I first found the idea of a hypercube in the Dragon Compendium. This article expands on that article’s both a fascinating and confusing idea. Here I will describe a hypercube and try to make a cheat sheet that lets you use a hypercube with out the headaches.
What Is a Hypercube?
It's a mathematical concept that doesn't work in a 3-D environment. A normal cube can be made of nine small uniform cubes. A hypercube, however, is made from only eight identical cubes? Fitting only eight cubes in a space meant for nine requires a warping of space. Removing the extra cube changes how you move through the area and makes it very complicated to use in game.
How to Draw a Hypercube
Draw a big cube with a smaller cube inside, with all plains facing the same directions. Connecting the corners of the two cubes forms a diagram of a hypercube. You should be able to see seven distinct areas within the cube. Despite the trapezoid shapes all represent an identical cube. Where is the eighth cube? If you are like me you won't see the eighth cube just looking at the diagram. The eighth cube is the outside of the diagram. It forms another identical cube. The parts of a hypercube as far as I can tell are the inner cube, the outer cubes and the outside cube. I'm sure these have proper scientific names, but I don't know them and using them will not be necessary to use a hypercube in a D&D game.
Gravity in a Hypercube
Within the structure of a hypercube gravity becomes subjective. The "floor" that you are standing on is down. As you travel between the cubes down changes allowing you to walk on the wall and ceiling meaning that each of the 8 cubes has 6 surfaces that you can walk on for a total of 48 surfaces.
The Outer Cubes
The outer cubes make an interesting way to travel. When walking (or climbing) from one of the outer cubes to another you reoriented so down is a different way. If you are walking between them then you stay on the same plain (see the diagram), but if you climb the plain you are on increases by 1, but skips 1 and 6 when it rolls over. When you descend in number reduce, but also skip 1 and 6.
Diagrams of the Outer Rooms
I based the diagram up d6’s, but all the number do not match up when up into this form. I have made the changes, but it you build your own 3-D model of d6’s you will need to renumber some sides. These diagrams can also be a guild for building individual rooms based upon the cubes.
The sequel, "Cube 2: Hypercube" dealt with a tesseract, though it was rather cheesily done.
Tesseracts have been a part of D&D for years, though they're rarely seen. Since they make the head of Game Masters spin as much as they do those of the players, this may explain the reason for their scarcity.
:P
http://en.wikipedia.org/wiki/Hypercube
It may help.
A little.
Inner: c = center cube
Middle: n, s, e, w, t, b = north, south, east, west, top, and bottom cubes
Outer: o = outer cube (space surrounding the other seven cubes)
Let's look at the relationships to the center cube only:
I'll enjoy using this one, and its given me some great ideas.
-Mage
First of all, let me try to make a diagramme:
_________
|\....f......./|
|.\______/.|
|.|.........|.|
|b....g....|e|
|.|______|.|
|/___d___\|
Okay so this is pretty rough but it's a good one-dimensional view from any side of the hypercube. G is the central cube, surrounded by B,D,E and F. Even though they all look like different shapes on the outside, inside they are perfect cubes. In addition, we are looking _through_ Cube A to look at my diagramme, and Cube C is on the far side where we can't see it. So far so good?
This is where I get foggy. Where is cube H? You said it is the outside of the cube. To me this means one of two things:
1) Cube H is a "negative" cube; is it the empty space surrounding the cube. I hope that's not what you mean. When I am standing outside my house, I don't call it "my other house" or "my negative house". I call it "the driveway".
2) Cube H is the "perimeter" cube. In other words, just as cubes A-F look trapezoidal (but are really cubic), Cube H looks bigger than the other cubes but, on the inside, is really the same size. If this is what you mean, then Cubes A-F appear to fit inside of Cube H in our diagramme. This is what you mean, right?
I have more thoughts and questions but I want to make sure I have this part right first.
It reminds me of a SF story I once read where a man built a home that was, basically, an unfolded hypercube. That is, it looked sorta like your diagrams but in 3d: one cube at ground level, a "+" shape of cubes on the second floor, then two more floors of a single cube above that.
In the story, a minor earthquake causes the house to ... change.
As one person in the story reasons it, the house was a stable 3d shape, but unstable in 4d, so when it got shaken it collapsed through the fourth dimension. Net result: the outside it looked like just the first floor, a single cube. But when you went in, you found the whole house of 8 cubes was there. You also found you couldn't get out, as the windows now connected to each other and the door you just entered through leads to the attic.
