Life In 19x19

I've been making a 4D goban as a school project, and decided it may be a good idea to share it here. It is written in C++ with Qt, so should compile on most systems (I used GCC on Linux). Requires CMake to build.

The program provides a 4x4x4x4 goban to play on. I've found games to last around as long as a typical 19x19 game. Unless someone makes a mistake, it's pretty much impossible to kill anything until the very end, but still very interesting to try to spot eyes and count liberties.

Installation depends on system; on Linux, the following should do (from the source dir):


No config files, server will listen on port 15493. Once the clients are loaded, !help explains available commands.

Link:
https://github.com/Sevis/4DGo

PS: The score counting is not currently implemented: chances are that one person will lose the capturing race. In the case of a seki, Japanese scoring should be easy to do by hand.

Feature requests are welcome, if anyone thinks this is worth developing further, and will add some comments to the code if anyone would like to read.

What is 4D? How many liberties does a stone have? Isn't go hard enough?

daal wrote:What is 4D? How many liberties does a stone have? Isn't go hard enough?

I'd guess it has 8 liberties.

The problem with high dimensional spaces is that creating a surface enclosing a territory becomes much harder as the number of dimensions increases. I'd expect many sekis.

Li Kao wrote:

daal wrote:What is 4D? How many liberties does a stone have? Isn't go hard enough?

I'd guess it has 8 liberties.

The problem with high dimensional spaces is that creating a surface enclosing a territory becomes much harder as the number of dimensions increases. I'd expect many sekis.

I would guess 6 liberties for a center stone, 5 for a middle side stone, 4 for a top/bottom side stone, 4 for a middle corner stone, and 3 for a top/bottom corner stone.

6 liberties for a center stone is 3D go. 4D go has 8 liberties in the center. Two liberties along each of the axes (w,x,y,z).

Surrounding territory or killing anything is already almost impossible in 3D go. In 3D go, there's not even a forced capture of the 1-1-1 point if the 1-1-1 point is played as the first move. But at least, having the 2-2-2 point means you can kill the 1-1-1 point if it is played thereafter, so maybe there's hope.

For 4D go, surrounding anything is even more difficult. Except on small board sizes, I don't think it's possible to capture a first move on the 1-1-1-1 point even if the second player gets 3 moves in a row! This suggests to me that there are so many liberties, you can almost play wherever you like, and nothing will get captured unless you make a terrible blunder.

For higher dimensional go that actually has territory, you probably want to find a higher-dimensional lattice that is a bit more restricted than the cubic lattice.

lightvector wrote:6 liberties for a center stone is 3D go. 4D go has 8 liberties in the center. Two liberties along each of the axes (w,x,y,z).

Ah, yeah. I guess I'd just sort of skimmed his description and assumed a 3D 4x4x4 (And just went for the 4d title for flair?)

Meh. Personally, I don't believe any dimensions beyond the three of space and one of time.

Would be interested to see a screenshot of this in action, as I can't really visualize what it would be like. (Definitely not my cup of tea, though [I don't drink tea, anyway])

You should think of it as 4 cubes stacked on top of eachother along the w axis. A bit like how every cube is a set of 4 squares stacked on top of eachother along an axis you can actually see.

Thank you for the comments, everyone. ^^

It is indeed eight liberties per stone, and things don't die quickly -- although I have made some errors that almost cost me a stone, and would have probably cost me the game (a ponnuki is worth 128 points, but my opponent didn't notice the opportunity).

I'll have to disappoint those who were expecting a 4D render engine; the board is shown as a 4x4 field of 4x4 boards (with letters used for stones ). I'll get a screenshot here soon, but it's not much to look at. I find the most interesting part to be the mental visualisation of the board, which still doesn't go all too well for me, seeing as just converting to 3D and then imagining that doesn't quite do the trick.

The `more restricted lattice' idea is good, and wouldn't be overly hard to implement technically with the way things work now. On the other hand, displaying it to the user would be a problem. The problem of the board having to be very small is still there, though; perhaps making it non-hypercube would be a good idea, such as a 9x9x3x2 board (this would also reduce every stone's liberties to at most 7).

PS: Interesting about the 2-2-2 stone in 3D go, hadn't realised.

I first played 4x4x4x4 tic-tac-toe at college. That was quite interesting, although after about 100 games I discovered a pretty foolproof simple winning strategy that made it trivial for the first player. At least against the opponent I had. . At school a couple of years earlier I made a little cardboard 3x3x3 tic-tac-toe board, in three dimensions. It was a trivial game. 4x4x4 would have been much better, but I didn't think of it. The 4x4x4x4 game is easily played in two dimensions — we chalked up a 4x4 lattice of 4x4 squares on the wall.

A few years back I played around with Rubik's Cubes. I was hopeless at speedcubing, but could solve a 3x3x3, 4x4x4, and 5x5x5 easily enough (after learning the methods!). Then I discovered this 3x3x3x3 applet. It made my head spin. One interesting way to study "reading" a 3D cube is to start with a solved one, then make several twists, change the orientation, then try to reverse the twists. This 4D applet I found mind-numbing. Even trying to undo two twists was almost impossible for me.

