# Comments/Ratings for a Single Item

All I did to the original diagram was change its formatting to make it easier to read. I got the idea for putting a box around the main board from Pritchard, who did the same, but unlike him, I left all the inner coordinates in.

While on its own the original diagram on this page was a bit obscure, in conjunction with Fergus' circular diagrams it really clarifies the rationale behind Nadvorney's interpretation of the diagonal move; a bit of thought also reveals why Miller's reasoning in keeping the bishop on its own colour is flawed: if the bishop stays on its own colour you would expect a rook stepping over the pole to change colour as it does on a normal square‐cell board, whereas here (or on any spherical/Klein‐bottle‐shaped board with a multiple of four files) it doesn't. On e.g. a 10‐file board, Miller's reasoning would line up with Nadvorney's.

As for Chess on the Dot, the change in the diagonal's handedness at the poles also keeps it on one colour (on a board of this parity), but isn't stirctly necessary for a closed loop: Nadvorney's version (as can be seen on its diagram) does it just as well, and even Miller's manages, albeit via a much more circuitous route.

Fwiw, here's the original diagram as salvaged from the Internet Archive:

```
c7 d7 e7 f7 g7 h7 a7 b7 c7 d7 e7 f7
c8 d8 e8 f8 g8 h8 a8 b8 c8 d8 e8 f8
g8 h8 a8 b8 c8 d8 e8 f8 g8 h8 a8 b8
g7 h7 a7 b7 c7 d7 e7 f7 g7 h7 a7 b7
g6 h6 a6 b6 c6 d6 e6 f6 g6 h6 a6 b6
g5 h5 a5 b5 c5 d5 e5 f5 g5 h5 a5 b5
g4 h4 a4 b4 c4 d4 e4 f4 g4 h4 a4 b4
g3 h3 a3 b3 c3 d3 e3 f3 g3 h3 a3 b3
g2 h2 a2 b2 c2 d2 e2 f2 g2 h2 a2 b2
g1 h1 a1 b1 c1 d1 e1 f1 g1 h1 a1 b1
c1 d1 e1 f1 g1 h1 a1 b1 c1 d1 e1 f1
c2 d2 e2 f2 g2 h2 a2 b2 c2 d2 e2 f2
```

I have updated this page with diagrams and more detailed descriptions.

I started on adding an introduction, but it's late, and I will have to wait until tomorrow to continue.

I have a better idea of how this works now. I was describing a torus in my last comment. Unlike a torus, the north and south poles are not adjacent on a sphere. I will write up a better description and provide some more visual diagrams when I have the time.

The description of this game is very confusing. The middle eight rows make sense. These include a regular Chess board with two files from the other side on each side. This much is like Cylindrical Chess.

```
g8 h8 a8 b8 c8 d8 e8 f8 g8 h8 a8 b8
g7 h7 a7 b7 c7 d7 e7 f7 g7 h7 a7 b7
g6 h6 a6 b6 c6 d6 e6 f6 g6 h6 a6 b6
g5 h5 a5 b5 c5 d5 e5 f5 g5 h5 a5 b5
g4 h4 a4 b4 c4 d4 e4 f4 g4 h4 a4 b4
g3 h3 a3 b3 c3 d3 e3 f3 g3 h3 a3 b3
g2 h2 a2 b2 c2 d2 e2 f2 g2 h2 a2 b2
g1 h1 a1 b1 c1 d1 e1 f1 g1 h1 a1 b1
```

But the two rows at the top and bottom do not fit the same pattern. Here is the pattern that would make sense to me:

```
g2 h2 a2 b2 c2 d2 e2 f2 g2 h2 a2 b2
g1 h1 a1 b1 c1 d1 e1 f1 g1 h1 a1 b1
g8 h8 a8 b8 c8 d8 e8 f8 g8 h8 a8 b8
g7 h7 a7 b7 c7 d7 e7 f7 g7 h7 a7 b7
g6 h6 a6 b6 c6 d6 e6 f6 g6 h6 a6 b6
g5 h5 a5 b5 c5 d5 e5 f5 g5 h5 a5 b5
g4 h4 a4 b4 c4 d4 e4 f4 g4 h4 a4 b4
g3 h3 a3 b3 c3 d3 e3 f3 g3 h3 a3 b3
g2 h2 a2 b2 c2 d2 e2 f2 g2 h2 a2 b2
g1 h1 a1 b1 c1 d1 e1 f1 g1 h1 a1 b1
g8 h8 a8 b8 c8 d8 e8 f8 g8 h8 a8 b8
g7 h7 a7 b7 c7 d7 e7 f7 g7 h7 a7 b7
```

I do not understand why the pattern on the page is not this.

The reason people are calling it a 'Klein bottle' rather than a sphere is because of the weird situation around the poles.

If you draw the eight spaces surrounding each pole as triangles, then the rook shouldn't pass -- it only crosses edges. If you draw them as squares and try to connect opposites, each pole becomes a 'cross-cap' rather than a point. A sphere with two cross-caps is topologically equivalent to a Klein bottle.

(Alternatively each pole could be a double-handle, then you have an orientable surface of genus 4.)

Contrary to what some have written, this game is indeed spherical. Firstly, the right and left edges are connected to each other, making a cylinder (or annulus for the mathematically pedantic). Then imagine shrinking the top and bottom circles on this cylinder until they become mere points. This means that the top eight 'squares' are really triangles, all joining at the top edge of the board, which is now a point. The same goes for the bottom. The black pieces are then in a circle around the north pole and the white ones around the south pole. For examples of what this looks like, see: http://69.90.174.250/photos/thumb_small/69461/69461,1165605152,1.jpg http://www.cs.unc.edu/~olano/papers/primitive/sphere_check.gif http://upload.wikimedia.org/wikipedia/commons/b/b3/UV_mapping_checkered_sphere.png (although they all have too many squares, it should be eight from pole to pole and eight around the equator) For reference, the ranks become lines of latitude and the files lines of longitude. A rook travelling up a file comes back down the 'opposite' file four squares to the side. The other pieces are harder to work out, but I assure you that this entry has got the geometry correct. I independently invented the rules about five years ago and made my own board out of a 10cm diameter polystyrene ball, with pins as pieces. This worked quite well and only took a few hours to make with a black marker and a piece of string for measurements. Before making it, I determined the rules by drawing the board as a series of concentric circles (roughly like circular chess, but 8x8 not 4x16), then make them go all the way into the center point, which is one of the poles. This can give a playable flat board for the game, and help you see how going through the poles works (although it is pretty poor for helping with the colour that begins around the outside of the circle). As to the game, it is quite similar to cylindrical chess. Indeed, it is identical until you go through a pole, which cannot happen until at least a few pieces leave the home squares. The end is quite different as the poles become vacated and are convenient for travel. My only rules quibble would be that you don't need to modify the castling rule. Indeed, I would play this and cylindrical chess without any castling as the entire point of it is void when there are no corners of the board, and it would never have been invented on such a board.

I think we need to start teaching topology in elementary schools.

Of course, chess on a Klein bottle has got to be at least as cool as chess on a sphere, right?

One assumes that a magnetic version (or perhaps a velcro version) exists. I have big trouble seeing the board layout. Starting position is the same as square chess. I think the polar problem is too complex. Better would be degenerate triangles perhaps.

Maybe a diagram of the starting position would be nice... Or at least, some notation...

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That's a good point. On a 10-file board, the files would surround the pole in a way that makes diagonal movement more intuitive, and once the board was defined, it would be easy to make spherical adaptations of various 10x8 variants like Grotesque Chess and other related games or 10x10 variants like Grand Chess.