[DF] Non-Euclidean Architecture in dungeons questions.

[Grouchy Chris]

Originally Posted by b-dog

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.

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?

[Flyndaran]

Originally Posted by b-dog

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.

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

[Anthony]

Originally Posted by Flyndaran

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

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

[Flyndaran]

Originally Posted by Anthony

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

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

[Kuroshima]

Originally Posted by Grouchy Chris

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.

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)

[Flyndaran]

Adventure in an Escher painting?

[Kuroshima]

Originally Posted by Flyndaran

Adventure in an Escher painting?

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

[Edges]

Originally Posted by Lamech

No, a line can't parallel itself.

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.

Originally Posted by Lamech

...its very important to not confuse people who assume that "non-euclidean geometry" means any geometry that is not euclidean.

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.

Originally Posted by Lamech

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.

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.

Originally Posted by Lamech

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.

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.
______________

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.

[sir_pudding]

Originally Posted by Edges

The OP asked about non-euclidean architecture.

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").

[Anthony]

Originally Posted by Edges

The OP asked about non-euclidean architecture.

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