rslt ← {x} (u ##.Cut case) y ⍝ Cut operator.
From Roger Hui:
Monadic Cases:
u Cut 0 ⊢y applies u to y after reversing y along each axis; it is equivalent to
(⍉⎕io,⍪-⍴y) u Cut 0 ⊢y .
The "fret" 1⌷y (the leading major cell of y) marks the start of an interval of
major cells of y ; the phrase u Cut 1 ⊢y applies u to each such interval. The
phrase u Cut ¯1 ⊢y differs only in that frets are excluded from the result. In
u Cut 2 and u Cut ¯2 the fret is the last major cell, and marks the ends of
intervals.
The monads u Cut 3 and u Cut ¯3 apply u to tessellation by maximal cubes, that
is, they are defined by their dyadic cases using the left argument (⍴⍴y)⍴⌊/⍴y.
Dyadic Cases:
x u Cut 0 ⊢y applies u to a rectangle or cuboid of y with one vertex at the
point in y indexed by v←1⌷x and with the opposite vertex determined as follows:
the dimension is |2⌷x , but the rectangle extends back from v along any axis j
for which the index j⌷v is negative. Finally, the order of the selected cells is
reversed along each axis k for which k⌷2⌷x is negative. If x is a vector or
scalar, it is treated as the matrix ⍉⎕io,⍪x .
The frets in the dyadic cases 1, ¯1, 2, and ¯2 are determined by the 1s in bool-
ean vector x ; an empty vector x and non-zero ≢y indicates the entire of y. If
x is the scalar 0 or 1 it is treated as (≢y)⍴x . In general, boolean vector j⊃x
specifies how axis j is to be cut, with a scalar treated as (j⌷⍴y)⍴j⊃x.
u Cut 3 and u Cut ¯3 yield (possibly overlapping) tessellations. x Cut ¯3 ⊢y
applies u to each complete rectangle of size |2⌷x beginning at integer multiples
of (each scalar of) the movement vector 1⌷x . As in u Cut 0 , reversal occurs
along each axis for which the size 2⌷x is negative. The case of a vector or
scalar x is equivalent to ⍉1,⍪x , and therefore provides a complete tessellation
of size x . The case u Cut 3 differs in that shards of length less than |2⌷x are
included.
Refs:
Iverson, K.E., Rationalized APL, 1983, Section K.:
http://www.jsoftware.com/papers/RationalizedAPL.htm
Hui, R.K.W., Some Uses of { and }, 1987, Section 3.2.:
http://www.jsoftware.com/papers/from.htm
Iverson, K.E., A Dictionary of APL, 1987, m⍤v.:
http://www.jsoftware.com/papers/APLDictionary.htm
Hui, R.K.W., and K.E. Iverson, J Introduction and Dictionary, 2011, Cut (;.):
http://www.jsoftware.com/help/dictionary/d331.htm
Examples:
⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝ 0-Cut
⍝ The 0-cut selects rectangular subarrays, reversing each axis for which the
⍝ specified length is negative.
⎕←x←4 10⍴⍳40
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
⍝ cut at indices 2 3 with sizes 3 4:
(⍉2 3,⍪3 4) ⊂Cut 0 ⊢x
┌───────────┐
│13 14 15 16│
│23 24 25 26│
│33 34 35 36│
└───────────┘
⍝ cut at indices 2 3 with sizes 3 ¯4 (the ¯4 indicates reversal of axis 2):
(⍉2 3,⍪3 ¯4) ⊂Cut 0 ⊢x
┌───────────┐
│16 15 14 13│
│26 25 24 23│
│36 35 34 33│
└───────────┘
⍝ cut at indices 1 1 (default) with sizes 3 ¯4 (the ¯4 indicates reversal of
⍝ axis 2):
3 ¯4 ⊂Cut 0 ⊢x
┌───────────┐
│ 4 3 2 1│
│14 13 12 11│
│24 23 22 21│
└───────────┘
⍝ reverse each axis:
⊢Cut 0 ⊢x
40 39 38 37 36 35 34 33 32 31
30 29 28 27 26 25 24 23 22 21
20 19 18 17 16 15 14 13 12 11
10 9 8 7 6 5 4 3 2 1
⍝ cut at indices 2 ¯2 with sizes 3 6 (the ¯2 indicates negative indexing):
(⍉2 ¯2,⍪3 6) ⊢Cut 0 ⊢x
14 15 16 17 18 19
24 25 26 27 28 29
34 35 36 37 38 39
⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝ 1 ¯1-Cut and 2 ¯2-Cut
⍝ The 1- and 2-cut partition according a "fret", 1s in a boolean left argument
⍝ in the dyadic case or the leading major cell of the right argument in the
⍝ monadic case. Frets specify the starts of partitions in the 1-cut and the ends
⍝ in the 2-cut. The ¯1- and ¯2-cuts differ in excluding the frets.
