[DF] Non-Euclidean Architecture in dungeons questions.
 

So in my dungeons there are Elder Things that live long with the rest of the dungeon tropes. They use Non-Euclidean architecture to hide their doors and living spaces so that they will not be noticed in the dungeon although areas where they have Non-Euclidean gates are often shunned by other monsters due to some innate danger sense. Deep Ones might have a Non-Euclidean door at the bottom of a pool in a cavern which most other dungeon denizens avoid. Other Elder Things may have Non-Euclidean doors in the walls of dungeons which are unknown to other monsters as they likely can not comprehend them or even if they could be able to navigate through them.

So here is the problem I am having, how to use GURPS skills to detect them and to be able to go through them? Spells like See Secrets might have some chance to detect them as they are designed to be secret doors to 3-space creatures. But on the other hand the Non-Euclidean doors are alien to the 3-space reality of the spell caster and thus it would be not seem right to have the See Secrets spell easily detect them IMO. But there should be a chance to know that something is not right about a certain wall or cavern I would think.

I don't think Detect Gate should work because Non-Euclidean doors are not gateways to other planes of existence like Astral Plane, Hell, Spirit etc. They are instead part of mundane reality albeit extra-dimensional. They also are not magical in any way IMO as they are part of reality that man was not meant to know. I think there should be some spells that can be learned to detect Non-Euclidean doors but these are secret spells that are found in forbidden texts. I am not sure how to make these Non-Euclidean doors be somewhat detectable (maybe just that there is a feeling that something is not right about a certain area) but that they are beyond the knowledge of normal spells like See Secrets. So if anyone has any ideas as to how to help me with this then please post.

There are also Non-Euclidean rooms and hallways that are difficult for 3-space beings to walk through and navigate. What kind of penalties would you assign to 3-space PCs who try to do so? There may be some secret spells that can help PCs to be able to navigate in extra-dimensional areas and PCs might be able to learn to navigate in them by reading texts as well although doing so might make them insane.

So what I would like is to have ideas to help to make my dungeon fair yet keep the Non-Euclidean architecture alien to 3-space delvers. Thanks.

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

How about having Hidden Lore (Elder Things) work to help notice this? Or give characters with Elder Gift the ability to perceive this (or a bonus to offset the undoubted Per penalty to notice that things just ARE NOT RIGHT).

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

Figures based on non-euclidean geometry are as easy to perceive as those based in euclidean geometry. A triangle drawn on a sphere is non-euclidean for example. Solving puzzles involving non-euclidean geometry might be tricky. Formally it is TL5 Mathematics, so you could give TL penalties when dealing with systems in without the fifth postulate.

What you actually seem to talking about is higher spatial dimensions, however. Really it ought to be flat out impossible for 3D beings to perceive anything other than the 3D portions of a higher dimensional object (just as it is for the Flatlander to perceive the height of a 3D object intersecting flatland). In DF I'd use Thaumatology or Hidden Lore (Elder Things) for attempts to model or predict the behavior of an n-dimensional object (in another game Mathematics is probably more appropriate).

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

I am the KING of POINT WORLD!!!! Go ME!!!

Sorry, couldn't resist. He was my favorite character.

If it's higher spacial dimensions, there still has to be some intersection. If so, then what it looks like might be perceivable by 3-D beings, though it might not make any sense. If it is completely invisible, though, then it is inherently unfair, as there isn't anything to discover it other than dumb luck. Though if it is an intersection to another slice of 3-space, then it is effectively a gate, and Detect Gate ought to be able to find it. Mathematics might do it. But I'd still think it would be a Hidden Lore skill (in keeping with DF).

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

This came up when a friend was writing a text adventure game for an advanced java class in the mid-90's. He made the mistake of asking me to beta test it.

Go north.
Take Sword.
etc.

Knowing early Java, and knowing how people were abusing it at the time, I tried the following command: "Take north".

Of course, to save time, he wrote every object to inherit from one big object called Object. So now the direction "north" was in my bag. As long as it was in there, I couldn't actually go north. Neither could wandering monsters. Rather than insert protection routines or re-architect the program, which was due in a day or two, he re-imagined the whole game as a surrealist exploration.

