Re: [DF] Non-Euclidean Architecture in dungeons questions.
 

Not quite. The hyper dimentional definition is simply a narrower version of the dictionary definition.
And still, I say that far more people use the word that way, than the mathematicians' way.

Re: [DF] Non-Euclidean Architecture in dungeons questions.
 

Only because that's what the OP seems to actually be talking about.

Yeah, this is probably more what Lovecraft meant when he described Non-Euclidean geometry.

Re: [DF] Non-Euclidean Architecture in dungeons questions.
 

He probably used the term correctly and just thought that that sort of thing looked weird. He might have felt very uncomfortable looking at this building.

Re: [DF] Non-Euclidean Architecture in dungeons questions.
 

That is both cool looking and slightly disturbing akin to a dissolving sand castle suddenly baked into perpetuity.

Re: [DF] Non-Euclidean Architecture in dungeons questions.
 

Huh. I would have just said "Cool, it's schwoopy". Made with beizer curves.

Re: [DF] Non-Euclidean Architecture in dungeons questions.
 

This is hurting my brain. First I had to read up on non-Euclidean geometry. And I think Lovecraft used the term incorrectly to imply higher dimensional existence. I agree with Sir Pudding.

But if what was originally looked for was a way to model a mind bending architecture that has components outside of what can be perceived by a normal 3D being... I think you're going to get into some really nutty rules that are hard to deal with.

I like the idea someone posted about expected navigation not working. Retracing a path being the only way you can "navigate"... but maybe that doesn't even work... as time passes the 4D object changes. Heck, just standing still might show the observer that things are changing as time passes.

I think that what I would do is just make it a penalty to the perception check, and maybe provide a spell that can give a bonus to perception checks specifically for that situation. Probably a penalty to navigation or checks to keep a bearing too (like someone else mentioned). It seems simplest rules-wise, and then you can make it as trippy as you want with descriptions as you play. People who practice finding their way around in these oddity-spaces might even learn the navigation skill specific to them, and get better at finding their way around.

Re: [DF] Non-Euclidean Architecture in dungeons questions.
 

There are infinite possible non-euclidean geometries but at any given point, they fall into one of two categories, elliptical and hyperbolic. Elliptical doesn't requite higher dimensional embedding space but hyperbolic does. Hyperbolic 3-space just won't fit in euclidean 3-space. So the discussions of extra dimensions and non-euclidean geometries may be more connected than "Lovecraft made a mistake" (note: I'm not an expert on Lovecraft).

Also, even though there are three categories of shape, a given non-euclidean space might be of any degree of departure from the euclidean. This means that there can be spaces that are only slightly non-euclidean (like where we live on the surface of the earth) all the way up to extremely non-euclidean (Lovecraftean if you will). DF delvers could gradually enter a world that is more and more non-euclidean. There can be gradually building penalties to spatial skills. In the beginning, they may not realize where the -1s and -2s are coming from. They could think they're drugged or have been mind zapped or there is magic in the rooms they're in. You could adapt RPKs rules in Psionic Powers for determining when your character has been influenced by psionics to see when the PCs realize that it's space itself that has gone wonky.

Also, penalties for functioning in unfamiliar geometries don't need to be restricted to Navigation-like skills. Really any skill that has a spatial component can be penalized. Anything from combat skills to Lockpicking could be distorted. How much something would be effected would relate to scale. In a moderately distorted space, long range things like Navigation and archery could suffer a -5 while short range things like melee and climbing could get -3 and very short range things like Lockpicking and Pickpocket could only have -1. In a slightly bent space, maybe only the long-range things get a small penalty and everything else works fine. And in very bent space, all these penalties could be much higher. There is a continuum of possibilities. The idea is that any action that incorporates movement through space can be penalized for those unfamiliar with that space and the size of the penalty is proportionate to the lengths involved in said movement. In a bent enough space, even spells with 0 range could be effected if they required gestures.

