How Many Different Types Of Symmetry In Go

Q. How many types of symmetry are there in Go Positions?

A. My answer is 26. I said 25 before but I was forgetting the empty pattern which is the only position which has all the symmetries including colour reversal symmetries.

Here are symbolic diagrams for all the non-empty pattern cases. (I hope I haven't made a mistake!)

The letters in the diagrams represent configurations of stones in an octant of the board Different letters mean different configurations. (Same letter means same configuration!) Lower case version of a letter means - the same configuration of stones but with Black/White reversed. Does this make any sense? Does anyone Care? Maybe useful for someone?

     C B
   D     A                                                     No Symmetries
   E     H
     F G

                                                               Horizontal
     B B         C B         B A         C B         C B       Vertical
   A     A     D     A     C     A     C     A     D     A     Rising Diagonal
   C     C     D     A     D     B     B     D     A     D     Falling Diagonal
     D D         C B         D C         A D         B C       Half Rotation

     B B         B A         A B                               Horizontal + Vertical + Half Rotation
   A     A     B     A     B     A                             Rising Diagonal + Falling Diagonal + Half Rotation
   A     A     A     B     A     B                             Quarter Rotation + Half Rotation
     B B         A B         B A

     A A
   A     A     Horizontal + Vertical + Rising Diagonal + Falling Diagonal + Half Rotation + Quarter Rotation
   A     A
     A A

                                                               Reverse Colour Horizontal
     b B         C B         B a         C B         C B       Reverse Colour Vertical
   a     A     D     A     C     A     c     A     D     A     Reverse Colour Rising Diagonal
   C     c     d     a     D     b     b     d     a     d     Reverse Colour Falling Diagonal
     D d         c b         d c         a D         b c       Reverse Colour Half Rotation

     b B         B B         b B       Reverse Colour Horizontal + Vertical + Reverse Colour Half Rotation
   a     A     A     A     a     A     Horizontal + Reverse Colour Vertical + Reverse Colour Half Rotation
   a     A     a     a     A     a     Reverse Colour Horizontal + Reverse Colour Vertical + Half Rotation
     b B         b b         B b

     B a         B a         B A       Reverse Colour Rising Diagonal + Reverse Colour Falling Diagonal + Half Rotation
   b     A     B     A     b     A     Reverse Colour Rising Diagonal + Falling Diagonal + Reverse Colour Half Rotation
   A     b     a     b     a     B     Rising Diagonal + Reverse Colour Falling Diagonal + Reverse Colour Half Rotation
     a B         A b         a b

     a B
   b     A                             Half Rotation + Reverse Colour Quarter Rotation
   A     b
     B a

     a a         a A         A a       Horizontal + Vertical + Reverse Colour Rising Diagonal + Reverse Colour Falling Diagonal + Half Rotation + Reverse Colour Quarter Rotation
   A     A     a     A     a     A     Reverse Colour Horizontal + Reverse Colour Vertical + Rising Diagonal + Falling Diagonal + Half Rotation + Reverse Colour Quarter Rotation
   A     A     A     a     A     a     Reverse Colour Horizontal + Reverse Colour Vertical + Reverse Colour Rising Diagonal + Reverse Colour Falling Diagonal + Half Rotation + Quarter Rotation
     a a         A a         a A

-- Paul Donnelly

Surely the number of possible forms of symmetry is just the set of 2D point groups (on an infinite board, it would of course be the set of all 2D space groups)?

Note: I haven't checked the above tabulation, which might be exactly that :)

- Andrew Walkingshaw, occasional crystallographer

Thanks for the reference to "2D point groups". I did a quick net search using this as a search phrase, but didn't find anything that could confirm how many symmetry types there are. Can 2D point groups include colour reversal? I remember a theory I skimmed through a few years ago called "Polya's Theory of Counting", which deals with counting the number of distinct instances of complex objects with various symmetries, however I lost my copy of the paper, and it seemed like it would require quite a serious effort to understand and correctly apply to a given situation.

- Paul Donnelly