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I really enjoy having non-Euclidean dungeons in my campaigns. My players also enjoy them. But the whole idea of warped geometry tends to rely on magic. How, if at all, could I explain a non-Euclidean dungeon in a different setting?

I'd need to be able to drop hints to my players about the nature of the dungeon beforehand without making it explicit that there will be non-Euclidean geometry.

Examples of the sorts of things I like to include are doors that lead to the ceiling of a room while going in a completely straight line, a hole in the floor that comes out of the ceiling, leading to a never-ending fall, and having five rooms in a pentagon, where each corner is a right angle.

The sorts of settings I mean are as varied as Science Fiction and Modern to Pokemon RPG, so answers that don't rely on setting or genre-specific factors would be better, but anything non-magic is good.

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Non-magic is an odd requirement, since magic is usually only fluff. I'm working on an answer to your question, but first I need to know, what answer are you looking for? Implementations of dungeons, or a fluff explanation of the "non-magic non-euclidean" dungeon? –  shatterspike1 Nov 15 '13 at 20:28
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@shatterspike1 Mostly good fluff explanations - the answer you've provided is exactly what I was looking for, thanks! –  Dakeyras Nov 15 '13 at 22:14
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You need to Think with Portals™. –  SevenSidedDie Nov 15 '13 at 22:19
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Just as a note: square-based maps are already non-Euclidean, since diagonal movement is faster than non-diagonal movement but takes the same amount of time for the same effort. –  Aesin Nov 16 '13 at 1:34
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How are you defining "magic" for purposes of this question? Is hand-wavey, particle-of-the-week superscience acceptable, or does it fall under "magic" via Clarke's Law? –  Dave Sherohman Nov 16 '13 at 11:24

7 Answers 7

up vote 28 down vote accepted

The Difference between Magic and Non-Magic is Usually Fluff

In plenty of settings and games, magic is objective, such as in D&D and Ars Magica. You can easily refluff "magic" to be clear "science" instead; after all, it's only the way reality works.

Use Warped Dimensional Spaces/Basement Universes as an Explanation
The hints that a dungeon works in a non-euclidean fashion can be explained as a warp space where the laws of reality apply differently, or a wormhole that leads to a basement universe in which physical laws are different. If you have the will for it, and your players enjoy this sort of thing, go ahead and look over the axioms of Euclidean geometry and change or remove them as you see fit: real life non-euclidean geometry doesn't have the Parallel Postulate, for example.

Distortion World from Pokemon Platinum is a good in universe example
The physics of distortion world are wonky, allowing traversal up and down certain areas through gravity, walking upside-down on certain platforms, and certain objects appearing as you walk near to them. This isn't "magic" flavorfully, as it's where Giratina resides as a kind of opposite number to Dialgia and Palkia, so if you're looking for Pokemon specific fluff to your non-euclidean maze, this is it. If you're looking for an oddly meta take on things, check out Glitch City, from a glitch in the Generation 1 games that leads to a city that is inescapable without flight.

Other suggestions for refluffing a non-euclidean dungeon could be

  • A breakdown in reality (physical laws function randomly in a different way, or are randomly non-functional.
  • A strange illusion, like a holodeck
  • A virtual reality simulation

Unfortunately, I can only provide so many examples of ways to refluff, but I'm certain you can come up with many on your own. In essence, physical laws work differently for reasons that are non-magical in nature, and the rest is up to you. I can expand this answer more if you're looking for examples of implementation as well. Good luck!

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“It's not magic! It's super science!” –  okeefe Nov 15 '13 at 21:10
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@okeefe Clark had it right. –  SevenSidedDie Nov 15 '13 at 22:18
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To explain: Clarke's Third Law says "Any sufficiently advanced technology is indistinguishable from magic." –  shatterspike1 Nov 15 '13 at 22:20
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You can also watch "Cube 2: Hypercube" for ideas on what a quantum pocket universe could look like. Or they could be stuck in cyberspace. Or a malevolent super AI might be reformatting reality to be more like cyberspace. Or they fell into Escher World. –  Gustav Bertram Nov 18 '13 at 8:28

The geometry of the surface of a sphere is non-euclidean. Its hard to tell that on sufficiently large spheres, but quite obvious on fairly small ones. So, in a sci-fi type setting you could have the dungeon occur on an asteroid.

It sounds like you are wanting something more Escher-esque than relistically non-euclidean, but if you mostly want non-euclidean then it is easy to find examples where it comes up in the real world.

One place Escher-esque situations come up often in Sci-Fi is in a mindscape. People can enter them either through some form of psionics or through various machines and the place is not bound by logic, much less Euclidean geometry. Though a savvy player will know there is at least a chance to encounter alien geometries in another persons mind before they enter.