In the story, they did discover a way to get out. If you walked between two rooms through one of the "new" junctions, like by going out a window on the second floor, and you pointedly did not look where you were going, you would "fall" out of the house.
Which, scientifically, ranks up there with Douglas Adams' notion that flying was the art of throwing yourself at the ground and missing, so a sudden distraction while tripping might accidentally lead to flight. But, as D&D is more about fantasy than reality, I thought people might like it as an alternative idea.
Gravity in a Hypercube
Within the structure of a hypercube gravity becomes subjective. The "floor" that you are standing on is down. As you travel between the cubes down changes allowing you to walk on the wall and ceiling meaning that each of the 8 cubes has 6 surfaces that you can walk on for a total of 48 surfaces.
This isn't really necessary. In fact, you can probably create an interesting experience for your players if each of the 8 cubes acted like a normal room, with gravity in one direction, definite compass directions like I assumed in my topology diagram, and ladders or stairways connecting floor and ceiling openings.
If the party decides to go through the west door of each room, they would find themselves in the same room they started in. Even more dramatic, if a player falls through the hole in the floor, after a little while the same player will fall through the ceiling of the same room, and keep falling forever unless someone catches the poor fellow.
I can imagine such a hypercube being the home of a powerful hermit wizard, who enters and exits by way of a portal. Each room could have a specific function: bedroom, privy, bath, kitchen, laboratory, dining room, storage room, guest receiving room -- all appropriately trapped and alarmed against intruders, with a well-hidden exit portal, and a harrowing encounter with the owner intent on defending his home. -A
Thanks, that helps. But how do you draw the cube made of nine small uniform cubes? I can't seem to visualize it.
Sorry about that my visualization of a cube was wrong. It should be 24 for a regular cube.
The hypercube can be mapped out by by using seven dice.
one on the bottom
a cross of five in the middle
and one on top.
If you line up the numbers in all the same direction and them fold them out in a 2-D way you can use unmodified d6's as a map but you will need to follow the model to tell where you end up.
Using hyperbolic space for the astral plane makes it possible to have boundaries that appear to get larger as you approach them, for instance. Conversely, using hypercubes (i.e., the "world" is the 3-d boundary of the cube) gives us a spherical geometry.
Anyone remember the "castrovalva" episode of Dr Who?
However, because the cube to the right of the centre borders both the centre cube and the one above it, if you go up and right and your friend goes just right, then you will appear to be coming out of the ceiling in the same room he is in. Just look at the picture to the right and imagine the sloping walls are normal vertical walls with doors and you should get the hang of it. Springing a hypercube on people is really weird, and I wonder how it would go! There is a lot of potential for fun there, and if you make the gravity of the "central" cube dominant, you only have confusion doing the up->right trick I mentioned, although that would allow you to drop a summoned elephant on someone, as on one side of the door its safely supported, and on the other its plummeting through the roof of the room. All you need now is a good bull-rusher!
Thanks, that helps. But how do you draw the cube made of nine small uniform cubes? I can't seem to visualize it.
Draw a point.
Copy the drawing (i.e. of a single point) out to the left.
Between each point in the original drawing and each point in the corresponding new drawing, draw a line.
Copy the drawing (i.e. of a single line segment ending in points) out to the northwest.
Between each point in the original drawing and each point in the corresponding new drawing, draw a line.
These lines will form the boundaries of (one) square.
Copy the drawing (i.e. of a single square bounded by line segments ending in points) out to the northeast.
Between each point in the original drawing and each point in the corresponding new drawing, draw a line.
These lines will form the boundaries of 4 new squares, and the 6 squares in total will bound a cube.
Copy the drawing (i.e. of a single cube bounded by squares bounded by line segments ending in points) out to the north north east.
Between each point in the original drawing and each point in the corresponding new drawing, draw a line.
These lines will form the boundaries of 6 new cubes. The 8 cubes in total (the original drawing, the copy, and the 6 new ones) will bound a hypercube.
.... and so on, up to as many dimensions as you wish.
...
_________
|\....f......./|
|.\______/.|
|.|.........|.|
|b....g....|e|
|.|______|.|
|/___d___\|
Okay so this is pretty rough but it's a good one-dimensional view from any side of the hypercube. G is the central cube, surrounded by B,D,E and F. Even though they all look like different shapes on the outside, inside they are perfect cubes. In addition, we are looking _through_ Cube A to look at my diagramme, and Cube C is on the far side where we can't see it. So far so good?
