Re: [computer-go] Big board. Torus ?
On 2/23/07, Heikki Levanto [EMAIL PROTECTED] wrote: Sure, but not all such boards are equivalent anyway! Add a stone to the board. Add another stone to one of its liberties. Add a third stone to any (empty) liberty of the last stone. There are three possibilities. Choose the one that maximises the liberties of the string. You have now defined a straight line. Continue this line until you meet a black stone (which must be part of the original line). I guess you meet the beginning of the line, where it all started. How big portion of the board is now filled with black stones? That can vary depending on the properties of the grid. In the simple case you have drawn a circle of a fairly small size (say 19). In another simple case you have filled the whole board, and used many more stones (say 361). In some cases you have filled half the available points, or some other fraction. How big will this fraction be on a totally random grid? What, exactly, do you mean by a totally random grid? There is no single obvious (to me) way of distributing vertexes between nodes. I can think of several interesting ways, but no single obvious way. a) start with a conventional board, add random wraps at the edges (makes for convenient visualization) b) start with an empty graph with n^2 nodes and pick random pairs of nodes and add a vertex between them if neither already has 4 vertexes (hard to visualize, risks disjoint boards) c) start with a conventional board, pick a random pairs of nodes and a a random vertex in each node. Switch the end points of the two vertexes if the result is not a disjoint graph. Repeat N times. ... It could easily be argued that only (a) results in a grid at all... cheers stuart ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
Re: [computer-go] Big board. Torus ?
Magnus Persson wrote: ... it is impossible to make eyes when attacks on the eyes has so many directions to escape. Every reasonable well played game will end in seki. I totally agree. In 2D a free stone has 4 liberties. In 3D, 6. In nD, 2n. The higher n, the less interesting. You could give 8 liberties to a stone by including the diagonal neighbors, but that would just produce that everything reasonable survives in seki. All games are a draw. 4 liberties is the magic number, but playing without edges mat be fun. Jacques. ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
Re: [computer-go] Big board. Torus ?
On Thu, Feb 22, 2007 at 07:19:52PM -0600, Matt Gokey wrote: Here is a thought experiment to test: define the board only logically using a graph (nodes and neighbor nodes). No topological shape and no mesh layout over any shape is needed. If all nodes have exactly four neighbors, there is no method or algorithm that you can run to find an edge. All nodes will look equivalent. Sure, but not all such boards are equivalent anyway! Add a stone to the board. Add another stone to one of its liberties. Add a third stone to any (empty) liberty of the last stone. There are three possibilities. Choose the one that maximises the liberties of the string. You have now defined a straight line. Continue this line until you meet a black stone (which must be part of the original line). I guess you meet the beginning of the line, where it all started. How big portion of the board is now filled with black stones? That can vary depending on the properties of the grid. In the simple case you have drawn a circle of a fairly small size (say 19). In another simple case you have filled the whole board, and used many more stones (say 361). In some cases you have filled half the available points, or some other fraction. How big will this fraction be on a totally random grid? -H -- Heikki Levanto In Murphy We Turst heikki (at) lsd (dot) dk ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
Re: [computer-go] Big board. Torus ?
The number of liberties is not the same measure as dimensionality. You need to look at a area/boundary ratio. At some point I adapted libEGO to hexagonal topology, and the game - Hex Go ( Ho? :-) ) was actually very interesting. Major features are: - almost no capture tactics - no ko - a lot of cut/connect tactics - a high ration strategy/tactics - interesting nakade :) Best, Łukasz On 2/23/07, Jacques Basaldúa [EMAIL PROTECTED] wrote: Magnus Persson wrote: ... it is impossible to make eyes when attacks on the eyes has so many directions to escape. Every reasonable well played game will end in seki. I totally agree. In 2D a free stone has 4 liberties. In 3D, 6. In nD, 2n. The higher n, the less interesting. You could give 8 liberties to a stone by including the diagonal neighbors, but that would just produce that everything reasonable survives in seki. All games are a draw. 4 liberties is the magic number, but playing without edges mat be fun. Jacques. ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/ ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
Re: [computer-go] Big board. Torus ?
