Here's some more cool stuff about dimensions. It's about real dimensions that are not really real (it's funny how the word real has different meanings in English).
Usually when we think about dimensions, they are integer numbers, like 3 for our regular space, and 4 if we include time to make it space-time.
In physics we sometimes have to compute infinitely long sums (integrals if you know calculus). Sometimes these sums in 4-dimensional space are infinite. That's not what we want.
We can fix this by making the dimension slightly less than 4, for instance 3.99, which is a real number (and not an integer). The infinite sums now splits nicely in a finite term, which is the answer we seek, and an infinite term that we trash.
Then, in the end, we let the dimension go almost back to 4, say 3.999999999999.
It's magic, isn't it? And what does it mean? It's just a math trick that let us do computations, nothing more and nothing less. The dimension of our real space-time is still 4 >:)
(The trick outlined above is called dimensional regularization. It was invented by the physicists t'Hooft and Veltman who won the Nobel Prize in 1999. I have no idea how to picture a 3.99 dimensional space, and there's no need to do it as long as the math works. The picture above is a 4-dimensional hypercube from Wikipedia)
I wish I'd had somebody give cool applications to math when I took it. I went through calculus, but nobody ever backed up the stupid process with what you could theorize when you did it all... I think I might have stuck with it better (oh, I know... I do statistics, now--but i didn't give in to that, even, until grad school... (where I finally had applications that mattered to me)
ReplyDeleteOkay, I tried to read and follow this post, but my head started whirling...wayyyy earlier than it probably should have. Wish my brain did numbers, but it's not that kind of brain model, apparently. :)
ReplyDeleteYup. Way over my head. I was with you until "we usually think about dimensions..." then I started drifting off on how "we" must mean you and your smarty pants physics buds. Not "we" as in me. Because me and dimensions start day dreaming about alternate dimensions. And then I zoned back in in time to see you were doing something with math and all hope was lost.
ReplyDeleteYeah, well, it seems like I heard that having a heliocentric solar system was only introduced because it made the math easier. Look what that led to.
ReplyDeleteHelp me! Help me! My head's exploding. And that, my friends, is a 4-dimensional hyperbole.
ReplyDeleteUmmm...what? My husband was a physics guy in college, too, and I still don't get this. You're too smart for me!
ReplyDeleteMichele
SouthernCityMysteries
People are different. Some like math and physics, some like language and writing, and some of you guys are very good writers.
ReplyDeleteI enjoyed math and physics ever since high-school days, and today I'm happy to have physics as my daytime job. I also like literature and writing. Probably it means I've split my limited talent on too many different things. I won't become really good neither as physicist nor writer.
The main message in this post and the previous one, is that you shouldn't take the extra dimensions that show up in physics too literary. The extra dimensions are introduced when necessary, to make theories consistent (all physics theories have to obey certain fundamental principles, like conservation of energy).
The dimensions beyond our space-time are not accessible to us, neither for telepathy nor angels and gods nor communication with the dead >:)