This is where I get foggy. Where is cube H?
...
Pretend you're in one of the cubes. Call it G. On each face of cube G is another cube (A, B, C, D, E, and F). Cube H is two cubes away in any single "direction." From Cube H, the cube you are in is two cubes away in any single "direction." If there are no obstructions, you should be able to see your own back four cubes away.
On the subject of gravity: why would there be any? The hypercube probably isn't on a planet; there is no down. All there is is crazy.
The outer-most edges of the "trapezoidal" rooms are each a "side" of this "outer cube." Thus, for example, if you travel from "c" to "N" (the North room), then you are oriented as such:
"c" is to the South, and "o" is to the North. For simplicity sake, ignore the other rooms for now, as you mostly want to know how "o" interacts with the others. This is achieved by the following logic:
The north wall of "c" is the south wall of "N", and the north wall of "N" is the south wall of "o."
Similarly, if you moved from "c" to "S", the South room, then the south wall of "c" is the north wall of "S", and the south wall of "S" is the north wall of "o."
As such, you now see how two outer walls of the hypercube are simultaneously the north- and south- walls of the outer cube, "o". Thus, the east wall of "E" is the west wall of "o," and the west wall of "W" is the east wall of "o."
So far we've neglected "t" and "b", the top and bottom cubes, as adding these is what complicates things if you don't first discern the relationship of N-E-W-S-o-c (new sock... no it's not really that clever but it helps).
Now that you have these relations determined, you can start to shift the "blocks" (wherein a "block" is a 3-dimentional room with ceilings floors and four walls denoted by the various letters). Let's now assume you are in "o" - then relations listed above are reversed:
Going west in "o" leads to "E"
Going east in "o" leads to "W"
Going north in "o" leads to "S"
Going south in "o" leads to "N"
Going twice in any direction leads you back to "c," though you'll be approaching from the opposite direction in which you walked. If you walked 2 rooms east from "c," and you stop as you cross the threshold of the door, you're on the west wall of "c."
The next step of understanding comes in moving north-south while occupying "E" or "W," or moving east-west while occupying "N" or "S." This gets a little more complicated, because the linear nature of the rooms no longer holds true.
Consider the room "E." For orientation sake, "E" is now being treated as if it were in the center. As such, the following are true:
Going east in "E" leads to "o."
Going west in "E" leads to "c."
However, "N" and "S" retain their planar-dimensional relation to C. This is where the planar-shift (some call it a gravity-shift) concept of a tesserect comes into play. Consider these truths:
If "N" is still north of "c" it cannot possibly be north of "E"
If "S" is still south of "c" it cannot possibly be south of "E"
If "c" is to to the west of "E" it can't be elsewhere.
If "o" is to the east of "E" it can't be elsewhere.
Room "W" is now opposite "E," much as "o" was opposite "c."
As such...
You have to move north twice to get to "W."
You have to move south twice to get to "W."
You have to move east twice to get to "W."
You have to move west tice to get to "W."
Keeping in mind that "N" and "S" are still in their relative position to "c", then dimensionally they must be above and below you, respectively. The reasoning? "t" is to the north and "b" is to the south. As such...
You move north to "b" from "E," and north to "W" from "b."
You move south to "t" from "E," and south to "W" from "t."
You still move east to "o" from "E", and east to "W" from "o."
You still move west to "c" from "E," and west to "W" from "c."
This is where most people get lost, the concept of the planar shift.
While walking east from "c" to reach "E", reality 'twists' 90-degrees downward; essentially, a shift in your plane of movement. This may or may not be reflected in looking through the doorway back to "c," which is why it is best to not consider the players as 'inside of' room "E" until they (or a magical force) closes the door behind them... or until tesserects are as natural to you as rolling d20. The only real way to explain visually looking back with the same orientation while simultaneously actually being in a new orientation is a dimensional wormhole (or magic portal) that "corkscrews" a quarter-turn in reality which cannot be perceived with the eyes (or mind). While these 'reality twists' are somewhat confusing from any of the Cardinal Rooms (N-E-W-S), they're downright perplexing for "t" and "b."
The reason "t" and "b" are so perplexing is that reality pulls an 'S-curve' trick when you pass through them, performing two different quarter-turns in 3-dimensional reality as you pass through them. Consider this truth, assuming you moved into "b" by going north from "E:"
"E" is south of "b" (appropriately)
"W" is north of "b" (!!!)