On Wed, Feb 21, 2007 at 07:55:15PM -0600, Matt Gokey wrote: Whether it is a torus or not is irrelevant. The only thing that matters from a go game play perspective is the graph topology. If all points have 4 neighbors the actual physical shape or layout doesn't matter. There can still be subtle differences. In a standard torus, a ladder comes back to its starting point. If you twist the torus enough, it will miss, and fill the whole board... Hardly relevant to random players that don't understand ladders, but anyway... -H -- Heikki Levanto In Murphy We Turst heikki (at) lsd (dot) dk ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
Re: [computer-go] Big board. Torus ?
Quoting Tapani Raiko [EMAIL PROTECTED]: In 3D Go, you need a surface of stones to surround space but just a string of stones peeking in to ruin it. In normal 2D Go, you surround area by strings and ruin area by strings, so there is a nice balance. My guess is that Go in any other dimensionality than two would be dull. Playing on the surface of a ball, a torus, or a Klein bottle might be fun, though. I once played 4-D go 4x4x4x4 with myself which is of course very hard to visualize. But igonring that, the pfundamental roblem is that it is impossible to make eyes when attacks on the eyes has so many directions to escape. Every reasonable well played game will end in seki. -Magnus ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
Re: [computer-go] Big board. Torus ?
Heikki Levanto wrote: On Wed, Feb 21, 2007 at 07:55:15PM -0600, Matt Gokey wrote: Whether it is a torus or not is irrelevant. The only thing that matters from a go game play perspective is the graph topology. If all points have 4 neighbors the actual physical shape or layout doesn't matter. There can still be subtle differences. In a standard torus, a ladder comes back to its starting point. If you twist the torus enough, it will miss, and fill the whole board... Hardly relevant to random players that don't understand ladders, but anyway... I'm not sure I agree with this. I hypothesize that 2d, 3d, 4d, torus, or any other shape is completely irrelevant with regard to game play. The only thing that matters is the graph topology. A corollary is that on any board that is completely balanced at the beginning with identical number of neighbors for all nodes, any 1st play is equivalent and therefore optimal. ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
Re: [computer-go] Big board. Torus ?
I'm not sure I agree with this. I hypothesize that 2d, 3d, 4d, torus, or any other shape is completely irrelevant with regard to game play. The only thing that matters is the graph topology. it is true that the only thing that matters is graph topology. it is also true that graph topology cannot be separated from the actual topology of the surface. dimensionality of the embedding space is irrelevant -- topology of the embedded surface is quite important. you should be able to extract the topology of the graph from any embedded surface upon which it can be drawn and vice-versa. s. 8:00? 8:25? 8:40? Find a flick in no time with the Yahoo! Search movie showtime shortcut. http://tools.search.yahoo.com/shortcuts/#news ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
Re: [computer-go] Big board. Torus ?
If I take a plane, I can draw a 9x9 board on it or a 19x19 board on it. I can also draw the previously mentioned circular / cylindrical board on it. Could you explain how you propose to extract the topology of these, given only the fact that I have drawn them on a plane? excellent point. :) i overstated my point quite a bit. let me be more specific and more careful and say that if you draw a grid that completely covers the surface of, say, a torus, where every crossing point in the mesh represents a point on the board, you cannot hope to do so in a way that makes the board act like a board drawn with those same restrictions on, say, the surface of a sphere. there is torus-ness embedded in your game board that you can't remove. the same goes for a covering mesh on a projective plane, a sphere with two handles and one cusp, etc.. the topology of the surface is important. on the other hand, the full graph description of a board as a (V,E) set is all that is required to define a game board. in the other direction, there are (V,E) sets that can't be drawn such that the only intersections drawn are those that are board points while keeping the drawing on any one topological object. these will have to be drawn on a different topological object to have this property. i'll answer your objection by noting that if you draw the board as a covering mesh, you can definitely tell whether or not it was drawn on a cylinder or a torus or a sphere. s. Don't get soaked. Take a quick peak at the forecast with the Yahoo! Search weather shortcut. http://tools.search.yahoo.com/shortcuts/#loc_weather ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
Re: [computer-go] Big board. Torus ?
