I've recently been watching a series of lectures on DVD (borrowed from Pa) called The Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas. It's pretty well done, and I've been enjoying it. The lectures on the fourth dimension got me thinking, of course, about magic.
Most of what they covered was pretty much what you'd get from reading Flatland, which is worth doing if you haven't already. In particular, think about a two-dimensional world where a two-dimensional being is looking at a square. Maybe the square is his safe, and he thinks his money is securely locked up there. But we, as three-dimensional beings, can simply reach inside the square and remove the money without the two-dimensional being seeing anything. The disappearance of his money would seem magical from his perspective. Similarly, a four-dimensional being could perform analogous feats that seem impossible from our three-dimensional perspective.
Now, something else mentioned in the lecture was that any attribute can technically be called a "dimension" (it's just fun and tricky to try to think about an extra dimension of space). For example, you could call color your fourth dimension, and then use x, y, and z coordinates plus a color to uniquely identify any point in that particular view of the universe. That mention of color reminded me of something I learned way back in one of my only two linguistics classes, and that is how people's perception of color isn't a fixed, physical thing. People from cultures with fewer words for color actually perceive fewer colors than people with more words for them. (This was tested by finding some obscure jungle tribes somewhere, with only three or so color words, and asking them to tell whether certain swatches of colors were the same or different. Many colors or shades that we would perceive as different actually looked the same to them.) I think this is interesting because it's a mental limitation (or enhanced ability if you take it the other way), and therefore something you can learn, as opposed to something physical, like color blindness.
So connecting these two ideas of perception and dimensions, makes me wonder if there are other dimensions we can't perceive now but can learn to. A fourth dimension of space would be a good start, but other unusual attributes would be intriguing as well. Taking it to an extreme, perhaps, what if you could consider "reality" a dimension? Would someone who learns to perceive that dimension be able to see alternate universes?
Of course, simply perceiving a dimension or attribute doesn't necessarily mean you have power to change it. Just because we can perceive colors doesn't mean they change at our will, and just because we might learn to perceive four-dimensional space doesn't mean we could actually do anything in it. On the other hand, we don't have complete control over our standard dimensions either, but we get by alright. We can't change the location of a large building in space, but we can move our own bodies to a reasonable degree, and use them to move smaller objects. And if we have paint at our disposal, we can change the colors of things. So it's probably worth just figuring out what else we can perceive and then working from there.