The Inherent Symmetry of Lines of Action

The initial setup of Lines of Action is symmetrical. In fact, it contains at least 2 types of symmetry: rotational and reflectional.

Rotational symmetry is when you can turn an object and wind up with the same pattern. In LOA, you can rotate the board 180 degrees and still have the same opening setup. Reflectional symmetry is when you can divide an object in half, fold one half over the other and wind up with the same pattern.  It's as if a mirror was placed down the center of the object, and the mirror reflection was the same as the original half. LOA has axes of symmetry running through the center of the board horizontally, vertically and diagonally.

While this is all very good for math geeks, what does it mean for players? It means that the same game can be played 4 different ways in LOA. As an example, here are 4 pictures of my first “standard opening” that I discussed in the last post.

1. c8-c6 a2-c2

2. b1-b3

1. c1-c3 a7-c7
2. b8-b6

1. f1-f3 h7-f7
2. g8-g6

1. f8-f6 h2-f2
2. g1-g3

All of these are, technically, the same opening. Go has the same properties – the same game can be played in 4 different ways. Since Go has been popular for centuries, a standard has been established wherein the black player makes his first move in the upper right-hand corner from where he is sitting. LOA has no such standard, unfortunately. Therefore, it may be helpful to be aware of the way certain openings are symmetrically equivalent.

As a final example, here is the shortest LOA game playable without suicide moves, as created by Mannish Charosh in the 4 possible variations. The original is listed first.

1. d1-b3 h5-g4

2. b1-b4 g4xg1

3. b4-e7 a3xc1

4. b3-e6 h3xf1

5. e1-e5#

1. e1-g3 a5-b4

2. g1-g4 b4xb1

3. g4-d7 h3xf1

4. g3-d6 a3xc1

5. d1-d5#

1. d8-b6 h4-g5

2. b8-b5 g5xg8

3. b5-e2 a6xc8

4. b6-e3 h6xf8

5. e8-e4#

1. e8-g6 a4-b5

2. g8-g5 b5xb8

3. g5-d2 h6xf8

4. g6-d3 a6xc8

5. d8-d4#