alain Baeckeroot wrote: Le jeudi 22 février 2007 14:11, Matt Gokey a écrit : The only thing that matters is the graph topology. A corollary is that on any board that is completely balanced at the beginning with identical number of neighbors for all nodes, any 1st play is equivalent and therefore optimal. Yes. But the round board at http://www.youdzone.com/go.html is not isotropic, it is a cylinder. You can build it with a square garden wire netting cut at 45°, and add one wire on each border to have 4 neighbors everywhere. If you start from any point and go straight you end on a border. If you start from a border and go straight you stay on the border. I don't understand. I think everyone is thinking too visually. What does straight mean in the context of go? Only liberties are meaningful. It is isotropic if you stop visualizing the shape and only consider the graph. Here is a thought experiment to test: define the board only logically using a graph (nodes and neighbor nodes). No topological shape and no mesh layout over any shape is needed. If all nodes have exactly four neighbors, there is no method or algorithm that you can run to find an edge. All nodes will look equivalent. If it were a cylinder, there would still be a top and bottom edge indicated by nodes with fewer neighbors. ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
Re: [computer-go] Big board. Torus ?
I don't understand. I think everyone is thinking too visually. What does straight mean in the context of go? Only liberties are meaningful. It is isotropic if you stop visualizing the shape and only consider the graph. I think straight would mean that when moving from one node to an adjacent one, you make the move that maximizes the distance from your origin. ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
Re: [computer-go] Big board. Torus ?
Le vendredi 23 février 2007 01:19, Matt Gokey a écrit : Here is a thought experiment to test: define the board only logically using a graph (nodes and neighbor nodes). No topological shape and no mesh layout over any shape is needed. If all nodes have exactly four neighbors, there is no method or algorithm that you can run to find an edge. All nodes will look equivalent. I was partly wrong, but i maintain this board is anisotropic : It contains squares and triangles, not equally spaced, all triangles are on the borders. Here is a simple algorithm to define the borders: Starting from one node and moving to the next, you can go to the Left, Front, Right or Back. - Insides nodes : if you go always to the Right, in 4 steps you are back to the initial position. - Near border nodes: there is one starting direction where you need only 3 steps to go back if you always turn to the Right - Border nodes: 9 steps. QED number of neighbors is not enought to define the topology, (i suspect this is well known ;-) So i maintain it is a cylinder, with 2 borders which correspond to our visual feeling. Alain. ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
Re: [computer-go] Big board. Torus ?
Tapani Raiko wrote: Matt Gokey wrote: I'm not sure I agree with this. I hypothesize that 2d, 3d, 4d, torus, or any other shape is completely irrelevant with regard to game play. The only thing that matters is the graph topology. A corollary is that on any board that is completely balanced at the beginning with identical number of neighbors for all nodes, any 1st play is equivalent and therefore optimal. It is true that the graph topology is the thing that matters, but having an identical number of neighbors for all nodes does not mean that the graphs are similar (isomorphic). For instance in the 3D diamond graph, each node (disregarding the edges) has 4 neighbors as usual, but there are 12 neighbor's neighbors, whereas normal Go board has only 8 (4 diagonals and 4 one-point jumps). I'd say there is a huge difference. OK, I see. That is of course a big and very significant difference. -So the graph topology is crucial not really the shape. -And if a board is empty and balanced with equal numbers of neighbors and uniform (e.g. same connectivity properties throughout) then any 1st play should be equivalent and therefore optimal. Right? A toroid board (defined as a normal square board with the top-bottom and left-right edges connected) would fit this condition, however, the round board does not appear to. Therefore they can't be isomorphic or structurally identical. When describing boards as mapped onto different shapes it would be helpful to describe the graph connectivity properties being imagined since so many different boards can be constructed on the same geometries. As for Don's original question about the toroid board, I think it does change the game significantly since it will be harder to make territory without edges and perhaps less interesting, but the first play is easy since any move is optimal and would not require running even one simulation ;-) ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
Re: [computer-go] Big board. Torus ?
