/usr/share/perl5/Math/PlanePath/CellularRule57.pm is in libmath-planepath-perl 113-1.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 | # Copyright 2011, 2012, 2013 Kevin Ryde
# This file is part of Math-PlanePath.
#
# Math-PlanePath is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by the
# Free Software Foundation; either version 3, or (at your option) any later
# version.
#
# Math-PlanePath is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
# or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
# for more details.
#
# You should have received a copy of the GNU General Public License along
# with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
package Math::PlanePath::CellularRule57;
use 5.004;
use strict;
use vars '$VERSION', '@ISA';
$VERSION = 113;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
use Math::PlanePath::Base::Generic
'round_nearest';
use Math::PlanePath::CellularRule54;
*_rect_for_V = \&Math::PlanePath::CellularRule54::_rect_for_V;
# uncomment this to run the ### lines
#use Smart::Comments;
use constant class_y_negative => 0;
use constant n_frac_discontinuity => .5;
use constant parameter_info_array =>
[ { name => 'mirror',
display => 'Mirror',
type => 'boolean',
default => 0,
description => 'Mirror to "rule 99" instead.',
},
Math::PlanePath::Base::Generic::parameter_info_nstart1(),
];
use constant sumxy_minimum => 0; # triangular X>=-Y so X+Y>=0
use constant diffxy_maximum => 0; # triangular X<=Y so X-Y<=0
use constant dx_maximum => 3;
use constant dy_minimum => 0;
use constant dy_maximum => 1;
sub absdx_minimum {
my ($self) = @_;
return ($self->{'mirror'} ? 0 : 1);
}
use constant dsumxy_maximum => 3; # straight East dX=+3
use constant ddiffxy_maximum => 3; # straight East dX=+3
use constant dir_maximum_dxdy => (-1,0); # supremum, West and dY=+1 up
#------------------------------------------------------------------------------
sub new {
my $self = shift->SUPER::new (@_);
if (! defined $self->{'n_start'}) {
$self->{'n_start'} = $self->default_n_start;
}
return $self;
}
# left
# even y=3 5
# 5 12
# 7 23
# 9 38
# [1,2,3,4], [5,12,23,38]
#
# N = (2 d^2 + d + 2)
# = (2*$d**2 + $d + 2)
# = ((2*$d + 1)*$d + 2)
# d = -1/4 + sqrt(1/2 * $n + -15/16)
# = (-1 + 4*sqrt(1/2 * $n + -15/16)) / 4
# = (sqrt(8*$n-15)-1)/4
# with Y=2*d+1
# row 19, d=9
# N=173 to N=181 is 9 cells rem=0..8 is d-1
# 1/3 section 3 cells rem=0,1,2 floor((d-1)/3)
# 2/3 section 6 cells
# right solid N=191 to N=200 is 10 of is rem<d
#
# row 21, d=10
# 1/3 section 4 cells rem=0,1,2,3 floor((d-1)/3)
# 2/3 section 6 cells
#
# row 23, d=11
# 1/3 section 4 cells rem=0,1,2,3 floor((d-1)/3)
# 2/3 section 7 cells
#
# row 25, d=12
# 2/3 section 8 cells
#
# row 27, d=13
# 2/3 section 8 cells
#
# row 29, d=14
# 2/3 section 9 cells floor(2d/3)
#
# row 31, d=15
# 2/3 section 10 cells floor(2d/3)
#
#
# row 18 d=8
# odd 1/3 section 4 cells (d+4)/3
#
# row 20 d=9
# odd 1/3 section 4 cells
#
# row 22 d=10
# odd 1/3 section 4 cells
#
# row 23 d=11
# odd 1/3 section 5 cells
sub n_to_xy {
my ($self, $n) = @_;
### CellularRule57 n_to_xy(): $n
$n = $n - $self->{'n_start'} + 1; # to N=1 basis, and warn if $n undef
my $frac;
{
my $int = int($n);
$frac = $n - $int;
$n = $int; # BigFloat int() gives BigInt, use that
if (2*$frac >= 1) {
$frac -= 1;
$n += 1;
}
# -0.5 <= $frac < 0.5
### assert: 2*$frac >= -1
### assert: 2*$frac < 1
}
if ($n <= 1) {
if ($n == 1) {
return (0,0);
} else {
return;
}
