Conquer Club

The graph style Classic and Middle Earth maps got me thinking: which maps on CC are non-planar? That means if you tried to draw them on a flat surface (a plane), with territories as points and attacking paths as lines between points, then a non-planar graph would always have attack lines that cross.
Classic and Middle Earth are planar maps, because you can draw them (as someone did) without any lines crossing. The connection between Kamchatka and Alaska is not a problem, because you could always redraw it as a line that went over Greenland and Europe and didn't cross any others.
A famous example of a non planar map is K(3,3). This means a graph with 6 points, where 3 of them are each attached to the other three. It comes up in this way: imagine you are laying utilities lines, and you need to get water, electricty and gas to three different houses. How can you lay the lines so that none of them will cross? (Ans: you can't, but it's very entertaining to watch people try).
Another non-planar graph is the complete graph on 5 points. That means 5 points, where each one is connected to the other 4.
If any graph contains something non-planar, then it is non-planar.
So Waterloo is a non-planar map, in a lot of places. 2 sets of 3 artillery that face each other make a K(3,3). The 4 cavalry in Zeithan + Picton 09 (or 11) make up a complete graph on 5 points. That makes it more complicated, but you'll never see the Waterloo map drawn as a bunch of points with straight lines between them.

Most of the maps on CC are planar, beacause they either represent real places with adjacent attacks, or they are very simple concepts with standard layouts. So you could draw them all like the new Classic. Are any maps other than Waterloo non-planar? Could mapmakers make some more? Would they be as good as Waterloo? (Ans: yes, yes, maybe).

Any map with ports would be non-planar, I'd guess.

pancakemix wrote:Any map with ports would be non-planar, I'd guess.

I believe that isn't true, but don't know whether I'm able to formulate rules. Must think...

(It would be MUCH harder to formulate rules for maps with bombarding)

This is something I consider to be very relevant to distinguishing gameplay. Some maps considered "simple" are, in fact, able to have very complex strategies without all types of crazy bonus logic, by using this principle (excluding bombardments).

Here is my list:

Madness
Solar System
Space
King of the Mountains
D-Day I think (someone check my math on this)
Sydney Metro
Prohibition Chicago
NYC

I also think that AoR1 & 3 fit the bill, but that AoR 2 would be planar because of the ice blocking some of the ports. (again, someone check my math here)

There may be more that I haven't played... so I won't claim this is the all compressive list...

Hey Bald,

Interesting ideas, I think they belong here more appropriately (cartos, forgive me if I moved it to the wrong place)

Hopefully some discussion gets going on this

Twill

Twill wrote:Hey Bald,

Interesting ideas, I think they belong here more appropriately (cartos, forgive me if I moved it to the wrong place)

Hopefully some discussion gets going on this

Twill

But here (in GD) is much bigger audience (i.e. it's a higher probability that we'll discover something really worth)

Am I actually posting in FOUNDRY, first time in my life? Well, my sincere greetings, good folks!

And after the first greetings - something I want to tell you for long time:

Somebody should write a really good (hiking ) guide
(I entered your sub-forum few times and every time ran away - since I felt very dizzy...)

pancakemix wrote:Any map with ports would be non-planar, I'd guess.

Not exactly. The New World has ports, but they don't all attack each other, and there aren't enough of them. A lot of maps like this though (with special attack rules that aren't drawn on the map) will qualify.

gdeangel wrote:This is something I consider to be very relevant to distinguishing gameplay. Some maps considered "simple" are, in fact, able to have very complex strategies without all types of crazy bonus logic, by using this principle (excluding bombardments).

I think complexity has a lot more to do with the average number of attack lines from each territory (the "connectivity" of the map), but in order to get this number higher, it's necessary to create non-planar maps, so they certainly do help.

gdeangel wrote:I also think that AoR1 & 3 fit the bill, but that AoR 2 would be planar because of the ice blocking some of the ports. (again, someone check my math here