The infocom games did much the same thing once in a while... illogical things that could be described textually but not actually shown in a graphic game. So I say, if it's That Kind of Game, use synesthesia and paradoxes and logical inconsistencies to your advantage.

I think the way to pull this off is to let the players' imagination run wild; don't limit yourself overmuch with game mechanics. Reward players for wild ideas. If they insist, then give them a Hidden Lore roll once in a while. Double the bonuses and penalties for things like Hidebound and Versatile.

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

Sure that's what I was saying. It'll look like something three dimensional and if it moves through any higher dimensions it may appear to change shape. In which case you have a clue to it's actual shape, but can't see or interact directly with the n-dimensional parts. Which can make for a pretty cool puzzle if you, for instance need to predict when it's going to be "open" again.

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

Another thing to consider for Non-Euclidean doors is that they may be locked. The silver key may open them but if there is no key would Thieves have any way to pick them? These locks might be too alien for them and only those with Hidden Lore (Elder Things) might be able to open them. Of course a Thief might also have Hidden Lore (Elder Things).

An example of Non-Euclidean architecture in Hall of the Fire Giant King might be the wall of tentacles. This seems to be an ordinary wall until PCs pass by it and then 20 tentacles lash out at them along with two beaked mouths. With Non-Euclidean architecture this "wall" might be constructed with enough extra dimensional geometry to be able to house some sort of tentacled eldritch horror. The fact that the drow who serve the Elder Eye (a D&D version of a Lovecraftian deity) can freely pass through it might indicate that those drow can comprehend the Non-Euclidean geometry of the wall.

An example from a dungeon I am running would be a pit where goblin-kin, trolls and ogres throw sacrificial victims into for good luck is actually a Non-Euclidean door to another dimension built ages past by the Sakyss in order to try to allow an Elder God into 3-space reality so they could destroy it. But the decline of the Sakyss led to this temple being abandoned. The temple was left in ruin until goblin-kin, trolls and ogres thought it would be a good place for a lair. The Elder God was able to emanate psionic telepathic powers that urged the new occupants to be "feed" their god by dropping an occasional sacrifice victim down into a pit inside the temple. The bottom of the pit has as extra-dimensional door that allow the Elder Gods to be able to extrude some tentacles and pull the victim through the door to be devoured. The Elder God spends most of his time sleeping but when certain chants and rituals are performed then he wakes up and waits for his food. In exchange he grants some favors to the goblin-kin, trolls and ogres.

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

Well, I would think then that in order to be able to know about Non-Euclidean doors you would need a spell cast upon you to be able to perceive extra-dimensional geometry? But still I like the idea of people having a feeling that something is not right. Maybe only those with Danger Sense might know?

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

To perceive a non-Euclidean door (like a triangular door in a sphere shaped wall) you'd need no such thing. Really there's nothing that spectacular about non-Euclidean geometry. It just refers to systems that don't use the fifth Euclidean postulate.

As far as n-dimensional features are concerned I'm not sure I'd allow a spell to allow someone to actually perceive the higher dimensions directly. Certainly no existing spell does this. If you do have a spell that does this, and since you are running cosmic horror, you might have it work something like Professor Tilinghast's device in "From Beyond".
It could (probably) should look really weird. A door through n-space, might look like a weird set of three dimensional objects just suspended immobile in the air until something opens it, in which case the shapes move and disappear, new shapes appear. If left open it stops in a new configuration. If closed it returns to the original configuration.

EDIT:
You could have a puzzle where an object rotates in n-space when certain things are done in 3-space. The 3-space parts of the object can actually be in completely different rooms of the dungeon and you need to move them for some reason.

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

Okay what would be some effects of non-euclidean geometry for a room?
It occurs to me that if the whole room had non-euclidean geometry it would mess with the visuals pretty hard.
For example normally when looking at something far away your two lines of site from your eyes are at 180 degrees. Now to not see double your eyes would need to less than 180. So one piece of info you would be getting is that monster 500 feet away is right in your face. A bunch of other things would be wrong too I'm pretty sure, but I don't feel like carefully thinking about non-euclidean geometry and how lens work to figure it all out.