⍝ cut on leading blanks; apply ⊂ to each cut:
⊂Cut 1 ⊢' Cogito, ergo sum.'
┌────────┬─────┬─────┐
│ Cogito,│ ergo│ sum.│
└────────┴─────┴─────┘
⍝ cut on leading blanks; apply {⊂⌽⍵} to each cut:
{⊂⌽⍵}Cut 1 ⊢' Cogito, ergo sum.'
┌────────┬─────┬─────┐
│,otigoC │ogre │.mus │
└────────┴─────┴─────┘
⍝ cut on leading blanks; apply ⊂ to each cut, excluding the leading cell:
⊂Cut ¯1 ⊢' Cogito, ergo sum.'
┌───────┬────┬────┐
│Cogito,│ergo│sum.│
└───────┴────┴────┘
⍝ cut on leading 1s in the left argument; apply ⊂ to each cut:
1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 ⊂Cut 1 ⊢' Cogito, ergo sum.'
┌────────┬─────┬─────┐
│ Cogito,│ ergo│ sum.│
└────────┴─────┴─────┘
⍝ cut on leading blanks; apply ⊂ to each cut:
⊢Cut 1 ⊢' Cogito, ergo sum.'
Cogito,
ergo
sum.
⍝ cut on leading blanks; apply ⊢ to each cut, excluding the leading cell (the
⍝ common vtom (vector to matrix) utility):
⊢Cut ¯1 ⊢' Cogito, ergo sum.'
Cogito,
ergo
sum.
⍝ cut on leading 1s in the left argument (←→ right argument because of the ⍨);
⍝ apply ≢ to each cut:
≢Cut 1⍨ 1 0 1 0 0
2 3
⍝ cut on leading 1s in the left argument; apply ≢ to each cut, excluding the
⍝ leading cell:
≢Cut ¯1⍨ 1 0 1 0 0
1 2
⍝ cut on trailing /; apply ⊂ to each cut:
⊂Cut 2 ⊢'Cogito,/ergo/sum./'
┌────────┬─────┬─────┐
│Cogito,/│ergo/│sum./│
└────────┴─────┴─────┘
⍝ cut on trailing 1s in the left argument (which specify the _ends_ of the cuts)
⍝ apply ⊂ to each cut:
1 0 1 0 0 1 0 0 ⊂Cut 2 ⊢'chthonic'
┌─┬──┬───┐
│c│ht│hon│
└─┴──┴───┘
⍝ cut on trailing 1s in the left argument (which specify the _ends_ of the cuts)
⍝ apply +⌿ to each cut:
1 0 1 0 0 1 0 0 +⌿Cut 2 ⊢8 3⍴⍳24
1 2 3
11 13 15
39 42 45
⍝ cut on leading 1s in the left argument; apply ⊂ to each cut:
x←5 3⍴⍳15
1 0 1 0 0 ⊂Cut 1 ⊢x
┌─────┬────────┐
│1 2 3│ 7 8 9│
│4 5 6│10 11 12│
│ │13 14 15│
└─────┴────────┘
⍝ cut using 1 0 1 0 0 on axis 0 and 1 1 0 on axis 1 (not yet implemented by the
⍝ model):
(1 0 1 0 0) (1 1 0) ⊂Cut 1 ⊢x
┌──┬─────┐
│1 │2 3 │
│4 │5 6 │
├──┼─────┤
│ 7│ 8 9│
│10│11 12│
│13│14 15│
└──┴─────┘
⍝ cut using ⍬ on axis 0 and 1 1 0 on axis 1 (not yet implemented by the model):
⍬ (1 1 0) ⊂Cut 1 ⊢x
┌──┬─────┐
│ 1│ 2 3│
│ 4│ 5 6│
│ 7│ 8 9│
│10│11 12│
│13│14 15│
└──┴─────┘
⍝ cut using ⍬ on axis 0 (not yet implemented by the model):
⍬ ⊂Cut 1 ⊢x
┌────────┐
│ 1 2 3│
│ 4 5 6│
│ 7 8 9│
│10 11 12│
│13 14 15│
└────────┘
⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝⍝ 3 ¯3-Cut
⍝ The 3-cut provides tessellation.
