[DF] Non-Euclidean Architecture in dungeons questions.

[Ulzgoroth]

Originally Posted by Lamech

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

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

[BaHalus]

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

[Grouchy Chris]

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.

[vicky_molokh]

Originally Posted by Grouchy Chris

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.

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.

[Kuroshima]

Originally Posted by Grouchy Chris

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.

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?

[Edges]

Originally Posted by Lamech

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

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

Originally Posted by Ulzgoroth

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

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.

[Flyndaran]

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

[Lamech]

Originally Posted by Edges

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.

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.

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

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.

[Grouchy Chris]

Originally Posted by Kuroshima

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?

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.

[b-dog]

Originally Posted by Grouchy Chris

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.

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.