I’d like to mention first that I don’t study any science. I used to be quite good at physics and mathematics, including vector mathematics, at school/college, but I didn’t pursue any studies in this direction on university level.
I’d be interested to know if any dopers can recommend books that introduce the reader to multi-dimensional thinking. I read a variety of popular science books that addressed this topic as a part of their discussion of relativity, but I’d like to go a bit deeper into that specific aspect. I know that it’s not really possible for humans to visualize an object of more than three dimensions, but there are approaches to help you understand better which properties multidimensional objects have. Can anyone recommend introductory literature here?
The idea, incidentally, came up when a few friends and I started playing Connect Four as a pastime. We had the idea of drawing several grids, each representing a different “storey,” to produce a “three-dimensional” Connect Four grid. Putting several of those three-dimensional grids next to each other ought to result in a “four-dimensional” game, right?
There are some SF books I really enjoyed on this subject.
Flatland, by Edwin Abbott, together with its much later sequel Sphereland, by Dionys Burger, should be required reading for any student of the subject, IMHO. They are available in one volume. I had many “AHA!” moments while reading Sphereland. It definitely gave me a conceptually stronger grasp of GR than I could get from my textbooks!
Less rigorous but still fun is The Boy Who Reversed Himself, by William Sleator. It’s not deep, but it was written for teenaged readers, so it’s a quick read.
Probably easiest would be to have two 2D 16X16 squares. You could then say that the one on the left is x and y, and the one on the right is w and z (or whatever.)
I’m not sure of the rules of Connect Four, so I’m not sure if/how you could relate that back into something understandable/playable.
A classic, that introduces the idea of a fourth dimension by looking at how our three-dimensional world might appear to someone who only exists in two dimensions—mixed in with social satire of Victorian England, and stuff like that. Frankly, I found some parts kind of tedious. If you do read it, it may be worth tracking down the Annotated Flatland, with annotations by Ian Stewart.
Though I haven’t read it, my previous experience with Rucker suggests this might be a good place to start.
I also second the recommendation of The Boy Who Reversed Himself. You’d also probably enjoy the Robart A. Heinlein short story “—And He Built a Crooked House—”.
If you had four 4X4 grids laid out side by side, you could say each one was a layer in a 3D game. If you then copied that down so you had four rows of four 4X4 grids, you would have a 4D game.
Looking that over to see if you had a straight line of four anywhere (including diagonals) would be a good pain the ass.
Imagining the Tenth Dimension is an interesting little flash tutorial that walks you through easy visualizations of ten dimensions. Since I am pretty inept at advanced physics, I have no idea how accurate the ideas presented are. But it’s still an interesting thing to look at.
I’ve played exactly this game. You really have to watch the board since your opponent can mark a square in some seemingly random location that’s actually part of a winning row.
Rudy Rucker also wrote a novel called Spaceland that’s about a character who comes into contact with creatures who live in a four-dimensional world. Obviously, from the title, it’s partly a tribute to Flatland. And it does a pretty good job of making understandable the idea of four dimensions.
If you want to have the background to study higher-dimensional geometry, you should look at Brannan, Esplen and Gray’s Geometry. It doesn’t specifically cover higher-dimensional geometry, but the approach it presents–a geometry is a space and a set of mappings from that space to itself–is the fundamental approach that higher-dimensional geometry uses. It’s a very accessible text if you have some familiarity with vector/matrix algebra and group theory (and there are appendices that give quick overviews of those topics, but it’d be nice to have a more detailed reference). Also, see if you can find a paperback edition, as it’s about half the price of the hardback linked above.
Thanks to everybody here for their input regarding books, and thanks also for the dopers referring to four-dimensional games. Flatland was referred to in a book by Martin Gardner I read years ago, but I never bothered to get the novel. Maybe I’ll do it; I’ll certainly buy and read some of the non-fiction books recommended here. Again, thanks guys!