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Diagonals

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Created page with "'''Home * Chess * Diagonals''' FILE:kleinbottle_inside.jpg|border|right|thumb|link=http://www.cs.berkeley.edu/%7Eug/slide/gallery/kleinbottle/index.shtml|..."
'''[[Main Page|Home]] * [[Chess]] * Diagonals'''

[[FILE:kleinbottle_inside.jpg|border|right|thumb|link=http://www.cs.berkeley.edu/%7Eug/slide/gallery/kleinbottle/index.shtml|[https://en.wikipedia.org/wiki/Nexus Nexus] of a [[Chessboard|chessboard]] [https://en.wikipedia.org/wiki/Klein_bottle Klein bottle] <ref>[http://www.kleinbottle.com/whats_a_klein_bottle.htm What is a Klein Bottle?]</ref> <ref>[http://www.cs.berkeley.edu/%7Eug/slide/gallery/kleinbottle/index.shtml Nerius Landys and Shad Roundy Klein Bottle Project]</ref> <ref>[http://www.geometrygames.org/TorusGames/ Torus Games] - players who master the [[Games|games]] on the [https://en.wikipedia.org/wiki/Torus Torus] may move on to try them on the more challenging Klein bottle</ref> ]]

'''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.

=Main Diagonal=
<fentt border="a1" style="font-size:24pt">.......(+)/......(+)./.....(+)../....(+).../...(+)..../..(+)...../.(+)....../(+).......</fentt>

=Square Mapping Notes=
A [[Flipping Mirroring and Rotating#Rotationby90degreesClockwise|90 degree rotation]] of the [[Chessboard]], as well as [[Flipping Mirroring and Rotating#FlipVertically|flipping vertically]] (reversed [[ranks]]) or (exclusive) [[Flipping Mirroring and Rotating#MirrorHorizontally|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 [[Square Mapping Considerations#LittleEndianRankFileMapping|little-endian rank-file mapping]].

=Square Difference=
Within a 0..63 square index range and the mentioned [[Square Mapping Considerations#LittleEndianRankFileMapping|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 [[Ranks|rank]] from [[Files|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'''
{| class="wikitable"
|-
! r/f
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
|-
! 7
| style="text-align:right;" | 7
| style="text-align:right;" | 6
| style="text-align:right;" | 5
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
|-
! 6
| style="text-align:right;" | 6
| style="text-align:right;" | 5
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | -1
|-
! 5
| style="text-align:right;" | 5
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | -1
| style="text-align:right;" | -2
|-
! 4
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | -1
| style="text-align:right;" | -2
| style="text-align:right;" | -3
|-
! 3
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | -1
| style="text-align:right;" | -2
| style="text-align:right;" | -3
| style="text-align:right;" | -4
|-
! 2
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | -1
| style="text-align:right;" | -2
| style="text-align:right;" | -3
| style="text-align:right;" | -4
| style="text-align:right;" | -5
|-
! 1
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | -1
| style="text-align:right;" | -2
| style="text-align:right;" | -3
| style="text-align:right;" | -4
| style="text-align:right;" | -5
| style="text-align:right;" | -6
|-
! 0
! 0
| style="text-align:right;" | -1
| style="text-align:right;" | -2
| style="text-align:right;" | -3
| style="text-align:right;" | -4
| style="text-align:right;" | -5
| style="text-align:right;" | -6
| style="text-align:right;" | -7
|}

'''7 + rank - file'''
{| class="wikitable"
|-
! r/f
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
|-
! 7
| style="text-align:right;" | 14
| style="text-align:right;" | 13
| style="text-align:right;" | 12
| style="text-align:right;" | 11
| style="text-align:right;" | 10
| style="text-align:right;" | 9
| style="text-align:right;" | 8
! 7
|-
! 6
| style="text-align:right;" | 13
| style="text-align:right;" | 12
| style="text-align:right;" | 11
| style="text-align:right;" | 10
| style="text-align:right;" | 9
| style="text-align:right;" | 8
! 7
| style="text-align:right;" | 6
|-
! 5
| style="text-align:right;" | 12
| style="text-align:right;" | 11
| style="text-align:right;" | 10
| style="text-align:right;" | 9
| style="text-align:right;" | 8
! 7
| style="text-align:right;" | 6
| style="text-align:right;" | 5
|-
! 4
| style="text-align:right;" | 11
| style="text-align:right;" | 10
| style="text-align:right;" | 9
| style="text-align:right;" | 8
! 7
| style="text-align:right;" | 6
| style="text-align:right;" | 5
| style="text-align:right;" | 4
|-
! 3
| style="text-align:right;" | 10
| style="text-align:right;" | 9
| style="text-align:right;" | 8
! 7
| style="text-align:right;" | 6
| style="text-align:right;" | 5
| style="text-align:right;" | 4
| style="text-align:right;" | 3
|-
! 2
| style="text-align:right;" | 9
| style="text-align:right;" | 8
! 7
| style="text-align:right;" | 6
| style="text-align:right;" | 5
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
|-
! 1
| style="text-align:right;" | 8
! 7
| style="text-align:right;" | 6
| style="text-align:right;" | 5
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
|-
! 0
! 7
| style="text-align:right;" | 6
| style="text-align:right;" | 5
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
| style="text-align:right;" | 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 [https://en.wikipedia.org/wiki/Nexus Nexus] in the center of the range:

'''(rank - file) & 15'''
{| class="wikitable"
|-
! r/f
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
|-
! 7
| style="text-align:right;" | 7
| style="text-align:right;" | 6
| style="text-align:right;" | 5
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
|-
! 6
| style="text-align:right;" | 6
| style="text-align:right;" | 5
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | 15
|-
! 5
| style="text-align:right;" | 5
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | 15
| style="text-align:right;" | 14
|-
! 4
| style="text-align:right;" | 4
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | 15
| style="text-align:right;" | 14
| style="text-align:right;" | 13
|-
! 3
| style="text-align:right;" | 3
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | 15
| style="text-align:right;" | 14
| style="text-align:right;" | 13
| style="text-align:right;" | 12
|-
! 2
| style="text-align:right;" | 2
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | 15
| style="text-align:right;" | 14
| style="text-align:right;" | 13
| style="text-align:right;" | 12
| style="text-align:right;" | 11
|-
! 1
| style="text-align:right;" | 1
! 0
| style="text-align:right;" | 15
| style="text-align:right;" | 14
| style="text-align:right;" | 13
| style="text-align:right;" | 12
| style="text-align:right;" | 11
| style="text-align:right;" | 10
|-
! 0
! 0
| style="text-align:right;" | 15
| style="text-align:right;" | 14
| style="text-align:right;" | 13
| style="text-align:right;" | 12
| style="text-align:right;" | 11
| style="text-align:right;" | 10
| style="text-align:right;" | 9
|}

=See also=
* [[Anti-Diagonals]]
* [[Ranks]]
* [[Files]]
* [[Squares]]
* [[Intersection Squares]]
* [[Nexus]] the chess program

=References=
<references />

'''[[Chess|Up one Level]]'''
Retrieved from "https://www.chessprogramming.org/Special:MobileDiff/1520"

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