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Or a sufficiently large spherical dungeon structured as concentric shells, with lower layers closer to the center of the sphere, could be built underground. –  RobertF Jan 16 at 18:29
    
By the way, in the center of a homogeneous sphere we have no gravity. Not that a sufficiently small sphere would have gravity on it's surface ... or maybe some form of stable super-matter. Yey, now we have gravitation, varying with depth! –  Vorac Feb 13 at 10:22
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For similar examples, consider the Möbius strip (most simple non-oriented manifold that is otherwise reasonable) or a torus (simple example of a manifold with non-trivial homology). –  Thanuir Feb 18 at 11:37

I planned, but never ran, a sci-fi based game on the inside of a spaceship which was a rotating cylinder.

It could be relatively small - perhaps 1 or 2 km across, and only need to rotate at a slow number of revolutions per minute to generate 1 G on the inside surface. The "sky" would be the opposite side of the cylinder, and you could walk there in a couple of hours (or choose whichever size suits the scale of adventure). If you got up high enough into the air, gravity would reduce rapidly, allowing for some interesting devices (human-powered flying machines etc), and in the middle, things would be essentially weightless.

In addition, if you feel up for the physics of it, projectile weapons would have odd trajectories due to the strong influence of the Coreolis effect

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For more about operating on the inside of a spinning cylinder, look for the novel "Rendevous with Rama" and any associated on-line dialogs you may find about it. Here's a 3-d rendering on YouTube: google.com/… –  Zimul8r Nov 18 '13 at 15:56
    
Isn't this like Discworld? –  Lohoris Apr 28 at 7:55

A lateral suggestion, inspired by Numenera — whose books I've recently thumbed through — just don't explain things that closely.

This is appropriate for settings where a baseline level of strangeness is expected, and which exact weird things that happen are part of the content you as a GM provide. Keeping an air of mystery is what you go for.

So, focus on making the place navigable and not mazelike, drop enough hints as to what's going on to make the noneuclidean aspect come to life, don't bother with an answer to "why" beyond "it's a fun puzzle".

As for the hints, you could use features like decorations. Like for your corridor leading to the ceiling of a room, mention that the ceiling of the incoming corridor is carpeted and the floor is painted white. The falling portal would be easy since a player can see themselves looking through it. The right angled pentagonal room could have a (rectangular) piece of furniture placed in its corners.

Another recommendation I'd have is playing Antichamber for inspiration – not everything will transfer from graphics to narration, but the game runs on noneuclidean geometry, there's bound to be something.

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How about a dungeon containing shifting or spinning rooms and corridors via giant wheels and levers?

After the player characters exit a room or corridor & close the door behind, a switch is flipped and the rooms and passages outside of the PCs current location are shuffled around. Since the PC's current location isn't being moved, they won't feel the ground shifting under their feet, although they may hear giant gears grinding behind the walls depending on how well oiled the gears are.

So a room which the PCs have just exited could be moved and rolled ahead of the PCs so that the next door they open reveals ... the same room they just left!

It may also be possible to create seemingly impossible bottomless pits by rotating a shaft or torus in which the hapless PC is falling, so that he keeps going around and around without hitting the floor or walls. I think this would require carefully balancing the centripital forces and geometry of the torus. Other ways to simulate bottomless shafts might involve giant fans blowing upward. If the pit is dark and the PC doesn't have a light source, the illusion of an endless fall would be complete as long as the fan continues spinning at optimal speed.

Another possibility - the dungeon is built as a series of concentric, spherical layers embedded deep under the ground. Rooms & corridors on the underside of the dungeon would be upside down of course, but if the entire dungeon was built to slowly spin like a giant ball at the same speed which the PCs are walking, then perhaps the adventurers wouldn't notice. A bit far-fetched but it could work if the PCs don't move around too quickly, or split up.

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A.K.A the cube :) –  Guillaume Jan 16 at 21:55

In a SF world, you can have this kind of 'non-euclidean dungeons' by placing your players inside a Virtual Reality world (Matrix).

The odd parts and glitches can come from various sources: bugs, backdoors, hacks that will alter the perceived reality of the players.

There is a good anime from the Matrix anime set about this with kids playing inside a weird house.

For example your player team can have been hired to retrieve a rogue agent (human or AI) in a VR world. This agent has modified the structure of the world and has created a kind of a lair that is protected with traps and layers using weird geometry.

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Call of Cthulhu setting (including but not limited to the same-named RPGs) has notes on non-Euclidian geometry, and a few of the Lovecraft novels (that the setting is based on) include vivid descriptions of such; I can't recall which ones, though.

For a detailed example of a non-Euclidean Call of Cthulhu RPG setting, take a look at the Original D&D Discussion forum, in a thread called "Map of R'lyeh (sort of)"

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Can you expand on what the ideas presented therein are? As is, this doesn't actually answer the question on how to do it. –  Jonathan Hobbs Feb 12 at 7:23

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