"N" is east of "b" (!!!)
"S" is west of "b" (!!!)
"c" is up from "b" (appropriately)
"o" is down from "b" (appropriately)
"t" is opposite of "b" (appropriately)
While the relationship of "W," "N," & "S," are certainly confusing, the appropriate relationships exist in all rooms.
If you derive the (true) opposite statements from the description of "b" room-relations above, you've almost got the tesserect concept.
"E" is north of "t"
"W" is south of "t"
"N" is west of "t"
"S" is east of "t"
"c" is down from "t"
"o" is up from "t"
"b" is opposite of "t"
Examining these relationships, the way of thinking in relation to room "c" is such:
Moving in a straight line N-E-W-S (directions, not the rooms) from "c" does not cause a planar-shift. You can walk in the same direction continually and pass through the same rooms.
Moving east from "c" into "E" causes a 90-degree planar shift downward when moving in a north-south orientation. That is, while a compass would read North you are facing downward in relation to "c." This can be thought of as a northern orientation towards "b."
Moving west from "c" into "W" causes a 90-degree planar shift upward when moving in a north-south orientation. That is, while a compass would read North, you are facing upward in relation to "c." This can be thought of as a northern orientation towards "t."
Moving in a straight line up or down does not cause a planar shift. You can climb in the same direction continually and pass through the same rooms.
Moving up from "c" into "t" causes a 90-degree planar shift clockwise when moving in a north-south orientation. That is, while a compass would read North, you are facing East in relation to "c." This can be thought of as a northern orientation towards "E."
Moving down from "c" into "b" causes a 90-degree planar shift counter-clockwise when moving in a north-south orientation. That is, while a compass would read North, you are facing West in relation to "c." This can be thought of as a northern orientation towards "W."
Now you have the 6 basic rules needed to navigate a tesserect. Further northern orientations can be derived, but with these six rules, you can proceed in any direction you like and achieve any orientation you like by merely retracing your steps and turning clockwise or counter clockwise in "non-central" rooms.
In each ruleset, moving in any one NEWSoc direction continually will lead you in one of three repeating loops; by turning a quarter-turn clockwise or counter-clockwise in any room other than "c" or "o" you can re-orient yourself to be walking on walls or ceilings of rooms that you could have & should have mapped using your original orientation.
This doesn't necessarily make them any easier to understand, but they're now considerably easier to navigate.
-Geregald
On another note: I've always used the concept of a hypercube for when my players enter Limbo. Just seemed to make sense.
Keeping in mind that "N" and "S" are still in their relative position to "c", then dimensionally they must be above and below you, respectively. The reasoning? "t" is to the north and "b" is to the south. As such...
You move north to "b" from "E," and north to "W" from "b."
You move south to "t" from "E," and south to "W" from "t."
You still move east to "o" from "E", and east to "W" from "o."
You still move west to "c" from "E," and west to "W" from "c."
Whoops. Made a bit of a typo - In the above paragraph I say "t" is north and "b" is south, but in the four following statements I reverse it.
When I was originally posting I had "south" and "north" reversed in order, but I changed that and forgot to move the letters. "b" is indeed North of "E", and "t" is South of "E."
Have fun mapping-out your tesserects
-Geregald
Here's something that helped me understand it. http://upload.wikimedia.org/wikipedia/commons/2/22/Hypercube.svg In this picture, (lifted from wikipedia) the center cube, the six skewed cubes surrounding it, and the entire space outside the cube are the eight rooms that make up the hypercube. The important thing to realize is that, in reality, each of these rooms is exactly the same size. They just look skewed because of the image. The center cube is the only one that is not distorted. The surrounding cubes look skewed, and the outside cube looks infinitely large, but they are really the same as the center cube.
To figure out why the cubes don't look the same size that they actually are, look at this picture. http://en.wikipedia.org/wiki/Image:Cubic_graph.png If you look at this cube, only the front and back sides are true size. The left, right, top, and bottom sides look like trapezoids even though they are squares. This is because the picture is a 2-D representation of a 3-D object. Similarly, a hypercube's rooms are so distorted because it is a 3-D representation of a 4-D object.
Go here. When you're done with that, I wanna say that it sounds like some people sound confused by the whole "Outside Cube" thing.