So now we have 4D go. Without a GUI. . Maybe I'll play with it a bit . . . .

Paul

I looked at that download link. I'm not a computer nerd. I use Windows 7 and don't write code beyond relatively simple HTML. So I have no idea what the program looks like. But maybe I can help people visualize 4D Go, by referring to 4D tic-tac-toe.

2D tic-tac-toe 4x4:

Apart from using a 4x4 board instead of the usual 3x3 board, this is trivial to imagine. On paper, draw a square 4x4. Add the noughts and crosses, functionally equivalent to black and white stones placed alternately. Let's label the playing points on a 4x4 grid in a somewhat familiar fashion:
Top row: Aa Ab Ac Ad
2nd row: Ba Bb Bc Bd
3rd row: Ca Cb Cc Cd
4th row: Da Db Dc Dd

A winning line would be stones placed along any of the rows, any of the columns, either of the diagonals, 10 in all. Examples would be Aa Ab Ac Ad, Aa Ba Ca Da, and Aa Bb Cc Dd.

3D tic-tac-toe 4x4x4:

In three dimensions, this would look like a cube. But let's look at it in two dimensions, certainly easier to draw out on paper. On paper, draw out a lattice of four 4x4 grids. You then effectively play 16 simultaneous games of tic-tac-toe, as there are 16 different ways you can stack up a single 4x4 grid from a 4x4x4 cube (include the 4 "diagonals").

1Aa 1Ab 1Ac 1Ad | 2Aa 2Ab 2Ac 2Ad | 3Aa 3Ab 3Ac 3Ad | 4Aa 4Ab 4Ac 4Ad |
1Ba 1Bb 1Bc 1Bd | 2Ba 2Bb 2Bc 2Bd | 3Ba 3Bb 3Bc 3Bd | 4Ba 4Bb 4Bc 4Bd |
1Ca 1Cb 1Cc 1Cd | 2Ca 2Cb 2Cc 2Cd | 3Ca 3Cb 3Cc 3Cd | 4Ca 4Cb 4Cc 4Cd |
1Da 1Db 1Dc 1Dd | 2Da 2Db 2Dc 2Dd | 3Da 3Db 3Dc 3Dd | 4Da 4Db 4Dc 4Dd |

Examples of 4x4 grids would be:
1Aa 1Ab 1Ac 1Ad | 2Aa 2Ab 2Ac 2Ad | 3Aa 3Ab 3Ac 3Ad | 4Aa 4Ab 4Ac 4Ad |
1Aa 1Ab 1Ac 1Ad | 1Ba 1Bb 1Bc 1Bd | 1Ca 1Cb 1Cc 1Cd | 1Da 1Db 1Dc 1Dd |
1Aa 1Ba 1Ca 1Da | 2Aa 2Ba 2Ca 2Da | 3Aa 3Ba 3Ca 3Da | 4Aa 4Ba 4Ca 4Da |
1Aa 1Ba 1Ca 1Da | 2Ab 2Bb 2Cb 2Db | 3Ac 3Bc 3Cc 3Dc | 4Ad 4Bd 4Cd 4Dd |


4D tic-tac-toe 4x4x4x4:

This gets tricky to annotate similarly, as I don't have "upper case numbers". But the idea is just an extension of the 3D case, where instead of playing 16 simultaneous games of 4x4, you will be playing 16 simultaneous games of 4x4x4. Or 256 (?) simultaneous games of 4x4.

-----

I have no idea yet if this is a good game or trivial, but it would seem to me that turning all this into Go would result in playing 256 simultaneous games of "regular" Go. In tic-tac-toe, where the object of the game is to get four stones in a row before your opponent, that is a relatively simple goal, and the game is possible to play and see what is going on. Just! In a game where you surround territory, and capture, it gets more complex. Doh.

Let's take the question of liberties. Using the same reasoning as above, one single stone is playing on 256 (?) boards at once. If it's an edge stone, on any one board it has three liberties. So in those 256 games, it could be that on 32 boards it has 0 liberties, on 64 1 liberty, on 32 2 liberties, and on the remaining 128 boards it has 3 liberties. What are the criteria for it to be considered captured?

If we're talking about a board chalked up on the wall, one single stone either has to be on the board(s) or not on the board(s). It can't be in some quantum state of both at once. However, software could render it in both states simultaneously. Hmmm.

My thoughts right now are to formulate the rules for the game in three dimensions, so it makes sense to play 16 games simultaneously. Then merely jack it up a dimension. I think the hard questions would have been answered by that time.

Paul

One highly convenient thing, if you happen to want to play 16 simultaneous games of 4x4 Go, is that a 19x19 board is the perfect size. Just get some masking tape or similar to put over every 5th line and it neatly divides the board up into 16 4x4 smaller boards. And of course you can use regular stones for the moves.

That bit was easy. Taking the move on the 4x4x4 grid and placing the matching stones in the correct positions on the 4x4 grids would be easy for software but is hard for me. It seems that one stone in 4x4x4 generates 4 or 5 stones in 4x4: one for each of the main axes and one or two out of the four diagonals. I'm sure it will click soon. It was easy at college 40 years ago.