⍝ The ¯3-cut differs in excluding shards, cells less than the full size.
⎕←x←5 7⍴⍳35
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31 32 33 34 35
⍝ tessellate using movements 2 1 and sizes 3 2:
(2 2⍴2 1 3 2) ⊂Cut 3 ⊢x
┌─────┬─────┬─────┬─────┬─────┬─────┬──┐
│ 1 2│ 2 3│ 3 4│ 4 5│ 5 6│ 6 7│ 7│
│ 8 9│ 9 10│10 11│11 12│12 13│13 14│14│
│15 16│16 17│17 18│18 19│19 20│20 21│21│
├─────┼─────┼─────┼─────┼─────┼─────┼──┤
│15 16│16 17│17 18│18 19│19 20│20 21│21│
│22 23│23 24│24 25│25 26│26 27│27 28│28│
│29 30│30 31│31 32│32 33│33 34│34 35│35│
├─────┼─────┼─────┼─────┼─────┼─────┼──┤
│29 30│30 31│31 32│32 33│33 34│34 35│35│
└─────┴─────┴─────┴─────┴─────┴─────┴──┘
⍝ tessellate using movements 2 1 and sizes ¯3 2
⍝ (¯3 indicates reversal of axis 1 of each cut):
(2 2⍴2 1 ¯3 2) ⊂Cut 3 ⊢x
┌─────┬─────┬─────┬─────┬─────┬─────┬──┐
│15 16│16 17│17 18│18 19│19 20│20 21│21│
│ 8 9│ 9 10│10 11│11 12│12 13│13 14│14│
│ 1 2│ 2 3│ 3 4│ 4 5│ 5 6│ 6 7│ 7│
├─────┼─────┼─────┼─────┼─────┼─────┼──┤
│29 30│30 31│31 32│32 33│33 34│34 35│35│
│22 23│23 24│24 25│25 26│26 27│27 28│28│
│15 16│16 17│17 18│18 19│19 20│20 21│21│
├─────┼─────┼─────┼─────┼─────┼─────┼──┤
│29 30│30 31│31 32│32 33│33 34│34 35│35│
└─────┴─────┴─────┴─────┴─────┴─────┴──┘
⍝ tessellate using movements 2 1 and sizes 3 ¯2
⍝ (¯2 indicates reversal of axis 2 of each cut):
(2 2⍴2 1 3 ¯2) ⊂Cut 3 ⊢x
┌─────┬─────┬─────┬─────┬─────┬─────┬──┐
│ 2 1│ 3 2│ 4 3│ 5 4│ 6 5│ 7 6│ 7│
│ 9 8│10 9│11 10│12 11│13 12│14 13│14│
│16 15│17 16│18 17│19 18│20 19│21 20│21│
├─────┼─────┼─────┼─────┼─────┼─────┼──┤
│16 15│17 16│18 17│19 18│20 19│21 20│21│
│23 22│24 23│25 24│26 25│27 26│28 27│28│
│30 29│31 30│32 31│33 32│34 33│35 34│35│
├─────┼─────┼─────┼─────┼─────┼─────┼──┤
│30 29│31 30│32 31│33 32│34 33│35 34│35│
└─────┴─────┴─────┴─────┴─────┴─────┴──┘
⍝ tessellate using movements 1 1 (default) and sizes ¯3 2:
¯3 2 ⊂Cut 3 ⊢x
┌─────┬─────┬─────┬─────┬─────┬─────┬──┐
│15 16│16 17│17 18│18 19│19 20│20 21│21│
│ 8 9│ 9 10│10 11│11 12│12 13│13 14│14│
│ 1 2│ 2 3│ 3 4│ 4 5│ 5 6│ 6 7│ 7│
├─────┼─────┼─────┼─────┼─────┼─────┼──┤
│22 23│23 24│24 25│25 26│26 27│27 28│28│
│15 16│16 17│17 18│18 19│19 20│20 21│21│
│ 8 9│ 9 10│10 11│11 12│12 13│13 14│14│
├─────┼─────┼─────┼─────┼─────┼─────┼──┤
│29 30│30 31│31 32│32 33│33 34│34 35│35│
│22 23│23 24│24 25│25 26│26 27│27 28│28│
│15 16│16 17│17 18│18 19│19 20│20 21│21│
├─────┼─────┼─────┼─────┼─────┼─────┼──┤