I'm not sure about D-Day, but it's got a lot of territories, so it will take some time to collapse the unimportant ones to check it. AOR 2 is non-planar, because although the ice blocks some ports, it leaves 5 in the north, all connected.
You could also include Age of Merchants (more than 5 ports, all attack each other) and (surprisingly!) Duck & Cover (East, West, Central US can each attack Moscow, Central Russia and Siberia, through different attack lines).
Space also slipped by me because of having so few "ports". The attack lines portrayed are deceivingly simple: it looks like there are only two line crossings, which can be reduced to one by drawing the inner planets in the opposite direction, and quite often one line crossing can be removed. Turns out there are at least 3 crossings in any planar representation.
Some more for the list:
City Mogul, Drug War, Supermax Prison Riot (5 "ports")
Gazala (4 airfields and practically any other territory make a complete graph on 5 points, with attack routes substituted for direct attack to the latter)
Conquerman (similar to Gazala, 4 ports is enough as long as they are not isolated from each other).

I also suspect Pearl Harbor (Can FIN AA hit all those planes without crossing any lines?) but I'm not certain.
And again, probably still not a comprehensive list.

Twill wrote:Interesting ideas, I think they belong here more appropriately

I don't. Here, mapmakers will see them, but none one else will. I don't think they're very interested with the theory of game play, but casting a wider net (in GD) might catch someone who is. In fact, I think you've already managed to scare away someone who was thinking about it.

Zemljanin wrote::idea: Somebody should write a really good (hiking ) guide
(I entered your sub-forum few times and every time ran away - since I felt very dizzy...)

!!!

There is one obvious thing I want to say loudly, since not everybody is familiar with topology:
If a certain map has more than one sea, it is planar IF and only IF each it's sea is planar.

EDIT:
Ups, it has already been said, more general and more elegant:

BaldAdonis wrote:If any graph contains something non-planar, then it is non-planar.

/EDIT

@BaldAdonis
I see that you already have some theory (five nods), but I must note that four ports are also non-planar, unless at least one of them is on an island (and at least one of them is out of that island)
(also virtually obvious, but I think it's practical to have it explicitly written)

Zemljanin wrote:I see that you already have some theory (five nods), but I must note that four ports are also non-planar, unless at least one of them is on an island (and at least one of them is out of that island)
(also virtually obvious, but I think it's practical to have it explicitly written)

I know, I didn't want to say it that way, because it might cause confusion where it does come up. For example, in Age of Merchants, the port on Pirate Cove is an island (in fact, the whole map is islands, but it's more clear if you don't have to consider attack lines that go off the map), and so it and three others are planar.
I thought it was easier to say 4 ports AND another attacking route between them, as in Conquerman.

I found another: Bamboo Jack is non-planar, because of the free-for-all attacks that go on in the underground railways stations.

Zemljanin wrote:

Twill wrote:Hey Bald,

Interesting ideas, I think they belong here more appropriately (cartos, forgive me if I moved it to the wrong place)

Hopefully some discussion gets going on this

Twill

But here (in GD) is much bigger audience (i.e. it's a higher probability that we'll discover something really worth)

Am I actually posting in FOUNDRY, first time in my life? Well, my sincere greetings, good folks!

And after the first greetings - something I want to tell you for long time:

Somebody should write a really good (hiking ) guide
(I entered your sub-forum few times and every time ran away - since I felt very dizzy...)

Apologies for making you more dizzy, but this thread is probably of more use in the Foundry Discussion forum... [moved]

this talk makes me think of this thing I had to dl to look at something widowmakers did. I think it was Adobe After Effects. you could show the image just as squares and lines (or that's how wm did it) and you could move groups or single squares. kinda cool

edbeard wrote: I think it was Adobe After Effects. you could show the image just as squares and lines (or that's how wm did it) and you could move groups or single squares.

This is what I'm doing now. Although mapmakers might not want to hear this, the images are completely superfluous. In fact, aside from the initial pleasing effect, they usually get in the way and make the game harder to play, especially for anyone with trouble seeing (consider Brazil or Middle Earth, with all of the very slight line shading changes between blocked/adjacent territories). A map with just squares and lines (which you can get by feeding in the xml to any suitable vector imaging program) is much more practical. The only problem is that non-planar maps are very difficult to represent. An easy to follow rule (like "all the prisoners in the yard can attack each other") makes for an impossible to follow representation (a complete graph on 21 points has 210 lines in it). But if I had a list of the non-planar graphs, I could just leave them out and the original map image would stay on as the default.