After my diatribe a couple of days ago about the misuse of the word dimension, I got into a discussion with a friend that can be summed up as, "Okay, then how are we supposed to picture spaces with more than three dimensions?"
Well, the simple answer is that we can't. Our brains are equipped to manage pictorial representations of three dimensions or fewer. We can try to get a handle on it via analogy -- a particularly masterful example is Edwin Abbott's Flatland: A Romance of Many Dimensions, which considers a two-dimensional character named A. Square, who has as hard a time picturing a third dimension as we do a fourth. When a three-dimensional sphere passes through Flatland, A. Square perceives it as a series of successive two-dimensional slices -- a circle that appears out of nowhere, grows larger, then shrinks and finally vanishes. The implication is that if a four-dimensional object -- a hypersphere, perhaps -- were to pass through our three-dimensional world, we'd see something similar; a projection of successive "slices," a sphere popping into existence, expanding, then contracting and vanishing.
But the fact remains that these are ways of thinking about a concept that is, honestly, beyond our ken. It's the problem that plagues many of the deep models of physics -- something that can be described clearly and accurately by the math is nevertheless impossible to visualize. It's a bit like the situation with quantum mechanics; the math is astonishingly precise and makes spot-on predictions, but if you ask most physicists, "So what physical reality is the math describing?" the answer you'll get is a slightly embarrassed "we don't know." (If they don't say "Shut up and calculate.")
It's a serious sticking point with people like myself, who understand best when we can picture what's going on. It was when I hit that spot in my undergraduate studies -- when the professor said, basically, "The math is what's real, here, don't bother trying to visualize it because you can't" -- that I decided that a career in physics was not in the cards for me.
Despite that, I have continued to be intrigued with notions like quantum indeterminacy and higher-dimensional space, even though when I read about them I often have an expression on my face like the one my puppy has when I explain a complex concept that is beyond his comprehension, such as why he shouldn't eat the sofa. I'm currently reading a wonderful book about the topic of extra dimensions, by the brilliant theoretical physicist Lisa Randall, called Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions, which does an outstanding job of bringing the topic down to a level we eager-but-not-so-bright puppies can understand. (And if you want more, she has an appendix with mathematical notes elucidating the topic in a deeper and more precise fashion.)
One of the more fascinating topics she goes into is the concept of a brane -- a cross-section of a higher-dimensional space a bit like A. Square's expanding-and-contracting circles. The name comes from the word membrane, because (like a cell membrane) a two-dimensional brane can be a boundary on a three-dimensional space. The surface of the Earth's ocean, for example, can be seen as a two-dimensional brane (not only acting as a boundary, but oscillating up and down into the three-dimensional space on either side).
Of course, you're not limited to two-dimensional branes in three-dimensional space. A generalized name for branes in p dimensions is called a p-brane, which was one of my father's favorite insults (albeit spelled differently).
Where it becomes more interesting, and unfortunately far harder to picture, is when you consider the idea from some physicists -- Randall has been one of the lead researchers in this field -- that our own three-dimensional universe is a three-brane within a higher-dimensional space. There is a tantalizing suggestion that this model may explain some of physics's most persistent mysteries, such as why the gravitational force is so weak compared to the other three. If we are actually living in a three-dimensional slice, the gravitational force within our bit of space may leak across into the higher dimensions, weakening its intensity and perhaps influencing other branes within the space (which might give physicists a way of finding evidence for the conjecture).
There's even the suggestion that the Big Bang may have occurred because of collision between two three-branes in a multi-dimensional hyperspace -- a model called ekpyrotic cosmology.
But we're still up against the problem that it's impossible to answer the question, "But what does it actually look like?" The mathematics is crisp and clear; any picture we come up with is, by comparison, incomplete and inaccurate. Take, for example, a hypercube, a symmetrical four-dimensional structure that can be described mathematically but is impossible to visualize. All we can do is consider what projections of it -- shadows, so to speak -- look like in three dimensions. Here's a particularly mesmerizing projection of a rotating hypercube:
So we're kind of ending where we started. All of this is just a teaser, really -- a brief excursion into a subject that is just now being investigated by some of the most brilliant minds on the planet. If the mathematics of branes and higher dimensions and whatnot is beyond you -- it certain is me -- we're left with trying to get a faint glimmer of understanding via analogy. Which only gets you so far.
But at least it gives us something our branes -- um, brains -- can handle.
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