Home * Chess * Diagonals
kleinbottle_inside.jpg

Diagonals
go from south-west to north-east on a chess board, the main diagonal is a1/h8. There are 15 diagonals, with line-length from 1 to 8. A Diagonal is monochrome, all their squares are either white or black.
Nexus of a chessboard Klein bottle [1][2][3]

Square Mapping Notes

A 90 degree rotation of the Chessboard, as well as flipping vertically (reversed ranks) or (exclusive) mirroring horizontally (reversed files), change the roles of diagonals and anti-diagonals. However, we define the main diagonal on the chess board from a1/h8 and the main anti-diagonal from h1\a8. Whether the square difference of neighbored squares on a diagonal or anti-diagonal is either 7 or 9, depends on the square mapping. We further rely on little-endian rank-file mapping.

Square Difference

Within a 0..63 square index range and the mentioned square mapping (a1 = 0), the difference of two neighbored squares (if any) on a diagonal is nine.

Enumeration

If we follow a diagonal from south-west (a1) to north-east (h8) step by step, we increment the rank, and increment the file, which yields in same difference. Thus, subtracting rank from file indices (or vice versa) enumerates all Diagonals, unfortunately with negative numbers involved, which can easily adjusted by adding seven.
  • The main diagonal a1/h8 with index 0 and length 8.
  • The diagonals a8/ and h1/ with length 1, are index 7 and -7 respectively.
  • All even indices are the diagonals with dark squares
    rank - file
\f  0  1  2  3  4  5  6  7
r_________________________
7 | 7  6  5  4  3  2  1  0
6 | 6  5  4  3  2  1  0 -1
5 | 5  4  3  2  1  0 -1 -2
4 | 4  3  2  1  0 -1 -2 -3
3 | 3  2  1  0 -1 -2 -3 -4
2 | 2  1  0 -1 -2 -3 -4 -5
1 | 1  0 -1 -2 -3 -4 -5 -6
0 | 0 -1 -2 -3 -4 -5 -6 -7
 
    7 + rank - file
\f  0  1  2  3  4  5  6  7
r_________________________
7 |14 13 12 11 10  9  8  7
6 |13 12 11 10  9  8  7  6
5 |12 11 10  9  8  7  6  5
4 |11 10  9  8  7  6  5  4
3 |10  9  8  7  6  5  4  3
2 | 9  8  7  6  5  4  3  2
1 | 8  7  6  5  4  3  2  1
0 | 7  6  5  4  3  2  1  0

Alternative Enumeration

Some alternative enumeration of diagonals to keep the main-diagonal index 0, by anding the difference with 15. This yields in a 0..15 range with 8 as gap or Nexus in the center of the range:
   (rank - file) & 15
\f  0  1  2  3  4  5  6  7
r_________________________
7 | 7  6  5  4  3  2  1  0
6 | 6  5  4  3  2  1  0 15
5 | 5  4  3  2  1  0 15 14
4 | 4  3  2  1  0 15 14 13
3 | 3  2  1  0 15 14 13 12
2 | 2  1  0 15 14 13 12 11
1 | 1  0 15 14 13 12 11 10
0 | 0 15 14 13 12 11 10  9

See also


References

  1. ^ What is a Klein Bottle?
  2. ^ Nerius Landys and Shad Roundy Klein Bottle Project
  3. ^ Torus Games - players who master the games on the Torus may move on to try them on the more challenging Klein bottle

What links here?


Up one Level