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

[Richards]

It would be pretty cool if instead of a tire, donut, or torus, the city of Sigil were a Moebius strip. You'd still get back to where you started from if you headed off in one direction, but you'd be making two "laps," and you might even be able to take a shortcut by going down into the sewers on one side and popping up on the other. (Of course, then we have the problems of gravity planes shifting 180 degrees in the middle of the shortcut....)

Johnathan

 

[Shemeska]

Also to consider: things don't have to make sense. Sigil sits at the "top" of an infinite spire. Logically it doesn't make sense and shouldn't exist, yet it does regardless. The planes don't always fit with mortal or even logical conceptions of space.

 

[Lord Xtheth]

Being a DM of sound mind I hereby declare that MY Sigil is FULLY CLOSED IN. Meaning there are no openings to the ouside world whatsoever.
This doesn't have to reign true for anyone else.

I DO remember the day when you could look out of Sigil, see the top of some mountain, and sometimes the other side of Sigil (and I looked it up). However, this is a new edition, with fluff purposely left out so that DMs like myself can make whatever fluff we feel fits our adventures and campaigns.

I proudly support my closed-in Sigil!

[paid for by the Tourist Administration of Sigil]

 

[Rel]

I don't even have a Sigil in my campaign! I'm just here for the physics discussion!

If I DID have Sigil in my campaign, I'd have a closed torus. Then, when the players dug down in the basement to see what was on the outside of the torus (and the guys that I game with would absolutely do this), I'd tell them, "You pry up the flagstones in the basement and underneath is dirt. You dig down through a few feet of dirt and come to what appears to be...the surface of what must be an ENORMOUS turtle shell."

 

[Shemeska]

I don't even have a Sigil in my campaign! I'm just here for the physics discussion!

If I DID have Sigil in my campaign, I'd have a closed torus. Then, when the players dug down in the basement to see what was on the outside of the torus (and the guys that I game with would absolutely do this), I'd tell them, "You pry up the flagstones in the basement and underneath is dirt. You dig down through a few feet of dirt and come to what appears to be...the surface of what must be an ENORMOUS turtle shell."

FWIW, attempting to tunnel down through the ring was addressed in some of the material. The rock was wierd, it was brittle and crumbly upon being removed from the surrounding matrix. And people did tunnel down into so-called UnderSigil or The Great Below with some regularity. Except just like the city streets above, things moved around and rearranged themselves with bizarre frequency, with things often being forgotten, abandoned, or lost. Who knows what all might be down there, but it's rife with all sorts of wierd stuff, wandering undead, fiends, the warring cranium rat hives known as the Two Great Minds, etc.

However while tunneling down is possible, nobody has ever managed to reach the bottom of the ring or break through to the outside of the city - something everyone expects given the enclosed shape of the city as seen from the base of the Spire in the Outlands. Perhaps it keeps going down forever? Perhaps beyond a certain point you keep dropping through portals and you simply can't burrow down any deeper?

In my campaign I had the PCs delve down into the tunnels, and eventually they came across a staircase going down. Two miles of downward spiraling stairs later they stepped through a gateway (which did not show up as an active portal) and into a huge courtyard... open to the sky... underground. Pale green sky, featureless and stretching up and out beyond the boundary of the courtyard. By any measure it was impossible, and it simply didn't fit any preconceptions about anything that anyone knew about the structure and position of Sigil. But yet, there it was.

 

[Fallen Seraph]

I decided in my Sigil to try and make Sigil seem even older and more mysteries to create numerous layers.

Essentially, if people get below Under Sigil they start to break through into underground chambers. These chambers are caverns where reside ruins of old "Sigils", how many of these layers that exist is unknown. Also not even the best Planar Geographer can understand how there can be so many layers and yet the surface of Sigil be thin.

The common theory in my Sigil was the city compressed the reality of these layers into its surface while still keeping their shape.

 

[arscott]

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

John Rogers is a Physicist (among other things). That's not far off.

 

[Shemeska]

John Rogers is a Physicist (among other things). That's not far off.

But I don't believe he wrote the Sigil section.

 

[tomBitonti]

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.

(Remaining text omitted.)

This is close, but not quite correct. I don't quite have the precise terminology on hand, but if each loop moves X as a fraction of the short direction and Y as a fraction of the long direction (modulus either maximum distance in either direction), then when X is a rational multiple of Y, a path will eventually reach its starting point. If X is an non-rational multiple of Y, the path will never reach back (but should traverse a dense subset of the space, so will get you as close as you want to the original point).

Note: A space can have the topology of torus, without actually being embedded in 3-space. The descriptions are contradictory as to whether this is the case. The berk who climbed to a roof and saw nothing lives in a Sigil that is not embedded in 3-space. Sigil would then be a flat plane with its edges attached and that is isomorphic to a torus, but which is not a torus. A berk who can see the far side by standing in the middle of the short dimension is in a Sigil that is a subset of a torus.

Note Also: Another question is, if you cleared out a clear line of buildings in either of the cardinal directions, would you see yourself? If street level is "down" in any case, then whether there is a 3-space embedding or not doesn't matter: You should see yourself. But, in the torus/3-embedding case, if you looked with an angle up or down, you would have odd optics because of gravity bending around in funny ways.

 

[Lord Mhoram]

Also to consider: things don't have to make sense. Sigil sits at the "top" of an infinite spire. Logically it doesn't make sense and shouldn't exist, yet it does regardless. The planes don't always fit with mortal or even logical conceptions of space.

Given I never played any Planescape (I tried reading some but the cant hurt my brain) - I didn't see anything about a spire in the 4E manual of the planes. Might not have read close enough there.

And for my tastes, I'm going for a closed Sigil as well.