The following are tables of the group of transpositions (reflections and rotations) of the square, using two sets of labels:
] |
|.@|: |
|."1@|. |
|."1@|: |
|. |
|: |
|."1 |
|.@|:@|. |
|.@|: |
|."1@|. |
|."1@|: |
] |
|.@|:@|. |
|. |
|: |
|."1 |
|."1@|. |
|."1@|: |
] |
|.@|: |
|."1 |
|.@|:@|. |
|. |
|: |
|."1@|: |
] |
|.@|: |
|."1@|. |
|: |
|."1 |
|.@|:@|. |
|. |
|. |
|: |
|."1 |
|.@|:@|. |
] |
|.@|: |
|."1@|. |
|."1@|: |
|: |
|."1 |
|.@|:@|. |
|. |
|."1@|: |
] |
|.@|: |
|."1@|. |
|."1 |
|.@|:@|. |
|. |
|: |
|."1@|. |
|."1@|: |
] |
|.@|: |
|.@|:@|. |
|. |
|: |
|."1 |
|.@|: |
|."1@|. |
|."1@|: |
] |
] |
\: |
\:@\: |
/:@|. |
|. |
/: |
/:@\: |
\:@|. |
\: |
\:@\: |
/:@|. |
] |
\:@|. |
|. |
/: |
/:@\: |
\:@\: |
/:@|. |
] |
\: |
/:@\: |
\:@|. |
|. |
/: |
/:@|. |
] |
\: |
\:@\: |
/: |
/:@\: |
\:@|. |
|. |
|. |
/: |
/:@\: |
\:@|. |
] |
\: |
\:@\: |
/:@|. |
/: |
/:@\: |
\:@|. |
|. |
/:@|. |
] |
\: |
\:@\: |
/:@\: |
\:@|. |
|. |
/: |
\:@\: |
/:@|. |
] |
\: |
\:@|. |
|. |
/: |
/:@\: |
\: |
\:@\: |
/:@|. |
] |
] |
] |
identity |
|.@|: |
\: |
rotate counterclockwise 90° |
|."1@|. |
\:@\: |
rotate counterclockwise 180° |
|."1@|: |
/:@|. |
rotate counterclockwise 270° |
|. |
|. |
reflect along x-axis |
|: |
/: |
reflect along main diagonal |
|."1 |
/:@\: |
reflect along y-axis |
|.@|:@|. |
\:@|. |
reflect along back diagonal |
The second set of labels are due to considerations discovered independently by Thomson [1979], Hui [1981], and Benkard and Seebe [1983]:
{= and i."1&1 are an inverse pair, mapping integer permutation vectors to boolean permutation matrices and vice versa. Let F be a composition of ] \: /: |. , a function on permutations, and T be a composition of ] |: |. |."1 , a transposition of the square. Identify F and T , if
(F -: T&.({=) ) p
(T -: F&.(i."1&1)) mIn other words, the group may be viewed as a group of transpositions of the square, or isomorphically as a group of functions on permutations.
For example:
NB. reflect along y-axis
F=: /:@\:
T=: |."1
] p=: ?.~ 5
1 4 0 3 2
({=) p
0 1 0 0 0
0 0 0 0 1
1 0 0 0 0
0 0 0 1 0
0 0 1 0 0
T ({=) p
0 0 0 1 0
1 0 0 0 0
0 0 0 0 1
0 1 0 0 0
0 0 1 0 0
i."1&1 T ({=) p
3 0 4 1 2
F p
3 0 4 1 2
(F -: T&.({=)) p
1
] m=: ({=) p
0 1 0 0 0
0 0 0 0 1
1 0 0 0 0
0 0 0 1 0
0 0 1 0 0
i."1&1 m
1 4 0 3 2
F i."1&1 m
3 0 4 1 2
({=) F i."1&1 m
0 0 0 1 0
1 0 0 0 0
0 0 0 0 1
0 1 0 0 0
0 0 1 0 0
T m
0 0 0 1 0
1 0 0 0 0
0 0 0 0 1
0 1 0 0 0
0 0 1 0 0
(T -: F&.(i."1&1)) m
1The tables are a compact presentation of numerous identities involving functions ] |: |. |."1 on square matrices, or ] \: /: |. on permutations: The entry (<i,j){G is equivalent to the result of composing (<i,0){G and (<0,j){G . For example:
i,j |
row |
column |
composition |
simplification |
1 6 |
\: |
/:@\: |
\:@/:@\: |
/: |
|.@|: |
|."1 |
|.@|:@:(|."1) |
|: |
|
5 5 |
/: |
/: |
/:@/: |
] |
|: |
|: |
|:@|: |
] |
|
2 1 |
\:@\: |
\: |
\:@\:@\: |
/:@|. |
|."1@|. |
|.@|: |
|."1@|.@|.@|: |
|."1@|: |
|
2 2 |
\:@\: |
\:@\: |
\:@\:@\:@\: |
] |
|."1@|. |
|."1@|. |
|."1@|.@:(|."1)@|. |
] |
References
Benkard, J. Philip, and John N. Seebe, Reflections on Grades, APL83 Conference Proceedings, 1983-04-10.
Hui, Roger, The N Queens Problem, APL Quote Quad, Volume 11, Number 3, 1981-03.
Thomson, Norman D., The Geometric Primitives of APL, APL79 Conference Proceedings, 1978-05-30.
See also
Contributed by RogerHui. Portions of the text previously appeared as part of Some Uses of { and } by Roger Hui, APL87 Conference Proceedings, May 10-14, 1987.