Jonathan Shock has been working with the SciTalks people to make their site a repository of high-quality, informative physics material. At the SciTalks blag, Dr. Shock provides an introduction to the idea of “extra dimensions.” These aren’t woo-tastic flights of fancy like some “dimensions of the spirit,” but rather ideas we can explore with mathematical rigor in order to understand both their properties as abstract concepts and, perhaps, some features of the physical world.
We’re treated to a description of how to build a four-dimensional hypercube by shifting a cube at right angles to itself, along a fourth direction perpendicular to the cube’s three axes. That’s a hard idea to fit into a brain! However, by projecting the process down — essentially “casting shadows” of the higher-dimensional shapes — we can make a video of it.
As always, the math matters:
One thing I should mention first is that trying to visualise objects in higher dimensions is good to feel comfortable with what you’re studying. However, sometimes if we try and visualise the things that we are calculating, we may stifle, or bias our calculations. In my opinion it is usually best to become familiar with the mathematics, the purest descriptions of our theory and try to ‘visualise’ in this language rather than trying to use common sense from what we see around us. A good example is quantum mechanics. If we try and visualise what is going on on very small length scales we will quickly tie ourselves in knots and not be able to progress as far as we can by exploring the mathematics of our system. Of course it’s important to be able to translate the mathematics back into what you will actually see in your experiment.
Dr. Shock — sorry, but I can’t get over how awesome a scientist name that is! — also explains a little about the “holographic” aspects of string theory. He points to a “superb video” about this, but it’s a RealPlayer file which I can’t watch on this computer. Curse you, RealMedia!
Incidentally, Ben Goldacre and the Factician point out that Richard Dawkins’ latest show, The Enemies of Reason is now available ontube.
Yeah, I agree with what he says about the math. Personally, I find it incredibly easy to deal with the math for multiple dimensions. (Conceptually, that is. Actually solving the systems can get hairy really quickly.)
Oh, I was going to continue by saying that attempting to imagine these high-d objects as physical things is another matter entirely. Even the projections barely make sense to me. It’s like I know what they’re supposed to represent, but something fails along the way.
I’d actually say that geometric reasoning only has an intuitive edge over symbolic reasoning when the objects you’re visualizing correspond to real world experience. I doubt there is a “common sense” to visualize an infinite dimensional vector space, for instance. Better to approach it through the raw symbolic reasoning behind the concepts and then prove what a geometric pimp you are by coming up with some way to visualize it.