│29 30│30 31│31 32│32 33│33 34│34 35│35│
│22 23│23 24│24 25│25 26│26 27│27 28│28│
├─────┼─────┼─────┼─────┼─────┼─────┼──┤
│29 30│30 31│31 32│32 33│33 34│34 35│35│
└─────┴─────┴─────┴─────┴─────┴─────┴──┘
⍝ tessellate using movements 1 1 (default) and sizes ¯3 2, excluding shards:
¯3 2 ⊂Cut ¯3 ⊢x
┌─────┬─────┬─────┬─────┬─────┬─────┐
│15 16│16 17│17 18│18 19│19 20│20 21│
│ 8 9│ 9 10│10 11│11 12│12 13│13 14│
│ 1 2│ 2 3│ 3 4│ 4 5│ 5 6│ 6 7│
├─────┼─────┼─────┼─────┼─────┼─────┤
│22 23│23 24│24 25│25 26│26 27│27 28│
│15 16│16 17│17 18│18 19│19 20│20 21│
│ 8 9│ 9 10│10 11│11 12│12 13│13 14│
├─────┼─────┼─────┼─────┼─────┼─────┤
│29 30│30 31│31 32│32 33│33 34│34 35│
│22 23│23 24│24 25│25 26│26 27│27 28│
│15 16│16 17│17 18│18 19│19 20│20 21│
└─────┴─────┴─────┴─────┴─────┴─────┘
⍝ movements 1 1 (default) and sizes 3 3, excluding shards; apply , to each cut:
life←{1⊖1⌽(-⍴⍵)↑5 6 7∊⍨(3 3 ,Cut ¯3⊢⍵)+.×9⍴1,⍨4⍴2}
b←10 10⍴0 ⋄ b[2;3]←b[3;4]←b[4;2 3 4]←1 ⍝ "glider"
{'.x'[1+life⍣⍵⊢b]}¨¯1+⍳10 ⍝ game of life for 10 generations
┌──────────┬──────────┬──────────┬──────────┬──────────┬──────────┬──────────┬──────────┬──────────┬──────────┐
│..........│..........│..........│..........│..........│..........│..........│..........│..........│..........│
│..x.......│..........│..........│..........│..........│..........│..........│..........│..........│..........│
│...x......│.x.x......│...x......│..x.......│...x......│..........│..........│..........│..........│..........│
│.xxx......│..xx......│.x.x......│...xx.....│....x.....│..x.x.....│....x.....│...x......│....x.....│..........│
│..........│..x.......│..xx......│..xx......│..xxx.....│...xx.....│..x.x.....│....xx....│.....x....│...x.x....│
│..........│..........│..........│..........│..........│...x......│...xx.....│...xx.....│...xxx....│....xx....│
│..........│..........│..........│..........│..........│..........│..........│..........│..........│....x.....│
│..........│..........│..........│..........│..........│..........│..........│..........│..........│..........│
│..........│..........│..........│..........│..........│..........│..........│..........│..........│..........│
│..........│..........│..........│..........│..........│..........│..........│..........│..........│..........│
└──────────┴──────────┴──────────┴──────────┴──────────┴──────────┴──────────┴──────────┴──────────┴──────────┘
See also: life
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