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Multiheaded comments on Rationality Quotes April 2012 - Less Wrong
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Try visualizing four spacial dimensions.
When I was 13 or so, my brains worked significantly better than they currently do, and I figured out an easy trick for that in a math class one day. Just assign a greyscale color value (from black to white) to each point! This is exactly like taking an usual map and coloring the hills a lighter shade and the low places a darker one.
The only problem with that is it's still "3.5D", like the "2.5D" graphics engine of Doom, where there's only one Z-value to any point in the world so things can't be exactly above or below each other.
To overcome this, you could theoretically imagine the 3D structure alternating between "levels" in the 4th dimension every second, so e.g. one second a 3D cube's left half is grey and its right half is white, indicating a surface "rising" in the 4th dimension, but every other second the right half changes to black while the left is still grey, showing a second surface which begins at the same place and "descends" in the 4th dimension. Voila, you have two 3D "surfaces" meeting at a 4D angle!
With RGB color instead of greyscale, one could theoretically visualize 6 dimensions in such a way.
Now, if only this let you rotate things through the 4th dimension.
Doing specific rotations by breaking it into steps is possible. Rotations by 90 degrees through the higher dimensions is doable with some effort -- it's just coordinate swapping after all. You can make checks that you got it right. Once you have this mastered, you can compose it with rotations that don't touch the higher dimensions. Then compose again with one of these 90 degree rotations, and you have an effective rotation through the higher dimensions.
(Understanding the commutation relations for rotation helps in this breakdown, of course. If you can then go on to understanding how the infinitesimal rotations work, you've got the whole thing down.)