Physicists and mathematicians who think about higher-dimensional spaces are, if they allow their interest to somehow become public knowledge, inevitably asked: "How can you visualize more than three dimensions of space?" There are at least three correct answers: (1) You can't. (2) You don't have to; manipulating abstract symbols is enough to help you figure things out. (3) There are tricks to help you pseudo-visualize higher-dimensional objects by cleverly projecting them into three dimensions; see here and here.

But really, why can't we visualize things in more than three dimensions of space? Could a Flatlander, living in a world with only two spatial dimensions, learn to visualize our three-dimensional world? Could we somehow, through practice or direct intervention in the brain, train ourselves to truly visualize more dimensions?

I can think of a couple of explanations why it's so hard, with different ramifications. One would be simply that our imaginations aren't good enough to project our consciousness into a constructed world so very different from our own. Could you, for example, really imagine what it's like to live in two dimensions? Sure, you can visualize Flatland from the outside, but what about asking what it's like to really be a Flatlander? The best I can do is to imagine a line, flickering with colors, surrounded by darkness on either side. But the darkness is still there, in my imagination.

The other possible explanation is that the process of visualization takes up a three-dimensional space in our actual brain, preventing us from "tuning a dimensionality knob" on our imaginations. The truth is certainly more complicated than that (and I'm not experts, so anyone who is should chime in); the visual cortex itself is effectively two-dimensional, but somehow our brain reconstructs a three-dimensional image of the space around us.

Maybe this could be a new tantric discipline: visualization in higher dimensions. Or maybe the Maharishi already offers a course?