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

In real life, non-Euclidean geometry refers to geometry that isn't on a flat surface. (Start at the North Pole, draw a line down the Prime Meridian to the Equator, make a 90-degree turn, draw a line to the 90th Meridian, make a 90-degree turn, and draw a line to the North Pole. You've just drawn a a really big triangle - the lines are as straight as the medium allows - with three 90-degree angles. If that triangle isn't non-Euclidean, I don't know what is.)

So, don't use flat surfaces... anywhere...

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

Yeah, I already pointed out the "triangle on a sphere" thing. He's not really talking about non-Euclidean geometry (even though he keeps saying it), he's talking about higher dimensions intersecting 3-space.
The surfaces will be Euclidean, but you can have all kinds of non-Euclidean geometries on them.

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

That is because that is the term Lovecraft used to describe Mythos geometry. OK I will use extra-dimensional instead.

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

He didn't use it to describe extra-dimensional geometry though (although it was often associated with extra-dimensional effects, like in " Dreams in the Witch House").

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

As far as spells go I would imagine some Wizard discovering them in something like the Necronomicon of the DF world. The would most likely be secret spells though.

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

Why are you acting as if it's a definition he made up? It's the first and most widely used definition I've heard outside of higher math classes.
I love being pedantic too, but one post is enough.

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

Run your fingers "straight" across the walls with closed eyes and friends watching. When they see you do something "impossible" or "wrong" you've found something.

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

Because otherwise it's about as free-form a word as "magic architecture", because it's a fantasy term.

One definition is easy to work with, the other is a puddle.

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

This is wrong. Non-euclidean geometry is any geometry that has different postulates from euclidean geometries (Or maybe changing the parallel postulate I'm not sure which). Second, you absolutely can model some non-euclidean geometries on a flat surface. Finally, you can totally expand a flat non-euclidean geometry to 3 dimensions similar to the way you expand euclidean geometry to 3 dimensions.
Its an example of normal euclidean three space. I suppose you could argue that the surface of any finite three dimensional object in euclidean three space is non-euclidean. However you could also actually have non-euclidean three space for the adventures to play in.


Well you could have n-space buildings, that just use standard euclidean three space. (What has already been described to you by others.)

You could also use non-euclidean three space. I would probably go with hyperbolic space. This is fun. Lets say the PC's walk 100 yards down a hallway, and then they make a 90 degree right turn. They do this three more times. Normal geometry they are back at the starting point. But with hyperbolic geometry they aren't back yet.

Navigation would be extremely hard unless they knew what kind of geometry they were in. They might not even be able to do it, other than remembering where they've been. If you read a text on non-euclidean geometries and then applied it to navigating it might be possible, but I would argue it requires a whole new skill, or a hefty penalty.

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

I made a maze in a MOO that was a hypercube - a 4 dimensional cube. You can walk in a "straight" line and end up in the same room you started in, only arriving by another wall. Requires relative gravity to "work".

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?

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

Yeah, but you can deduce some of that from observation as long as the behavior is repeatable.

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

Yes, but what does framing it as higher-dimensional geometry add? It's just weird stuff happening on a regular schedule.

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

A cool explanation? Immunity to dispel magic? Excuse for using the sanity rules from Horror?

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

Well, next time the Psi in my game gets a psi phenomina while in elder thing infested territory, the whole group is going to end up trapped in a hypercube, and the only way to exit it will be to trace an Euler cycle on it... of course, each room will have an encounter each time they go through it...

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

Actually, there seems to be an extra reason he used the term. In the first couple of decades of the 20th century, physics went all weird. We're somewhat used to ideas like wave-particle duality, gravity being a bending of space and Schrodinger's Cat, but at the time these ideas were revolutionary and totally counter to what people -even scientists - of the time were used to. And lots of them refused to have anything to do with them. Even in the thirties, the Deutsche Physik movement that the Nazis took up was supported by real and distinguished, if elderly, physicists.

