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Xah Lee, 2003-12
I have been thinking about the go board game played on various regular tiling. First experiment is the honeycomb grid (which has 3 connections per intersection), then i'm going to try triangular grid (6 connections), then i'll try some regular or semi-regular tiling with, many of which has 4 connections just like the rectangular grid but different topology.
In designing a honeycomb board, there are few issues:
above: Illustration that shows different board design issues. The left two shows two different types of edges. Also note that the overall shape can be hexagonal or triangular. And, some board have a vertex at the center, while others have a hexagon.
I played with my brother two games, and it is quite interesting. The game is overly tactical, since every stone has only 3 connections. It's extremely easy to kill.
above: Two games in progress. White to move on the left, black to move on the right. The edge points has strong advantage, because it takes 5 moves to capture in practice, while it takes 3 in center positions. If one tries to capture the edge point, he can be captured before he succeed. So, the base points are somewhat invincible until endgame. It is most easy to make two eyes based on the edge. A good strategy seems to play on the edge first, then snake in. Perhaps the board should be modified so that the angular edge cannot be played. This can be done by either cutting the edge flat or allow only stones on intersections. This way, all points has 3 liberties.
In drawing these boards, some math questions came up. Look at the triangular boards, which has sides from 2 to 7. Some of these has a intersection at the center, others a hexagon. What sizes will have a central intersection?
It turns out, boards with a vertex in center has sides of 2+n*3 and 3+n*3. All other sizes of triangular board will have a hexagon in the center instead.
Triangular shaped board with n sides has 1+ 4*n + n^2 number of points (intersections) counting the non-intersection points on the sides. If edges are not allowed to play, then the formula is -2+n+n^2. (See Derivations)
A hexagonal board with n sides has 6*n^2 number of points. If edges are not allowed to play, then -6*n + 6*n^2.
Triangular grid has six liberties. The hexagonal grid studies shows that best overall shape is probably as round as possible, so that edge-special play is minimized. So, we can just think about boards with a hexagonal border. The triangular grid can be deformed to a rectangular shape, equivalent to a rectangular grid with diagonals in one direction.
In contrast to the hex grid, triangular grid has 6 liberties, which makes it hard to capture.
above: One game in progress. (black moves first) Left: white's turn. The two white stones on the left probably cannot be saved. Right: game ended. Black's turn.
In sharp contrast to the honeycomb grid, game on triangular grid is slow. In the honeycomb grid, life-and-dead situations comes up often, and one has to think a lot. In triangular grid, this is not so. A good strategy in the honeycomb grid seems to be spreading pieces all over the board, while a good strategy on the triangular grid seems to be playing in just one connected group (snaking).
A game in sequence. Left, white to move. Middle: black to move and black wins. Right: the game ends. It seems that whoever builds a wall enclosing larger area wins. So, on a small board, playing center seems advantageous. The board should be increased in sized to increase its strategic appeal. The center spot might also be removed. One might also try to remove several spots on the board, like punching in holes. This however, is basically changing the topology of the board, which a systematic exploration can be taken. (That is, using regular tiling as grids.)
This board with edge size n triangles has -3 n + 3 n^2 + 1 points. (See Derivations)
above: Image shows how triangular lattice and the hexagonal lattice are duals of each other. Also note the star shaped lattice, formed by the mid-points of the triangular grid, or the mid-points of the hexagonal grid. The star-lattice makes a grid where each intersection has 4 degrees.
Here is a game on a grid called 33344 tiling that has 5 liberties per intersection. (game played using the freedgo↗ program, highly recommended.)
above: Black to play. c2.5 or d3 are good moves.
above: In this game, black just played b9.5. What should white do? Note the life-and-death situation on the right side. Both the upper right and lower right white groups could be in danger. White can either play b3.5 to gain massive territory, or lose that and play h9 to be on the safe side. Which is the better move? (see answer here)
Playing go on 33344 tiling feels more similar to traditional go, at least in local fight. Note the board above is a 9x9 board with a total of 81 points. This grid is equivalent to the square grid plus diagonal links on every other row.
The following are various tiling. (image courties of Joao Pedro Neto↗)
4;
4;
3,4;
5;
5;
The following is “round go” by Harald Schwarz. (from http://www.escape.de/users/shusaku/euro96/issue3/roundgo.htm , url defunct as of 2006-08-27.)
As we know, many games are played on various regular grids, such as checker, chinese checker, gomoku (tic-tac-toe, connect four), go. The cellular automata Game Of Life, it is played on a rectangular grid. Many variations has been experimented, such as triangular grid and torus board. We can think about a systematical exploration of variations by changing the grid topology. The grid can be any of periodic or semi-periodic tiling, and we can change the global topology from a plane to sphere, torus, and all the possible 2 dimensional surfaces including non-orientable ones like cross-surface (projective plane). The grid (connection) topology can be abstractly defined, to give us more variation and systematic exploration.
In writing a program that play these games, one may also make it modularized so that different boards, rules, topology, and AI can be written as a plug-in.
In B Grunbaum and G C Shephard's “Tilings and Patterns” book (amazon.com↗), it was proven that if we consider the topology of Archimedean tilings, there are 13 types only. (i may be off in terminology, see the book for detail) One should note that these are planar graphs. That is, when we consider only the topology aspects of a tilling, we are essentially regarding it as a graph, and a planar one. When we explore different boards of go or cellular automata, we are interested in the topological properties of the grid. So, for a abstract systematic exploration of possible (regular) grids, we have to consider not just (regular) planar graphs, but also non-planar ones. For example, consider the standard rectangular grid of the go game. Now, if suppose each point are connected to its diagonal neighbor as well. Now we have a (regular) graph where each vertex has 8 degrees. This grid cannot be drawn on a plane without crossings. So, here we are dealing with non-planar graphs that are “regular” in some sense. I think this is totally unexplored field.
Joao Pedro Neto exams go on various boards. ~2003. http://www.di.fc.ul.pt/~jpn/gv/boards.htm
Free Go (software) by Lewey Geselowitz. ~2004. It plays go on various 2D and 3D grids. Highly recommended. http://www.leweyg.com/lc/freedgo.html
Henry Segerman has some notes on Go on a 3D diamond grid, also a PC program. http://www.nrinstruments.demon.co.uk/diamond/diamintro.html
An article on go on hex and triangular grid, by Joao Pedro Neto and Bill Taylor. Abstract games issue 13 Spring 2003: “A Family for Go”. (http://www.abstractgamesmagazine.com/go_variants.html url defunct)
Related essays:
Page created: 2003-12. © 2003 by Xah Lee. (excluding mirrored pages or images.)
