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Post by dmgenisisect on Jul 6, 2016 5:23:15 GMT
Hi all, when I'm not spending time being geeky on the internet or at a gaming table I spend all of my time studying Theoretical Physics and Pure Maths. Surprisingly enough this started bleeding over into my gaming, much to my players anguish and delight. I though I would start a short series where I share some of these ideas, stating with my favourite; Building dungeons in 4 dimensions! Lets start with everyone's favourite 4D object, a Tesseract. Loved by fans of sci-fi and comic books (though rarely used correctly), the tesseract is the 4 dimensional abstraction of a Cube. For those who don't know what it is, here as a 2D projection of one.
As you can see, the tesseract contains 8 cubes: a central, top, bottom, north, south, east, west and outer. Lets consider these cubes, as its rather unreasonable to try and actually work in a 4D space. where each of the cubes are oriented as so: 
The lines represent adjacency between different cubes.
With our new found knowledge of the tesseract lets build a 4D dungeon! Each of the cubes of the tesseract becomes a room, and we connect up the rooms using the adjacencies above. It would be boring if we use cubes for each room, so we will relax that condition and just pretend that we can fit whatever shape we want in the spaces. It's also pretty boring if every room has the same number of edges, so we can delete some of the lines.
And there you have it. You've got the skeleton of an 8 room dungeon which is embedded in a 4 dimensional space. The best part is, it's probably going to be impossible for your PCs to map the dungeon easily, as in all probability you can go in a strait line in your dungeon and wind up where you started after four rooms!
In my next post we will add non-orientability to our 4D dungeon!
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Post by Deleted on Jul 8, 2016 0:01:14 GMT
I used to smoke a lot of weed, so I also am proficient in theoretical physics and math... Travelling in 4 dimensions of space requires a completely new axis that we don't have a word for. Nor can we really conceive of what travelling along that axis even looks like. There's a video game called Miegakure which bills itself as a 4D puzzle game, if anyone is curious about one artist's attempt to simulate 4D travel. The important thing to note in that game, and in any scenario involving 4D travel, is that 3D observers are still limited to perceiving 3 dimensions. That is, we can only perceive a single 3D layer of 4D space at a time. How would you even begin to describe what it looks like from the character's perspective as he travels along the new axis? Lacking that, you could just hand wave the description and say you've arrived at a different point in 4D space, but then, what actually makes this dungeon different from a 3d dungeon with 8 rooms? |
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Post by dmgenisisect on Jul 8, 2016 3:41:25 GMT
By building the dungeon out 3D components embedded in a 4D space you add some interesting effects. The simplest to see is the fact that if you walk in a straight line through 4 rooms you will arrive back at where you started. I suppose that you can just hand wave this embedding as a teleportation puzzle in eight rooms, but I feel that makes it lose some of it's charm. You need not have only eight rooms either, the individual 3D components can have more interesting structures then simple rooms.
You can though make things more interesting, embedding your dungeon using more interesting surfaces. We could consider what the effect of embedding on a non-orientable surface, then suddenly the players have the ability to move in that 4th spatial dimension as we 'rotate' there perspectives as the move around, as well as add extra layers of confusion.
I'll concede I am coping out on the 'moving in 4D' problem by focusing on embedding and adjacency, but hey I'm doing it by sneaking in some graph theory...
Even if you can make these structures by hand waving and magic then that's good too! I wanted to share some ideas that to inspire people to think outside familiar dungeon design, designing monstrosities which cannot be mapped, and challenging your players to simply move around.
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Samuel Wise
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Post by Samuel Wise on Jul 8, 2016 4:17:22 GMT
Mathematically speaking a 4d dungeon is impossible. Let me illustrate it. The tesseract is only a 3d model of the impossible 4d plane, in and of itself, it is not 4d. Here is how it is impossible: You know when you cut an orange in half and look at it right, it looks like a circle? Imagine you have a 3d mountain and you cut it in half and you stare at the cutout head on. You will get a 2-dimensional contour plot of points that might look something like this, right?:  So, what happens when you take a 4-Dimensional object and cut it in half and stare at that head on? Instead of a 2-Dimensional contour plot, you get a 3-Dimensional Contour plot, you can actually plot this:  Pretty cool, actually. However, can you imagine an object, that when you cut it in half and look at it "straight on" you get a shape like this? I am almost certain you cannot, because it is impossible to imagine it. It does not exist in our world. Still, I really like the idea of making a pseudo-4d dungeon and who knows, though we cannot imagine what 4d is like, perhaps some other being is able to create a place to live using it. At the very least, maybe something I said is of interest.  |
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Post by dmmoleman on Aug 30, 2016 2:05:55 GMT
I think this is a really awesome idea dmgenisisect. To me, a simple teleportation puzzle is totally different from a teleportation puzzle that has some logic behind it (even if the player's have to use trial and error). I think this idea could have a lot of meat to it. Personally, I might use a simple fractal, or even some basic algebra, but the idea of using mathematics to plot out a maze of sorts, is pretty neat. For your idea, had you considered having different conditional states in the room based on what direction they entered the room (as opposed to what room their in). So you enter any of those rooms from the south entrance and their filled with water, if you enter from the north everything weighs 20 times it's normal weigh, use those kind of direction conditionals, with different items needed from different rooms to pass other rooms --- dude, wow, screw the fractal thing, I'm stealing your idea(borrowing- I'll give it back) thank you...