The idea that space itself was curved and thus all geometry is non-euclidean was especially objectionable to people used to the idea that Euclid was fundamentally right and this was natural law. The president of Notre Dame University, a theologian, wrote a book proving by impeccable Catholic logic that space really was Euclidean and therefore Einstein was fundamentally wrong. The book makes no sense in mathematics or physics terms, of course.

So Lovecraft seems to have been exploiting the horror that certain ideas held for people with conventional educations at the time he was writing. The resistance rather went away with the advent of the nuclear age, and the revelation of the true horrors of WWII.

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

There's the occasional twisted metric (moebius strip), though they may require discontinuities.

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

That's so true. It just seem like most of his horror would have made modern people go, "Meh." rather than go gibbering insane... while running the heck away from monsters of course.

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

With discontinuities, you run into points with undefined curvature. Barring these points however, all other points are either elliptical, euclidean, or hyperbolic. A mobius strip is generally considered a topological structure and hence can have any curvature. The ones I had the kids cut out of paper in school, for instance, were euclidean. The mobius strip and its cousins are a product of tinkering with how space is connected rather than curved.

Come to think of it, elder horrors could not only have weirdly bent space but strangely connected space as well. There is no limit to how confused a devious GM can make the PCs.

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

For clarity's sake: the debate is still raging. There's good proof on both ends for and against the idea that we live in Hilbert space (which is what I take most people to mean by 'Euclidean') with some essentially local curvature. The difficulty comes with Minkowski describing Einsteinian relativity as an essentially changing curvature of a time/space continuum.

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

A mobius strip (if you make your space the surface of the strip) is going to violate other postulates of Euclidean geometry.

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

For instance, two lines intersecting at an arbitrarily large number of points...

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

Hmm a Moebius corridor! Make them hold the sword with the other hand at the end of it !

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

Fun fact: if you join the edges of two Möbius strips, you get a Klein bottle.

Other fun fact: if you were a two-dimensional creature that lived in a surface shaped like a Möbius strip or Klein bottle, and you left home and traveled all the way around the strip (or bottle), you would come back as a mirror image of yourself. That is, form the point of view of other people who stayed behind, you would appear reversed left-to-right. But from your point of view, everyone else would appear to have been mirror-reversed.

Surfaces like Möbius strips and Klein bottles are called non-orientable. If you put your party into a non-orientable space, and they pick the wrong route through it, they may come back from the dungeon left-handed where they used to be right handed. Also, they'd be able to read and write only with great difficulty. And if you want to be really mean, they'd start wasting away because they would now need some left-handed amino acids in their diets, while nature only provides right-handed ones.

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

This is b-dog we're talking about. They will.

BTW, it's actually a neat way of forcing PCs back into the Eldritch Place. Just need to find a way to explain it to them.

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

Well, all this talk has given me an EVIL idea: The next dungeon is going to be outside time and space. They will be trapped into a hypercube next time I roll a psi phenomena in elder-infested regions. I'm thus interested in how to describe the weirdness. I tried to map this, but I'm not sure how successful I was (map here).

Let me define the set-up:

The party starts in the white cube. They see that they are in a cubical room with one exit in each face. They see cyan down, blue up, orange in front, magenta behind, red left, and yellow right, ok? they decide to go forward, and so move into orange. They still have cyan down, blue up, red left, and yellow right. They have now green in front, and white behind. They continue straight, and still have cyan down, blue up, red left, and yellow right, but now have magenta in front, and orange behind. From orange they will move into white again and complete the cycle.

Would they end up mirrored at some point in their travels?

The escape route I plan for them is to travel in an Eulerian cycle, that is, if each cube is a node, and each cube interface is a vertex, they must travel through the cube and go through every vertex without going twice through the same one. Would this mirror them? would it depend on the trail selected?

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

This is not necessarily true. A mobius strip can certainly be made of euclidean space. Which postulates would you violate?

Which postulate is violated in this case?