Paul

Hikaru vs a Godhand?

Oh what horrors causality bestows upon us mortals!

I'm about halfway through a 4x4x4 game now and can make some comments about it. First off, it's actually quite interesting. I dealt with the difficulty in imagining how the boards stacked up by just writing down a matrix of coordinates. So if I play the first black stone at 221, I just glance at my coordinate map and find the 5 points on the 4x4 grids that 221 maps to. It's fairly quick and positive that way, although it would be much quicker with software.

I play the game on a 4x4x4 grid, and with each move I place the corresponding 3-5 stones on the respective points in the 16 grids, one for each of the orthogonal ones and 0, 1 or 2 for the four diagonal choices.

So after about 30 moves, here's what it looks like on the 16 little boards.

Click Here To Show Diagram Code

[go]$$c This is a label for the diagram.
$$ ---------------------------------------
$$ | . . O . S . . X . S . . . . S . . . . |
$$ | . X X . S O O O . S X X O X S . O X . |
$$ | O O O . S . X X . S O X O X S X O X . |
$$ | . O . . S . O X . S . . X . S . O . . |
$$ | S S S S S S S S S S S S S S S S S S S |
$$ | . . . . S . O X . S . O X . S . . . . |
$$ | . . X . S . O X . S X X . X S . O X . |
$$ | . O O . S O . O X S X . X . S . . . . |
$$ | . . . . S . O . O S . X O . S . O . . |
$$ | S S S S S S S S S S S S S S S S S S S |
$$ | . . . . S O . O . S . . X . S . . . . |
$$ | O X X . S . O . O S O X . X S . . X . |
$$ | . . . . S O . O O S . O X X S O X . . |
$$ | . . . . S . O . . S X O X . S . . . . |
$$ | S S S S S S S S S S S S S S S S S S S |
$$ | . O . . S . . . . S . O X . S . . X . |
$$ | O . O . S X O X . S X X . X S . O . O |
$$ | . O . O S O . O X S . O X . S . O X . |
$$ | . . X . S . X X . S . . X . S . . . . |
$$ ---------------------------------------[/go]

7 black and 5 white stones have been captured.

Sometimes a move is legal on some of the boards and illegal on others. I decided that would mean the move couldn't be played at all. Trying to retake in a ko didn't work, as on one board it meant placing a black stone on top of a black stone that was already there.

I started off playing somewhat randomly on 2-2 points on the 4x4x4 grid. After a bit I decided to go for a definite move on a couple of boards at once, maybe capture a stone or at least atari. What happened was on those two boards that occurred as planned, but on the other two (maybe) boards I ended up putting stones in sometimes ridiculous positions.

You can see that some boards are played out, like the one in the bottom row second from the right. The one directly above it is interesting. The obvious vital point that captures is illegal (on other boards) for both players.

I'm probably not going to play this out any more, and I'm certainly not going to do a similar thing with another dimension up and four more go boards. The rules need to be tweaked a bit, but not too much. As I said, it's fairly interesting, and not too hard to put the equipment together as described.

Paul

Yawnguy, I am not sure how you are treating the extra dimensions, but I it is not the way I am treating it.

It is indeed possible to split a game of 4^n go into a number of games of 4^2 go, with the games being played simultaneously. However, I see 4^3 as being split into three combinations of four boards each, not of sixteen simultaneous combinations.




This makes for 12 games of 4x4, in each of which each stone has some liberties.

However, you seem to treat liberties as local to that board, which I do not agree with. Take, for instance, this 4x4x2 board:

Click Here To Show Diagram Code

[go]$$c
$$ -----------
$$ | . X O . |
$$ | . O X O |
$$ | . . O . |
$$ | . . . . |
$$ | S S S S |
$$ | . X X . |
$$ | . . X . |
$$ | . . . . |
$$ | . . . . |
$$ ----------[/go]

I would say that C8 and C3 are connected, and that C8 is thus not captured.. C9 is in atari, and white must connect at D9, as A9 would not threaten the black stones.


Similarly, 4^4 go can be split up into four 4x4x4 cubes (in four way), each of which can be split into four 4x4 slices (in three ways, but we should probably split each in the same way). This results in 16 4x4 boards at once, and at most 12*16 boards in total. Please do correct me if I am mistaken, but it seems like if we want to split to individual positions, we will end up with n! arrangements, n being the number of dimensions.


Reviewing your post, I think the reason you're treating things this way is because in tic-tac-toe, things really work that way. I would argue that in go, they do not: 2D go is not merely `a number of 1D go games'. Adding a third and fourth dimension simply add connections up, down, strangewards and charmwards (the terms I use for the fourth dimension), and thus on the single, `unsplit' goban, every stone has at most eight liberties, and will live or die according to them. How many liberties a stone has when you cut things into cubes, slices, and rows is irrelevant for it.

Hope that cleared things up. Sorry for the lack of a follow-up on this so far; I've been assuming there is no interest.