Long time Sailing

[mtm21]

I really have to say it....!!!! CANTR IS A DONUT!!!! NOT a doughnut!

[SekoETC]

Has anyone ever stopped to think that knowledge is power and you might be better off not spreading those maps around in every harbour? Then you might actually sell them to people. Finding special resources and bringing a sample to prove people you've actually been to some place special might convince them they could profit from the information. Actually I know one guy who tried to conceal the location of an island, only that he was very late with it. Lately one of my characters saw a map that connects the Polish maps to the English maps so it has a lot of islands. And my only actively sailing character has the Callagher atlas which has pretty many islands already but that character has no incentive of going around to explore because people are hardly awake enough to trade, and you can get enough profit with plain things like wood, limestone and cotton so why go chasing foreign wonders when all you know those lands could be all similar to your own? Someone should go on an expedition to nuke the maps to keep the location of islands a secret, though it might be too late by now.

[MakeBeliever]

Haha...you think the maps actually help. Looks like simple sailing from the gallagher doesn't it, oh yes, well tell that to my doolally sailor who's finally cracking under the strain of following it. What should have been a plain sailing has now taken her to the ends of blue nowhere of the westerly direction for years and years, if it wasn't for her imaginary crew she would be totally insane. Not to mention she keeps thinking up knew idea's in leather and is quite happily flashing off her wingtips, bikini and spiked collar to her non existent invisible crew, they love it and they tell her land will come one day. I dread to think of those poor souls that may finally get to meet that nutty sailor in the end.

[Joshuamonkey]

Haha...you think the maps actually help. Looks like simple sailing from the gallagher doesn't it, oh yes, well tell that to my doolally sailor who's finally cracking under the strain of following it. What should have been a plain sailing has now taken her to the ends of blue nowhere of the westerly direction for years and years, if it wasn't for her imaginary crew she would be totally insane. Not to mention she keeps thinking up knew idea's in leather and is quite happily flashing off her wingtips, bikini and spiked collar to her non existent invisible crew, they love it and they tell her land will come one day. I dread to think of those poor souls that may finally get to meet that nutty sailor in the end. Wink

Role playing to yourself? I've said things to myself before.

And I don't care about the money or wealth. I have what I need, and it's really the fame, knowledge, and island naming opportunities that I'm interested in.

[formerly known as hf]

The correct expression is a Torus

This is OLD we've discussed this before:
http://www.cantr.net/forum/viewtopic.php?t=10270

But, as I pointed out, Cantr IS NOT A TORUS

Cantr is purely 2-dimensional.

Take a piece of paper, representing the Cantr world. Wrap it around, to form a hollow cylinder.

Now, take the open edges of that cylinder and wrap them around so that it makes a doughnut / torus.

Can it be done without crumpling the paper, over lap or warping? No it can't.

Unless Cantrian coordinates are such that the 2d image of the Cantr world is a 2d representation of a torus (as I doubt it, and see no evidence for that) and somehow a projection of 3d space onto 2d space, than cantr is not a torus.

[LimSup]

The mathematically inclined would express it this way:
The Cantr world is homeomorphic, but not diffeomorphic, to a torus.
The general shape is that of a torus, but with distorted distances.

An almost correct realization in 3d space was given by Nosajimiki in the other thread:
Anyway... visually think of it as a map folded in half. When you hit the edge you just have to hop over onto the other side and keep going: no pixel distortions, same rap around features

You have to fold the map twice (once vertically, once horizontally), and tape together the borders. After the first folding and taping, you get a "flat cylinder". Fold that again, and tape around the circular border, to get a "flat torus". The taping is easier, if you do not fold the map in the middle, but fold it twice (at 1/4 and at 3/4) so that the borders meet in the middle.

LimSup

[ceselb]

IS NOT A TORUS

Topologically it is a torus, a virtual one but still a torus.

Cantr is purely 2-dimensional.

Also true (see virtual above).

[formerly known as hf]

No it can't be. LimSup had the word I was looking for.

If, by virtual, you mean 'like a torus but not really a torus' you'd be right.

If the Cantr world is a torus, the map data would need to be held as data in three-dimensions. Cantr clearly only has two dimensions. Thus the Cantr world cannot be a Torus.

Neither is there any indication that the cantr 2d map is a projection of a torus.

[SekoETC]

A torus is actually 4-dimensional so it can't be portrayed properly in just three dimensions. There really is no stretching or folding involved and the inside of the "donut" is just as wide as the outside, this just cannot be repeated in a 3-dimensional world.

[LimSup]

Well, it seems that we have two kinds of torus involved here.

The torus Seko talks about is the cartesian product of two circles, living in 4d space.
This is the shape of the Cantr world, embeddable with correct distances (along the surface) in 3d space like in my previous post as a "flat torus". This is however not an embedding mathematicians would like much: It is not differentiable, not one-to-one, and it breaks the property of the torus that points cannot be distinguished geometrically.
A 2d person living in Cantr world can find two main directions, just by walking around long enough, as in the images of a post of ceselb. But he cannot distinguish the two directions. And he cannot find out where he stands in the world: All points in this world are equal, space looks the same from every point (not considering the distribution of land and water but only the geometrical properties). (Astrophysicians have a word for this property, and our universe seems to share it.)

The torus most of the others have in mind is the surface of a donut: A circle revolved around a line in 3d space.
A 2d person living on the surface of a donut can find out the two main directions, and he can find out if he is on the inside or outside of the donut hole, because the ways around have different lengths. Here, all points on a line parallel to the pink line in the torus image, posted by ceselb, are equal, but different points on the red line can be distinguished.

Looking at the torus image, we see little grey "squares". If we declare that the side lengths of all these "squares" shall be considered equal, we get the first kind of torus: The way around inside is as long as the way around outside, because it crosses the same number of squares. Here we see the "distorted distances" I talked about.

LimSup

[Joshuamonkey]

So basically...going north isn't good for exploring?