The ordinary mobius strip (one that anyone can make with paper, scissors, and tape) has Gaussian curvature of zero. Parallel lines drawn on it will neither converge nor diverge.

Now it is true that a line may parallel itself on a mobius strip. And this situation isn't ordinary plane euclidean geometry. But it isn't non-euclidean. "Non-euclidean" is a strictly defined term referring to spaces in which the 5th postulate is not true. Such spaces have non-zero Gaussian curvature.

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

I'm surprised no one has mentioned the horrible horror movie sequel called, "Hypercube".
It also had temporal oddities.

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

No, a line can't parallel itself. While you are correct "Non-euclidean" is a strictly defined term referring to spaces in which the 5th postulate is not true, its very important to not confuse people who assume that "non-euclidean geometry" means any geometry that is not euclidean.
Source

1)For every point A and for every point B not equal to A there exists a unique
line that passes through A and B.
This fails since on a Mobius strip if you draw a line going along the strip when you get back to the start (what you called a line parallel to itself) you'll start hitting points that you could hit by drawing a different a shorter line.

2. For every segment AB and for every segment CD there exists a unique point
E such that B is between A and E and such that segment CD is congruent to
segment BE.
This fails as well. A simple line going along the strip has a maximum length. A line that is 90 degrees to that also would have either a maximum length or a infinite length. Either way it becomes pretty easy to make it so one of the lines can't be extended far enough.

The surface of a Mobius strip is NOT a euclidean geometry.

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

This is a bit unclear. In graph theory, which is what you're talking about when you're talking about Eulerian cycles, node is synonymous with vertex. The links between nodes are called edges or links. I am guessing that you want the cubes to be nodes, and the single door between each pair of cubes to be an edge, and so you want the route to go through each door exactly once, ending back at the cube they started from.

This is certainly doable, and no, they will not end up mirror-imaged. Any hypercube has a definite inside and outside, and so is orientable, so the players won't end up mirror-reversed no matter what route they take. That's all right, though, because the real fun in a tesseract is the gravity. Start in any cube, go forward two cubes, then up two cubes. You're now back where you started, but but you did not come up through the floor -- you came up through the ceiling, or what appeared to be the ceiling when you were first in that cube. Does this mean your personal gravity is now re-oriented and your new floor is what used to be your ceiling, or does the old floor remain a floor and you fall on your head? Up to the GM, but I think the first option is both more fun and more elegant.

Dragon magazine had an article on a tesseract dungeon sometime in the 80s, I think, and Bruno mentioned one of her own design just a few posts upthread.

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

Why not also poison the PCs as well as make them waste away? Remember when they made Tryptophan and instead of having only the L-enantiomer they had a racemic mixture and a lot of people were sickened, permanently disabled or died? I really could not live with myself as a DM if the PCs were merely dying of starvation.

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

No, I'm not sure what case you're referring to. There was an outbreak of something called Eosinophilia Myalgia Syndrome in 1989 that was traced to L-tryptophan supplements from a particular manufacturer. Nothing I've read, though, has said that it had anything to do with D-tryptophan also being present. Can you give a reference? Or are you maybe thinking of thalidomide?

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

A lot of organic toxins like cyanide are completely harmless when non-natural isomers.

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

Far as I can tell, hydrogen cyanide only has one isomer.

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

I wonder where I heard it then. Maybe it got "telephoned" from a misunderstanding of isocyanides.

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

Sorry, that was posted late in the night, after spending a while trying to produce the PDF posted. Yeah, I meant Edges (it's been a while since I studied graph theory, and it was in Spanish, so the language barrier makes it a little harder). Could you check that I made no mistakes with it?

As for gravity, I have 2 alternatives, either personal gravity depends on the side you enter the cube from (so it depends on the path followed), or it is always perpendicular to to the face you're standing in (so down is always away from the center of the cube)

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

Adventure in an Escher painting?

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

Definitivelly escher inspired, but this (also escher inspired) also played a part...

