Escherian Go

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:

image

or rather, this Droste version of it, made by a group of Dutch mathematicians in 2003:


Toroidal Go

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!


Escherian Go

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:

9x9 Escherian Board
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:

Black White
1. C22 2. B19
3. A7 4. A23
5. A22 6. A16
7. A15 8. B15
9. C11


Conclusion

I’m dying to try it out, anybody who wants to play a game against me, here on the forum?

[Update]
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.

23 Likes

Also, these can become as complicated as you’d like, this one has 396 intersections, being comparable to 19x19, and having a few more edges:

But why stop:

(I’m drawing these using the texture painter in Blender, by the way)

8 Likes

Ok

Black White
1. C8 2.
3 Likes

Looks very interestng; I’ll hopefully read this later.

2 Likes

I’ll respond with A20

Black White
1. C8 2. A20

We can make it team go if multiple people want to play :slight_smile:

Would you prefer with or without colours? With or without coordinates?

Also, I’ll start working on an interactive version :stuck_out_tongue:

5 Likes

I like the second visualization (with both colors and coordinates). I need all the help that I can get.

Might as well make it several (>2) teams with multiple voting players in each :wink:

We could have Democratic Diplomatic Escherian Go

  1. B15
4 Likes

This is harder than I thought already! I’ll be (dis)playing my move D22 after finishing the interactive version, probably makes for the best experience.

3 Likes

To ease the burden of doing it by hand, probably just one layer of nesting is enough.

However, if you are already going to go to the trouble of implementing it in code, might as well make it infinitely zoomable (both in and out) via the mouse wheel… :wink:

Of course, I kid, but know that a true pedant wouldn’t back down from the challenge :slight_smile:

  1. A16
3 Likes

looks sceptically, thinks “should I be worried?”

8 Likes

And we’re live!

  1. C8
  2. A20
  3. B15
  4. D22
  5. A16
  6. C20

5 Likes

Short Digression on why this is still Toroidal Go

Below is how we have glued our surfaces together.

Our material is very malleable, so we can just rotate the center square to align the arrows on each side.

Now, if we make a cut, this is the exact opposite of glueing things together:

But after cutting we see that this diagram below looks exactly like the Torus diagram from before, so were actually still just playing Toroidal Go, but without a regular grid.


Of course, we could also consider glueing the outside square to the inside square in reverse. This is sadly not possible in the actual world, since it requires 4 dimensions to work. The surface that results from this is called a Klein bottle.

Here’s a sketch of how a Klein bottle Escherian Goban could look like:

Notice that the R in the smaller copy is mirrored.

5 Likes

D7

That looks great, however all of the stones I place are black. How do I change color on mobile?

Sad :sob: that there is no infinite zooming.

2 Likes

awesome!

1 Like

Click twice :slight_smile:

D3


I recommend PC over mobile, by the way, since coordinates that you hover your mouse over are marked in red (in all copies), which makes it a lot easier to track the stones.

3 Likes

D20

Save and load by URL would also be nice

1 Like

This looks to me more as fractal go, since the board is endlessly repeating with smaller size.

I wonder though if that is someway meaningful: the only interaction between “original” stones (the bigger ones) and their infinite clones is the edge between “original” squares and their first copy. So having infinite copies doesn’t add anything to the game. Just one border could be enough.

Or maybe few lines inside and outside an annulus of fancy squares (which actually is very similar to how a torus would work).

Anyway it’s very good looking! :grinning_face_with_smiling_eyes:

4 Likes

B23

I’ve added saving and the stones now alternate between black and white.

https://vsotvep.github.io/escheriango.html?st=ApEtBoFwChGtHcDgDtHv

4 Likes

What is it? First I see it I think I drink too big. It’s too magical. Thank you.

12 Likes

It’s not really a fractal if you consider that the several differently-sized boards are identical to each other. I would interpret Fractal Go to be Go on an infinite board, where each intersection is separate from each other. With infinitely many territory, it will become hard to find good rules to determine the winner, though.

It is really the warped surface of a torus, so it’s not only similar to how a torus would work, it is how a torus would work. Take the basic square with a square in the middle, and let’s cut along the purple line:

When we straighten the purple line and curve the straight edges of the squares, we’d get the following four arches:

If we take the actual board, and cut out one of these arches:

We would get a shape like this:

Straighten the lines:

And put four of them next to each other and repeat on all sides, and you have the toroidal grid that we’re playing on:

I must admit that it’s a lot wider than I thought, preliminarily.

[Edit]

Oh yeah, I forgot about the offset, the corner directly above A1 in this straight diagram is of course B1, so actually it should look like this:

With the rectangular board being:

6 Likes

D1

https://vsotvep.github.io/escheriango.html?st=ApEtBoFwChGtDaHcDgDtHv

1 Like