There could be a slow process of getting used to the new spaces. Maybe each day in a given space lets you roll IQ at a penalty equal to that spaces effect on very small activities. Success lets you shave a -1 off actions in that space. Or you could say there is a skill, Non-euclidean Maneuvering, that you can roll before acting that lets you eliminate space-distortion penalties for that turn. I'd probably use Blind Fighting as a precedent to call it a very hard skill. So IQ/VH. GMs can apply penalties to taste. It could have a default but someone would probably need to have some exposure to non-euclidean space before having access to the default.

The type of geometry and the degree of curvature need not be the same for neighboring locations either. There can be a gradual increase in curvature deeper in the dungeon but there can also be local distortions of any degree to suit the GM's fancy. Stepping from one space into another could be disorienting. It could give further penalties to actions as a character is applying what they learned from the last space to a space that behaves quite differently.

Presumably there would be monsters in these spaces that are at home there. Even a goblin could be dangerous if the party has huge space-distortion penalties and the goblin doesn't... not to mention elder horrors. If it's a place that non-euclidean creatures haven't visited before, there could be signs that they are used to reading that indicate the type of space they're entering. Clever PCs could gradually learn these signs to their benefit. The symbols on the walls could be more than decoration. Even more clever PCs could change the signs to get a few turns of a "taste of your own medicine" type ambush (which could of course backfire with those elder horror who know have a sense for the shapes). Also, risk-taking PCs could mind read non-euclidean natives to get insights into how to maneuver the strange, new spaces.

Finally, one should not overlook the fun that can be had with gravity in these spaces. Einstein's GToR relates gravity to the curvature to space. Maybe the DF world uses some kind of GToR too. So rules-lawyery-physics-expert-players notwithstanding, the GM can make up pretty much whatever kind of gravity changes they want. Maybe in one room, all the walls are really floors. Go crazy with Escher stairs, triple encumbrance penalties in a corridor, etc.

It sounds like a fun premise.

Re: [DF] Non-Euclidean Architecture in dungeons questions.
 

Actually that's pretty much the opposite of what I was saying. I think HPL used the term correctly. There's no story I can think of where "non-Euclidean geometry" is used to describe something that's clearly Euclidean. He does imply that non-Euclidean geometry can be used to open the way for higher dimensional intersections but that's not the same thing. I think he meant like hyperbolic geometry as Lamech says. Which is still pretty weird, but all in (hyperbolic) 3-space.

Re: [DF] Non-Euclidean Architecture in dungeons questions.
 

Yeah, that's where the suggestion (yours?) for a Mathematics roll with TL penalty would be useful. Given that it's DF, I'd be more comfortable with skills that exist on the templates of common adventurer archetypes. Hidden Lore (Elder Things) makes sense. And if it is acting this way, it behaves like a gate, so a gate detecting ability might work. Of course, part of this depends upon the cosmology that the GM has in mind for the world -- i.e., is it *really* (3+n)-D? Or is this just a special effect of the Elder Things? The default is probably just not to even think about it and call it a special effect. If so, then "conventional" delving skills (or magics) ought to work.

Re: [DF] Non-Euclidean Architecture in dungeons questions.
 

You'll have to know something about the shape and movement of the higher-dimensional in directions orthogonal to 3-space in order to make any useful prediction.

If you don't have some way to observe the additional dimensions, I don't think the fact that it's higher-dimensional geometry is going to be of any use.

A side effect of not seeing all the dimensions in play is that objects may move, or not move, in seemingly impossible ways. An 'object' with no visible support might be firmly attached to the structure outside of your current cross section.

Also, of course, there's always the fun of colliding with things along an axis you don't really have. Which is sort of problematic, really. I mean, if a hyper-cone hits someone point-first along the fourth dimension you can imagine a sphere appearing and expanding inside them, killing them messily in a manner I imagine the OP would approve of. But what happens if it hits base-first instead?