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

Form a mobius strip out of paper. Draw a line down the center all the way around and call this line A. Choose a point on line A and call it O. Now draw a ray from O at a very small angle to A. Call this ray B. You will note that as you draw line B it will slowly get farther from A. Once you have gone all the way around so that you are drawing next to O, you will find that you have missed O. But you are now drawing parallel to B. This is what I meant by a line paralleling itself. You can keep going around and laying B down next to itself over and over as it continues to get farther from A and closer to the edge each time.

Now if B is a line and not a ray, then you have the situation of B getting farther from A and closer to the edge in both directions. Now after your first pass, you can hold the paper up to the light and see that B has also intersected itself. This allows for the sort of situation Ulzgoroth was presumably referring to, that of multiple intersections. This occurs not only in the 2-line scenario that he brings up, but is found even in the one line situation.

Yes. This is why I gave the definition of what non-euclidean geometry was. It seemed that you and others were confused (not everyone... e.g. sir_pudding had the right of it). Forgive me if I presumed you were confused when you weren't. It just seemed that way. (In my defense, you did say in post 19 that you didn't know).

But this distinction between not euclidean and non-euclidean isn't trivial. It defines what the thread is about. The OP asked about non-euclidean architecture. To start talking about systems which might violate some postulates but not the 5th gets off topic and runs the risk of confusing people. I would think that bringing up such a system is the perfect time to point out what non-euclidean means.

I'm not sure what you're saying here. But it's true that mobius strips don't follow the first postulate. In a simple mobius strip (i.e. one in which the surface is not non-euclidean and is therefore called euclidean space at every point), there can be more than one line through a given two points. This is because arbitrarily small angles to the edge can be chosen leading to lines that pass themselves arbitrary-many times and because in mobius strips, betweenness isn't strictly defined.

This one doesn't work though. The 2nd postulate can hold for a mobius strip. The second postulate basically says that you can extend any segment by an arbitrarily-large, yet finite amount. It essentially assumes no edge to the plane. While the basic mobius strip that one cuts out of paper has an edge, in topology (and again, a mobius strip is a topological object), it is valid to set boundary points at infinity. One could mathematically construct an infinitely wide mobius strip. Like a klein bottle, it couldn't be embedded in euclidean 3-space without intersections. But that doesn't make it's surface non-euclidean.

What I was ineffectually getting at in my last post was that if it doesn't violate the 5th postulate, it is not non-euclidean.
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Basically it seems like we're talking past each other. You seem to be arguing that mobius strips are not euclidean. I conceded this in post 46. And I'm saying they are not required to be non-euclidean. You seem to acknowledge the 5th postulate's importance in your last post. Looking back it appears that we have both been guilty of thinking the other was arguing against something they weren't. Sadly, on my limited time on forums, this sort of thing seems to be the norm.

Regards.

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

Maybe. He appears to want to actually talk about intersections of Euclidean 3-spaces with Euclidean n-spaces (and NO, I do not think that's what HPL meant by using "non-Euclidean").

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

And, given the context, almost certainly meant the common-language understanding of the term, which would be 'not euclidian'.

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

I'm not even sure it's all that not euclidean. Most of the stuff he seems to be talking about would be euclidean in all spaces.

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

Ah. Well, that's another topic all together (albeit a fun and interesting one particularly in relation to DF).

Maybe I've been in academia too long. But in my experience, if people had any reference for the term, their common-language understanding turned out to be close to the actual definition. I know a lot more people that have heard of elliptical and hyperbolic geometry than have heard of say 7-point geometry or its obscure relatives or have bothered to consider whether or not non-orientable surfaces conformed to Euclid's Elements.

I genuinely thought the OP meant non-euclidean and really ran with it in post 27. I thought the idea was pretty cool actually. But if he was talking about something else... don't mind me.

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

FWIW, it gave me some really good ideas...

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

My experience is that people mean "weird, unfamiliar types of geometry". Which is a set that typically includes the mathematical definition, but is not limited to it.

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

I dig it. I guess I've been exposed to a more extreme educational divide. Most people I know either have no reference for the word at all, or they have a pretty good idea what it means.