There is no true Outside Cube. Squishing a Hypercube into 3 dimensions just makes it look like there is. Every cube touches another cube on each one of its surfaces, at a 90 degree angle in a direction we can't quite wrap our brains around.
Also, has anyone given any thought to the 4 dimensional space sectioned off inside the hypercube?
http://www.deviantart.com/deviation/54962701/
Go here. When you're done with that, I wanna say that it sounds like some people sound confused by the whole "Outside Cube" thing.
Cool. Nice explanation of the process.
For a DM running a game, though, all you need is the connection diagram I posted in [post=12153876]message 7[/post] of this thread. Just make a list of the rooms labeled T, B, N, S, E, W, C, O, and write a description of each room. When your players choose a direction and go through the door, you simply follow the connection diagram to describe to your players the next room in which they end up.
I think it would be fun to see how long before the players figure out they're in a hypercube. They won't be able to draw a map -- as the DM, you have the map with the key to identifying each room -- the map created by the players may get confusing.
-A
It's basically a giant sphere who's surface you walk on. Looking up, there is another sphere that is always directly overhead. It is, in fact, the other side of the sphere you are currently on rotated through 180 degrees.
Not sure that that exactly fits the physics, but it seems much more 'organic' than the cubes.
But I don't actually know anything about the subject so don't take me too seriously.
The key element, i think, is that you're effectively stuck since all 4 dimensions are completely 'circular'. whichever way you go, you will end up where you started.
In truth, i don't think it's really possible to translate a three dimensional being with perspective into a 4d object: you can trace their position easily enough (once you get your head around the maths) but trying to figure out what you actually "see" is another story.
For instance, would a 2d object on a 3d surface notice the curvature? How could you discribe the sudden lack of visibility given that its 'sight' is limited to yes/no (if it can exist in 2 dimensions, it only 'sees' in 1d parallel to the surface it is on).
And howmuch leeway do you give it's 'vision around corners' is it effectively 'blinded' by the curviture? Can it see a shorter distance than it normally would (on a perfectly flat plane), or can it see all the way around the sphere and see itself.
In both cases of losing a dimension, it seems like you've gotta be arbitrary in explaining how moving in a strait line gets you back to where you started.
Yes, it's somewhat difficult and intense to grasp, but that's hyperspace for you.
As for how to picture a hypersphere, well, if a hypersphere were to move through and thus intersect our three-dimensional relm, it would appear as a small circle that got bigger and then smaller again.
In hyperbolic space things get smaller as they get more distant (at a faster rate than in flat space), and in spherical space things get bigger as they become more distant, untill they come around the other side.
This is where I get foggy. Where is cube H? You said it is the outside of the cube. To me this means one of two things:
1) Cube H is a "negative" cube; is it the empty space surrounding the cube. I hope that's not what you mean. When I am standing outside my house, I don't call it "my other house" or "my negative house". I call it "the driveway".
2) Cube H is the "perimeter" cube. In other words, just as cubes A-F look trapezoidal (but are really cubic), Cube H looks bigger than the other cubes but, on the inside, is really the same size. If this is what you mean, then Cubes A-F appear to fit inside of Cube H in our diagram. This is what you mean, right?
I have more thoughts and questions but I want to make sure I have this part right first.
The "outside" of a hypercube ceases to exist when inside the hypercube. When you get to cube H you find another cube the same size of the others. The exits in this cube leads to cubes A-F.
The parts that I described in the original post can only be perceived from outside the hypercube.
I'm afraid hyperspheres are even more of a disaster... I don't even know if it's possible to work that out. AFAIK, a sphere is the collection of all points with the same distance from a point, so a hypersphere would only have two possible places (in and out). Both would be 4 dimensional space and the inner one would have a finite size, while the outside would be infinite (assuming the hypersphere is floating in infinite 4d space).
But I don't actually know anything about the subject so don't take me too seriously.
A hypersphere would be quite boring actually. Think of a sphere as two disks glued togehter along a circle. Then a hypersphere is two balls glued together along a sphere. So if you leave Sphere 1 going west on the northwest wall, you enter Sphere 2 going west on the northeast wall.
You could also make a hypercube dungeon with rooms at the points, with walls in 3 of the 6 standard directions (NESWUD) and one other direction (portal? Lose the fight?).
You could make one based on faces. You'd have 48 square rooms with a door in each, and a staircase up/staircase down. I think you might actually only end up in 24 of them because of orientation issues. You could have an entire hypercube campaign!
