I’ve come up with a new board to play Go on. It’s inspired both by Toroidal Go, which is like playing Go on the surface of a Torus, and M. C. Escher’s Prentententoonstelling:
or rather, this Droste version of it, made by a group of Dutch mathematicians in 2003:
Let’s start with how Toroidal Go works, which you’re probably already familiar with. To make a Torus, we take a rectangle, and then glue the top and the bottom to each other, and the left and the right sides, in the following way:
Here, blue arrow 1 gets glued to blue arrow 1, red arrow 2 gets glued to red arrow 2, and so on.
If you were to follow the green dashed arrow off the top of the board, you’d enter the board again at the bottom, and if you follow the purple arrow to the left, you’d end up on the right side of the board.
If you make your rectangle out of highly elastic rubber, so that you can stretch it, then by glueing together the blue arrows you get a cylindrical tube, and by glueing the red arrows afterwards, you’d get a donut shape:
Getting the surface of the Escherian Goban
Now, let’s take a rectangle, and stretch it a bit, to get this funky rectangle:
It’s a nice shape, since we can put four copies of them together to get a large square with a small square in the middle:
And that’s where we will start glueing. We glue the large square inside the smaller square. Since real-world objects are usually not stretchy enough for this to work, you might not only have to bend your imaginary rubber sheet, but also your imagination a bit.
In other words, if we follow the big red Hi, drawn below in the lower left funky rectangle, downwards off the page, we’d end up in the top left funky rectangle plus that we’ve shrunken in size and rotated by 135 degrees! Amazing!
Now, imagine that we draw a couple of lines in our funky rectangle, then we make a funky grid on which we could play Go!
The grid I drew contains 24 intersections per funky rectangle, giving a total of 96 intersections, hence a game on this board should be about as crowded as a regular 9x9 board. To make a bit more comparisons:
|Total number of intersections||81||96|
|Intersections with 4 liberties||49||64|
|Intersections with 3 liberties||28||32|
|Intersections with 2 liberties||4||0|
Of course the interesting part is that there is not really a well-defined edge: I’ve made sure that no intersections with 3 liberties are placed next to each other. I therefore expect it to be harder to live, like in Toroidal Go, because most of the board functions more like the centre than like the edge.
A coordinate system is a bit tricky, since it’s not a regular grid, but I thought it might be easiest to give each of the four funky rectangles a letter, and then simply number the intersections. I’ve also coloured the areas, to make it easy to recognise which parts are copies of each other:
As an example of how it would look with some stones on the board, here is the first 9 moves of a hypothetical game:
I’m dying to try it out, anybody who wants to play a game against me, here on the forum?
The first game has started, against yebellz.
I’ve also created an interactive site that makes it a bit easier to actually play this. Quick and dirty for now.