Think in 4 dimensions
Sunday, August 31, 2008
There is something that really bugs me about our human perception of space, which I consider one of many of my mathematical deficiencies: I don't understand why I can't think in, or even visualize, 4 dimensional space. And it's not just me; we, humans, are not wired to to do this.
Sure, we live in a 4 dimensional world (width, height, length, and time), but I mean 4 dimensions in the Euclidean sense.
I was taught to manipulate objects in countless dimensions (vector spaces), but I can't really see beyond 3 dimensions--i.e., we can't draw things on a piece paper that are accurate representation of 4 dimensional objects. Some people claim to be able to visualize 4 dimensions, but how can we prove they can't?
By analogy, we can project the shadows of a 4 dimensional object into to our 3 dimensional world. What's more, we can project these things into 2 dimensional screens (see the nice bubble looking image above?). I buy that and I have seen the "projections," but I just can't visualize real 4 dimensional objects. And I've tried very hard.
The latest step I've taken in my quest to understand higher dimensions is to watch a freely available movie titled, unsurprisingly, Dimensions
If you watch the movie and end up understanding 4 dimensions, drop me a line.
I can't say that I can visualize four (or more) dimensions, but after studying math at the university I started to "get a feeling" of how certain properties generalize to many dimensions. For instance that two planes can be orthogonal in four dimenensions. Figuring out how such properties generalize isn't very hard.
Then I started to "visualize" by simply combining what I know about those properties. And after a while it felt like I really understood how (simple) problems worked in many dimenstions just I understand how (simple and hard) problems work in one, two, and three dimensions.
I don't claim that this is visualize, though. In fact, I haven't thought about problems like this in many years to I totally have lost the knack. :)
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