. The sphere may appear to float, but in reality is being suspended in the 4th dimension.

Scale that up to 11 dimensions and you can visualise it, or at least appreciate that the dimensions exist.
 
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