Nick Apperson wrote: I considered making a version of go that plays with tetrahedral geometry. It is a 3D arrangment where all nodes have 4 neighbors and the angles between each are 109 degrees. Its connection properties though are very different because of the way it it layed out. Hence, I am going to have to disagree. Now I understand what you were saying here. But if what you mean is that all that matters is the graph representation of the go board, I will agree with you there. Well that _is_ really what I meant, I just wasn't thinking of all other possibilities and implications. ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
Re: [computer-go] Big board. Torus ?
Matt Gokey wrote: alain Baeckeroot wrote: Le jeudi 22 février 2007 14:11, Matt Gokey a écrit : The only thing that matters is the graph topology. A corollary is that on any board that is completely balanced at the beginning with identical number of neighbors for all nodes, any 1st play is equivalent and therefore optimal. Yes. But the round board at http://www.youdzone.com/go.html is not isotropic, it is a cylinder. You can build it with a square garden wire netting cut at 45°, and add one wire on each border to have 4 neighbors everywhere. If you start from any point and go straight you end on a border. If you start from a border and go straight you stay on the border. I don't understand. I think everyone is thinking too visually. What does straight mean in the context of go? Only liberties are meaningful. It is isotropic if you stop visualizing the shape and only consider the graph. You are right it is not isotropic - sorry - I didn't look at it closely enough. Here is a thought experiment to test: define the board only logically using a graph (nodes and neighbor nodes). No topological shape and no mesh layout over any shape is needed. If all nodes have exactly four neighbors, there is no method or algorithm that you can run to find an edge. All nodes will look equivalent. I was assuming the board was uniform or isotropic I guess when I wrote this. Mea culpa. ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
Re: [computer-go] Big board. Torus ?
Le mercredi 21 février 2007 02:10, Antonin Lucas a écrit : No need for those difficulties, you can play along this board : http://www.youdzone.com/go.html I think this is not a torus, even if each vertice has 4 neighbours. I can easily mentally transform this into a cylinder, with an rectangular lattice and additional connection on the borders to have 4 neighbours. But there is still a weird border topology, all direction are not equivalent. Alain ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
Re: [computer-go] Big board. Torus ?
Stuart A. Yeates wrote: On 2/21/07, alain Baeckeroot [EMAIL PROTECTED] wrote: Le mercredi 21 février 2007 02:10, Antonin Lucas a écrit: No need for those difficulties, you can play along this board : http://www.youdzone.com/go.html I think this is not a torus, even if each vertice has 4 neighbours. I can easily mentally transform this into a cylinder, with an rectangular lattice and additional connection on the borders to have 4 neighbours. I agree If this were a torus, there would be links between the inner ring and the outer ring of vertexes. Whether it is a torus or not is irrelevant. The only thing that matters from a go game play perspective is the graph topology. If all points have 4 neighbors the actual physical shape or layout doesn't matter. ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
Re: [computer-go] Big board. Torus ?
I have seen such a board for sale online. I would have to search to find it again. Cheers, David On 21, Feb 2007, at 9:29 PM, Nick Apperson wrote: I considered making a version of go that plays with tetrahedral geometry. It is a 3D arrangment where all nodes have 4 neighbors and the angles between each are 109 degrees. Its connection properties though are very different because of the way it it layed out. Hence, I am going to have to disagree. But if what you mean is that all that matters is the graph representation of the go board, I will agree with you there. - Nick ___ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
