Sigil Ciy of Doors (Topology of a Torus) - MATHS

Dire Bare said:

Ah, no. You've got it reversed still. Sigil, 2e thru 4e, is NOT fully enclosed but is open on the inside edge. Pretty much exactly like a car tire.

Click to expand...

I won't argue 2E.

4E seems pretty clear though if you have Manual of the Planes. Either the book is badly written, or Sigil was changed.

Page 9 says "Sigil is a recursive demiplane. The city fills the interior of a torus, so a traveler can’t help but circle back to where he started by continuing in a straight line."

First, it says torus. That's a very specific mathematical shape with no gaps.

Torus - Wikipedia, the free encyclopedia

Second, how does it define recursion? Again on page 9: "Does the plane have an edge that you can reach..."

So 4E Sigil, being recursive, has no edges.

Further it doesn't give any direction to the recursiveness, or state that you have to walk in a particular direction. It also doesn't mention any gaps (which would stop you recursing in a particular direction).

Pg25 "The city is built on the inside of a gigantic, hollow ring that has no outside."

That again is quite clear, there is no outside and it is a ring, not a tyre.

There is a picture on page 27 which is the sole evidence for a gap or edge in 4E Sigil but you would have to be able to prove that band is a hole and not a window or an artistic cutaway. All the text is very very clear about it not having holes, gaps or edges.

Unless the text is faulty. If you are right, then the book is wrong.
That is not impossible

MoppyDragon said:

I won't argue 2E.

4E seems pretty clear though if you have Manual of the Planes. Either the book is badly written, or Sigil was changed.

Page 9 says "Sigil is a recursive demiplane. The city fills the interior of a torus, so a traveler can’t help but circle back to where he started by continuing in a straight line."

First, it says torus. That's a very specific mathematical shape with no gaps.

Torus - Wikipedia, the free encyclopedia

Second, how does it define recursion? Again on page 9: "Does the plane have an edge that you can reach..."

So 4E Sigil, being recursive, has no edges.

Further it doesn't give any direction to the recursiveness, or state that you have to walk in a particular direction. It also doesn't mention any gaps (which would stop you recursing in a particular direction).

Pg25 "The city is built on the inside of a gigantic, hollow ring that has no outside."

That again is quite clear, there is no outside and it is a ring, not a tyre.

There is a picture on page 27 which is the sole evidence for a gap or edge in 4E Sigil but you would have to be able to prove that band is a hole and not a window or an artistic cutaway. All the text is very very clear about it not having holes, gaps or edges.

Unless the text is faulty. If you are right, then the book is wrong.
That is not impossible

Click to expand...

If the D&D designers were writing a textbook and used the term "torus" to describe Sigil's shape, I'd let them know that Sigil is almost-but-not-quite a torus. But they are not writing a textbook, but rather a gamebook and the use of the word "torus" is fine with me, even if it is not completely accurate.

Furthermore . . . ah, nevermind. My inner geek id wants to argue vociferously against your ideas, but my higher brain is tired and reminds me that it just doesn't matter that much.

I like the idea of Sigil being a fully enclosed torus rather than the shape the 2e designers gave it . . . but you're still wrong and I'm right!! HA!!! (sorry, I got nothing else but the childish end to all arguments).

Course, if you want the official answer, email "The Sage" on the WotC D&D website. He'll set you straight! (if anyone can be set straight on such a recursive idea!)

*Sorry about lame sense of humor. I should be working!

Moppy,

FWIW, I came to the same conclusion you did about Sigil after reading the 4E MotP. I've never played 2E Planescape. I did think that that "interior of hollow torus" is different from what I remembered from the 3E Manual of the Planes.

Anyway, I like the torus idea. In my game, Sigil will be exactly as you have described, because that's how I imagined it when I read it, previous editions be damned. I think the torus idea is cool!

Plus, I figure it'll be easier to explain this way, given that I don't have to elaborate upon, "What it's like to live inside of a tire," and, "What nothingness looks like and how does it work."

Shem,

I'm curious what you think of the raavasta, now that your namesake is one. Perhaps a new thread?

MoppyDragon said:

First, it says torus. That's a very specific mathematical shape with no gaps.

Click to expand...

For what it's worth, I'm pretty sure I saw the 2e version described as a torus as well. I think they're just being imprecise with it. ;-)

MoppyDragon said:

First, it says torus. That's a very specific mathematical shape with no gaps.

Click to expand...

And none of the authors of the 4e MotP are mathematicians.

And neither am I, and I've used the term to describe Sigil before. The concept being conveyed isn't exactly the same as the mathematical shape. The flavor text pre-4e is clear beyond a doubt on the situation however, and the 4e text just isn't written as clearly as it could be. Perhaps the 4e description assumes some familiarity with the older descriptions to convey the same, or possibly the author wasn't familiar with the earlier material (though I recall Wyatt saying he had read "In the Cage")?

Kunimatyu said:

Shem,

I'm curious what you think of the raavasta, now that your namesake is one. Perhaps a new thread?

Click to expand...

Start the thread and I'll share my thoughts. They won't be nearly as acerbic (if at all) as some might expect given my general 4e feelings however.

In my opinion Raavasta is a far better treatment for Arcanoloths than just trowing them like "Demons".

Hope Ultroloth gets the same treatment.

I don't have the Manual of the Planes, yet, but as the original question was directed towards mathematicians, I'll answer it under the assumption that we're talking about the inside surface of a torus (tori are the collection of points only on the "surface" of a donut -- it is analogous to a plane in that it is not solid and only comprises two dimensions in its own coordinate system; Rel's correct that a solid donut is, instead, a toroid).

You are correct, any direction would eventually loop back to the point of origin. The question of how many "laps" you'd do around either of the "circles" involved in the torus (laps of the tube or the ring, if you will) is a matter of how the angle you choose relates to the ratio of the dimensions of the torus; but there would be no angle which would not produce a path that eventually closed on itself if continued indefinitely.

The reason for this is similar to the reason that a pool ball in a geometrically ideal environment will always form a closed loop path. The only difference is that the torus, instead of reflecting the path at the edges as an ideal pool table would, loops around to the same angle on the opposite edge because both opposing edges are joined.

Of course, now that I type this, it occurs to me that I've been thinking about this from a topological sense, without really giving extensive thought to how one would define such terms as "straight" in a toroidal existence. Depending on how that is defined, my answer stands. And as my answer provides an interesting and elegant solution, perhaps the existence of it as an answer is sufficient justification to predispose one to define straight such that the torus would have that property.

One thing I thought of was that, if there is a gap in the torus then, depending on how big the whole thing is, it makes the idea that there is a "disturbing nothingness that is unsettling to look upon" kinda...cheap due to how visible that nothingness would be through the gap.

Let's assume that the shape is exactly like a car tire scaled up to very large size. If I'm standing almost anywhere on the inner surface of the tire, I can look up and see, not only across to the opposite side of the tire, but also up through the gap into the void. This effect varies in how pronounced it is based on how wide the gap is and also how big the donut hole is.

If the donut hole is very small compared to the total radius of the torus then your view from across the city, through the gap, is more likely to show some part of the other side of the city or some part of the outside of the city but still not the void. You have to get much closer to the gap in order to be at an angle to see past the other side and out into the void.

As the donut hole becomes larger compared to the total radius of the torus then it becomes easier and easier to see out into the void. If the torus is almost all "hole" and very little "donut" then you end up with the Ringworld where you can quite easily see out into the void on either side of "the arch".

Again, this effect can also be compensated for based on the size of the gap. The more donut hole you've got the narrower the gap has to be in order to conceal the void.