In any of these you can close off some doors to make the dungeon seem like a vast maze rather then a vast open space.
In the end of that story the protagonist managed to escape out a window.. seemed reasonable enough to me.
Here's a way to visualize the exterior cube: Look at all the space outside the cube. Take the operation (1/x,1/y,1/z). You get a smaller cube. That's the analogue of the deformations that make the other cubes trapezoidal.
Also, about your house: Imagine your house is a very small rectangle, on the world, which is a big sphere (let's say it's a balloon or an inflatable globe). Cut out your house. Now if you shrink the rest of the world, mostly in the middle, but keeping the edges the same, until it is almost flat. That will look just like your house.
I would suggest that in using this sort of cube for a dungeon, this might be a fun trick to try: don't put doors/hatches in every wall. Look at rooms O, N, S, E, W and C and put doors between them only where it will allow the players to feel like they are still in 3-D. All other doors (doors that would allow you to see yourself a few rooms away, doors to rooms T and B, etc.) are sealed. I don't mean a well-barred door, I mean solid brick wall. Unless players make sense of the hints in the dungeon and realize they are in a few select rooms of a hypercube they will never know there are two extra rooms full of loot hidden around them. They could leave thinking they really cleaned out the dungeon, and in fact never even made it to the most well-secured area.
ap
That looks like a pretty good explanation.
It reminds me of a SF story I once read where a man built a home that was, basically, an unfolded hypercube. That is, it looked sorta like your diagrams but in 3d: one cube at ground level, a "+" shape of cubes on the second floor, then two more floors of a single cube above that.
In the story, a minor earthquake causes the house to ... change.
As one person in the story reasons it, the house was a stable 3d shape, but unstable in 4d, so when it got shaken it collapsed through the fourth dimension. Net result: the outside it looked like just the first floor, a single cube. But when you went in, you found the whole house of 8 cubes was there. You also found you couldn't get out, as the windows now connected to each other and the door you just entered through leads to the attic.
In the story, they did discover a way to get out. If you walked between two rooms through one of the "new" junctions, like by going out a window on the second floor, and you pointedly did not look where you were going, you would "fall" out of the house.
Which, scientifically, ranks up there with Douglas Adams' notion that flying was the art of throwing yourself at the ground and missing, so a sudden distraction while tripping might accidentally lead to flight. But, as D&D is more about fantasy than reality, I thought people might like it as an alternative idea.
I loved that story. http://en.wikipedia.org/wiki/And_He_Built_A_Crooked_House
Anyway, shouldn't we be calling this what it is? A Tesseract?
Entry and exit is accomplished by having one room connected to the outside, with a single exit into the rest of the house. All other entry/exits from that one room from others are blocked off.
Yes, when I was in high school, I actually tracked every single room's connection to the other rooms and found out the logic. I also listed out the 'true' room connections of a proper tesseract.
It's easiest to visualize a tesseract with the 8 cubes model, and just folding and unfolding them to make connections to other cubes. Modern 'squish' geometry just doesn't do it justice.
===Aelryinth
Okay, from the explanations I've received I get the idea about the "outside" cube. It isn't outside of R, it's just an extra room that is necessary to the structure of the hypercube but which doesn't fit neatly into the distorted 3-D map. Cool.
Exactly.
[QUOET]I would suggest that in using this sort of cube for a dungeon, this might be a fun trick to try: don't put doors/hatches in every wall. Look at rooms O, N, S, E, W and C and put doors between them only where it will allow the players to feel like they are still in 3-D. All other doors (doors that would allow you to see yourself a few rooms away, doors to rooms T and B, etc.) are sealed. I don't mean a well-barred door, I mean solid brick wall. Unless players make sense of the hints in the dungeon and realize they are in a few select rooms of a hypercube they will never know there are two extra rooms full of loot hidden around them. They could leave thinking they really cleaned out the dungeon, and in fact never even made it to the most well-secured area.
ap
A room guarded on 4 sides by walls (or 6 in 3D) is just cruel. Very, very, cruel. I do agree there should be walls, just not enough that you can think you cleared out the dungeon but in fact there is a room you never got to.
From there, it's not too tricky to visualize a hypercube, because it just continues the pattern.
-A
I'm using this as the floor plan for the next dungeon. They'll get teleported into it. It's just a few rooms to fill in, but will be god awful confusing for them to map out.