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Post by dmgenisisect on Aug 30, 2016 9:39:04 GMT
I haven't done this per se, I have done something similar to this white complicated embeddings to have players 'arrive back' at a room and have it look different, but in these cases they are in fact different rooms entirely.
On of the fun things I have done is make the players orientation change the rooms, such as archways that only exist when the PC walks in a particular direction, or hallways which are infinite in length under certain orientations.
I've been trying to prepare another article on on on-orientability but making the graphics half decent has been a bit of an issue.
I'll probabably try and put up an article on embeddings first though, as it appears there is some confusion about the structures I'm working with.
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Post by donosaur on Sept 6, 2016 15:32:39 GMT
So taking the tesseract diagram at face value, we can maybe come with a workable model for this 4th dimension. Here's my thought process: - a one dimensional dungeon: you can move forward or backwards (sounds like a railroad doesn't it?)
- a two dimensional dungeon: you can move forward, backwards, and side to side (a grid)
- a three dimensional dungeon: you can move anywhere on the grid and also up or down
- a four dimensional dungeon: you can move forward or backward, side to side, up or down, and in or out.
This fourth dimension is abstract, and depending on the theme or story device that you're using this gimmick for, you can give it a different flavor, but the key concept that is you are somehow moving deeper in, or out to the shallows of this dimension. Time is a popular fourth dimension, so maybe your players retrieve an artifact that lets them shift through time, where they shift deeper into the past, or back out to the present (several Legend of Zelda games have dungeons that work like this, albeit in a binary fashion). Maybe the fourth dimension is a series of other planes: they could be moving through levels of the Abyss, going deeper or shallower through some artifact or spell. Maybe you simplify it and have the players enter on the Prime Material and shift down to the Shadowfell or up to the Feywild. Maybe the 4th dimension is a state of mind, like madness, or ecstasy. Maybe's it's dream levels, like Inception.
 Actually, I think Inception might be the perfect representation of a 4 dimensional dungeon. The way to have this work in practice as an extension of dimension travel on the regular three dimensions. If you're travelling in one dimension (x) and then suddenly travel in a new dimension (y), your x position stays the same but your relative surroundings change. Same if you are on a grid and suddenly travel straight up; you're in a new place, but you can still see your spot on the grid from where you are. So in a 4D dungeon, you exist in a discrete place in 3D space, but then travel outwards or inwards and everything around you changes.Let's talk about levels. Some dungeons are 3D, but there's only two floors, so the 3rd dimension has quanta, or strata. It's fine and probably will save you some headache to have the 4th dimension just consist of one or two discrete layers. If you look at the tesseract model as a literal cube shaped dungeon, your players maybe enter the room from the outside and see just a cube shaped room, but this is only the surface. If they have a means of travelling from the outer cube inward, deeper into the 4th dimension, they wind up on the inner cube. Where they travel to on the inner cube depends on where they traveled from on the outer cube.
You could also flip it and use the dungeon like a prison instead of a delve. The players could be trapped within the inner cube and have to escape to the outer cube, aka the real world. Or maybe the inner cube is the real world and the outer surfaces of the cube are projections. If I go to the west side of the inner cube and then travel outward on the 4th dimension, I go to the SUPER WEST side of the cube, where things are suitably weird or exaggerated.
This has been super fun to think about and now I really want to find a way to do a 4D dungeon where they explore a character's memories, so thanks for the exercise! |
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Post by donosaur on Sept 6, 2016 15:49:55 GMT
I just had another thought: if you're playing with the fourth dimension, you'll want to account for which way causality flows. In 3D, things roll downhill, fire spreads, rooms can flood and fill up, etc. 4th dimensional shenanigans can affect things around them as well.
If the 4thD is time, then doing things in the past will affect things in the future, but not the other way around (change travels upward). If you're dealing with echo planes, any changes made to the prime material will affect things on the Shadowfell and Feywild, but not the other way around (change travels outward). If you're dealing with Inception rules, changes only stick if you go deep enough (imagine a dungeon where there's a projection of the big boss on every level of the 4th dimension, but you can only really beat him if you go all the way to the deepest level and defeat the real one).
Maybe you construct a 4D dungeon where the inner levels are a death trap that obliterates the players when they enter it, but then it deposits them safe and sound back on the outer level of the 4th dimension because causality only flows inwards. They have to set things up in a particular way on the outer level (which trickles down and rearranges things on the inner levels) so that they don't instantly die when they progress inwards.
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Post by dmgenisisect on Sept 7, 2016 0:08:21 GMT
This is actually a really fun way of building a 4D dungeon (Well in truth a 3+1D dungeon but I'll skip over that technicality). I had a fun on where players could fade through different layers of etherealness. For added fun the dungeon was inhabited by some over levelled encounters; a series of Golemns, and a homebrewed creature that lived in the ethereal plane. So the PCs had to use there varying level of etherealness to solve puzzles as well as avoid the two encounters. In fact this +1D frame of thinking even works for the 4D embedding above, for multiple ways to move around in the dungeon. Then again, any dungeon could have a +1D solution... Your PCs simply sit down and wait for the dungeon to fall apart because of age  Don't even get me started on causality stuff... I do my research in that and it's a real head scratcher. The second I figure out how to implement quantum non-locality into a dungeon... My players will probably kick me out but hey it'd be a cool design. |
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where each of the cubes are oriented as so:
