Into My Hypercube!

So back in the day, there were some cool Dragon Magazine articles about how to use the four-dimensional geometric concept of the tesseract, or hypercube in a dungeon. The first one laid out the basic idea, the second one showed how you could use it to create Escherian gravity effects. There was also a neat Baba Yaga adventure featuring the same idea.

Problem is: it's really hard to explain just the geometry of it, much less the cool things you can do with it in a dungeon. For example, here's someone trying pretty hard:
All that and they don't even get the gravity distortion in there.

Anyway, since I am working on this Alice In Wonderland thing I figure, hey, as long as I'm down here trying to figure out the Doormouse's armor class, might as well figure out how to build four-dimensional dungeons, right?

So ok, this is a tesseract...
Here's an animated GIF of how one surface moves into the next. Don't worry if you don't understand yet.

So, in order to explain to myself and you how this works a little easier, I've just taken half of the thing and made a minidungeon out of it. You enter from normal space through the door where I've written a totally crunk "1": (side cutaway view)
Don't worry, better picture in a second...

What makes this a half-tesseract is I'm missing the cells in the front, back and on the left and one cell formed, I believe, by the faces of all the outer cells functioning as the inner walls of the last cell. Yes, this makes my brain hurt and forgive me if I've got some of the math or terminology details wrong.

But the good news is that using only this chunk of a tesseract we can not only kinda explain how it works but, much more importantly, get all the basic weird dungeon effects you'd possibly want out of this particular piece of math...

So, dungeon, side view, here we go...(click to enlarge and see how filthy my scanner is)

The only important thing not in the picture is: from the point of view of the peeps in any given room, the room looks like a regular cube, not a wonky wannabe trapezoid like it looks in the picture.

As far as Mr Pink, Ms Blue and Dr Green are concerned, the floor in room 1 meets the walls with the doors in them at right angles--and it's the same in every other room. Basically, each time they go through a door, they're unknowingly shifting through nonEuclidean space. Or something like that.

Also helpful to remember: people fall in the direction of their feet. So for example if Blue opened one of those doors on the floor in room 3 and went through, Blue would wall toward that vertical wall on the bottom left (the surface parallel to and opposite the door he just opened, since the room is "straightened out" from his point of view).

Now, chances are your players won't map this all out or need to (or will kill you if you make them) but having this little picture does make it easier for the GM running an adventure to track who should be sticking to which wall in which room. Just go: "Ok, did this gnoll follow Mr Pink's path or Dr Green's path?"

Anyway, point is you can kill your players on the ceiling this way, so that's cool.

In case you can't tell, I'm pretty excited about this Alice thing.