# Recommendable introduction to multi-dimensional thinking? **URL:** \ **Category:** Cafe Society\ **Created:** [August 19, 2006, 1:59pm UTC](https://boards.straightdope.com/t/recommendable-introduction-to-multi-dimensional-thinking/369339 "2006-08-19T13:59:08Z")\ **Posts on this page:** 12\ **Page:** 1
**Author:** ![Schnitte](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/schnitte/32/9033_2.png) [@Schnitte](https://boards.straightdope.com/u/Schnitte)\ **Post date:** [August 19, 2006, 1:59pm UTC](https://boards.straightdope.com/t/recommendable-introduction-to-multi-dimensional-thinking/369339/1 "2006-08-19T13:59:08Z")
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? ---
**Author:** ![Podkayne](https://avatars.discourse-cdn.com/v4/letter/p/edb3f5/32.png) [@Podkayne](https://boards.straightdope.com/u/Podkayne)\ **Post date:** [August 19, 2006, 2:31pm UTC](https://boards.straightdope.com/t/recommendable-introduction-to-multi-dimensional-thinking/369339/2 "2006-08-19T14:31:37Z")
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.](http://www.amazon.com/gp/product/0062732765/sr=8-1/qid=1155997430/ref=pd_bbs_1/102-2712308-8181714?ie=UTF8) 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. ---
**Author:** ![BrainGlutton](https://avatars.discourse-cdn.com/v4/letter/b/82dd89/32.png) [@BrainGlutton](https://boards.straightdope.com/u/BrainGlutton)\ **Post date:** [August 19, 2006, 2:31pm UTC](https://boards.straightdope.com/t/recommendable-introduction-to-multi-dimensional-thinking/369339/3 "2006-08-19T14:31:40Z")
[Flatland](http://en.wikipedia.org/wiki/Flatland) by Edwin Abott Abott [Sphereland](http://en.wikipedia.org/wiki/Sphereland) by Dionys Burger [The Planiverse](http://en.wikipedia.org/wiki/The_Planiverse) by A.K. Dewdney [The Fourth Dimension](http://en.wikipedia.org/wiki/The_Fourth_Dimension_%28book%29) by Rudy Rucker ---
**Author:** ![Sage\_Rat](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/sage_rat/32/399_2.png) [@Sage\_Rat](https://boards.straightdope.com/u/Sage_Rat)\ **Post date:** [August 19, 2006, 2:57pm UTC](https://boards.straightdope.com/t/recommendable-introduction-to-multi-dimensional-thinking/369339/4 "2006-08-19T14:57:02Z")
> [@Schnitte](#): > > 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? 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. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [August 19, 2006, 3:27pm UTC](https://boards.straightdope.com/t/recommendable-introduction-to-multi-dimensional-thinking/369339/5 "2006-08-19T15:27:27Z")
> [@BrainGlutton](#): > > [Flatland](http://en.wikipedia.org/wiki/Flatland) by Edwin Abott Abott 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. > [@](#): > > [The Fourth Dimension](http://en.wikipedia.org/wiki/The_Fourth_Dimension_%28book%29) by Rudy Rucker 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—”](http://en.wikipedia.org/wiki/And_He_Built_A_Crooked_House). ---
**Author:** ![Sage\_Rat](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/sage_rat/32/399_2.png) [@Sage\_Rat](https://boards.straightdope.com/u/Sage_Rat)\ **Post date:** [August 19, 2006, 3:35pm UTC](https://boards.straightdope.com/t/recommendable-introduction-to-multi-dimensional-thinking/369339/6 "2006-08-19T15:35:01Z")
> [@Sage Rat](#): > > 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.) Doh… Ignore that. _re-kickstarts brain_ 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. Here would be a 3D diagonal win: oxxx xxxx xxxx xxxx xxxx xoxx xxxx xxxx xxxx xxxx xxox xxxx xxxx xxxx xxxx xxxo A different 3D diagonal win: oxxx xxxx xxxx xxxx xxxx oxxx xxxx xxxx xxxx xxxx oxxx xxxx xxxx xxxx xxxx oxxx Here would be a 4D diagonal win: oxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xoxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxox xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxo Good luck… ---
**Author:** ![Foldup\_Rabbit](https://avatars.discourse-cdn.com/v4/letter/f/df788c/32.png) [@Foldup\_Rabbit](https://boards.straightdope.com/u/Foldup_Rabbit)\ **Post date:** [August 19, 2006, 11:50pm UTC](https://boards.straightdope.com/t/recommendable-introduction-to-multi-dimensional-thinking/369339/7 "2006-08-19T23:50:03Z")
[Imagining the Tenth Dimension](http://www.tenthdimension.com/flash2.php) 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. ---
**Author:** ![Terminus\_Est](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/terminus_est/32/3087_2.png) [@Terminus\_Est](https://boards.straightdope.com/u/Terminus_Est)\ **Post date:** [August 20, 2006, 12:41am UTC](https://boards.straightdope.com/t/recommendable-introduction-to-multi-dimensional-thinking/369339/8 "2006-08-20T00:41:06Z")
> [@Sage Rat](#): > > 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. 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. ---
**Author:** ![Joools](https://avatars.discourse-cdn.com/v4/letter/j/6de8d8/32.png) [@Joools](https://boards.straightdope.com/u/Joools)\ **Post date:** [August 20, 2006, 2:55am UTC](https://boards.straightdope.com/t/recommendable-introduction-to-multi-dimensional-thinking/369339/9 "2006-08-20T02:55:33Z")
Rudy Rucker also wrote a novel called [Spaceland](http://www.amazon.com/exec/obidos/ASIN/0765303671/cromagnoncom?dev-t=D2N9QJDNNWGKUE%26camp=2025%26link_code=sp1) 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. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 20, 2006, 2:58pm UTC](https://boards.straightdope.com/t/recommendable-introduction-to-multi-dimensional-thinking/369339/10 "2006-08-20T14:58:51Z")
If you want to have the background to study higher-dimensional geometry, you should look at Brannan, Esplen and Gray’s [Geometry](http://www.amazon.com/gp/product/0521591937/sr=8-1/qid=1156085707/ref=sr_1_1/103-8548434-5931000?ie=UTF8). 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. ---
**Author:** ![BrainGlutton](https://avatars.discourse-cdn.com/v4/letter/b/82dd89/32.png) [@BrainGlutton](https://boards.straightdope.com/u/BrainGlutton)\ **Post date:** [August 20, 2006, 3:51pm UTC](https://boards.straightdope.com/t/recommendable-introduction-to-multi-dimensional-thinking/369339/11 "2006-08-20T15:51:51Z")
Here’s a [four-dimensional variant of chess.](http://www.chessvariants.com/3d.dir/timeline.html) ---
**Author:** ![Schnitte](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/schnitte/32/9033_2.png) [@Schnitte](https://boards.straightdope.com/u/Schnitte)\ **Post date:** [August 21, 2006, 10:05am UTC](https://boards.straightdope.com/t/recommendable-introduction-to-multi-dimensional-thinking/369339/12 "2006-08-21T10:05:37Z")
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!