}
# d is the two-row group number, y=2*d+1, where n belongs
#
my $d = int ((sqrt(8*$n-15)-1)/4);
$n -= ((2*$d + 1)*$d + 2); # remainder
### $d
### remainder: $n
if ($self->{'mirror'}) {
if ($n <= $d) {
### right solid: $n
return ($frac + $n - 2*$d - 1,
2*$d+1);
}
$n -= $d+1;
if ($n < int(2*$d/3)) {
### right 2/3: $n
return ($frac + int(3*$n/2) - $d + 1,
2*$d+1);
}
$n -= int(2*$d/3);
if ($n < int(($d+2)/3)) {
### left 1/3: $n
return ($frac + 3*$n + ((2+$d)%3),
2*$d+1);
}
$n -= int(($d+2)/3);
if ($n < $d) {
### left solid: $n
return ($frac + $n + $d+2,
2*$d+1);
}
$n -= $d;
if ($n < int((2*$d+5)/3)) {
### odd 2/3: $n
return ($frac + int((3*$n)/2) - $d + - 1,
2*$d+2);
}
$n -= int((2*$d+5)/3);
### odd 1/3: $n
return ($frac + 3*$n + ($d%3) + 1,
2*$d+2);
} else {
if ($n < $d) {
### left solid: $n
return ($frac + $n - 2*$d - 1,
2*$d+1);
}
$n -= $d;
if ($n < int(($d+2)/3)) {
### left 1/3: $n
return ($frac + 3*$n - $d + 1,
2*$d+1);
}
$n -= int(($d+2)/3);
if ($n < int(2*$d/3)) {
### right 2/3: $n
return ($frac + $n + int(($n+(-$d%3))/2) + 1,
2*$d+1);
}
$n -= int(2*$d/3);
if ($n <= $d) {
### right solid: $n
return ($frac + $d + $n + 1,
2*$d+1);
}
$n -= $d+1;
if ($n < int(($d+4)/3)) {
### odd 1/3: $n
return ($frac + 3*$n - $d - 1,
2*$d+2);
}
$n -= int(($d+4)/3);
### odd 2/3: $n
return ($frac + $n + int(($n+((1-$d)%3))/2) + 1,
2*$d+2);
}
}
sub xy_to_n {
my ($self, $x, $y) = @_;
$x = round_nearest ($x);
$y = round_nearest ($y);
### CellularRule57 xy_to_n(): "$x,$y"
if ($y < 0
|| $x < -$y
|| $x > $y) {
### outside pyramid region ...
return undef;
}
if ($self->{'mirror'}) {
# mirrored, rule 99
if ($y % 2) {
my $d = ($y+1)/2;
### odd row, solids, d: $d
if ($x < -$d) {
return ($y+1)*$y/2 + $x + 1 + $self->{'n_start'};
}
if ($x < 0) {
### mirror left 2 of 3 ...
if (($x += $d+2) % 3) {
return ($y+1)*$y/2 + $x-int($x/3) - $d + $self->{'n_start'} - 1;
}
} elsif ($x > $d) {
return ($y+1)*$y/2 + $x - $d + $self->{'n_start'};
} else {
### mirror right 1 of 3 ...
$x += 2-$d;
unless ($x % 3) {
return ($y+1)*$y/2 + $x/3 + $self->{'n_start'};
}
}
} else {
### even row, sparse ...
my $d = $y/2;
if ($x >= 0) {
### mirror sparse right 1 of 3 ...
if ($x <= $d # only to half way
&& (($x -= $d) % 3) == 0) {
return ($y+1)*$y/2 + $x/3 + $self->{'n_start'};
}
} else { # $x < 0
### mirror sparse left 2 of 3 ...
if ($x >= -$d # only to half way
&& (($x += $d+1) % 3)) {
return ($y+1)*$y/2 + $x-int($x/3) - $d + $self->{'n_start'} - 1;
}
}
}
} else {
# unmirrored, rule 57
if ($y % 2) {
my $d = ($y+1)/2;
### odd row, solids, d: $d
if ($x <= -$d) {
### solid left ...
if ($x < -$d) { # always skip the -$d cell
return ($y+1)*$y/2 + $x + 1 + $self->{'n_start'};
}
} elsif ($x <= 0) {
### 1 of 3 ...
unless (($x += $d+1) % 3) {
return ($y+1)*$y/2 + $x/3 - $d + $self->{'n_start'};
}
} elsif ($x >= $d) {
### solid right ...
return ($y+1)*$y/2 + $x - $d + $self->{'n_start'};
} else {
### 2 of 3 ...
$x += 1-$d;
if ($x % 3) {
return ($y+1)*$y/2 + $x-int($x/3) + $self->{'n_start'};
}
}
} else {
### even row, sparse ...
my $d = $y/2;
if ($x > 0) {
### right 2 of 3 ...
if ($x <= $d # goes to half way only
&& (($x -= $d+1) % 3)) {
return ($y+1)*$y/2 + $x-int($x/3) + 1 + $self->{'n_start'};
}
} else { # $x <= 0
### left 1 of 3 ...
if (($x += $d) >= 0 # goes to half way only
&& ! ($x % 3)) {
return ($y+1)*$y/2 + $x/3 - $d + $self->{'n_start'};
}
}
}
}
return undef;
}
# left edge ((2*$d + 1)*$d + 2)
# where y=2*d+1
# d=floor((y-1)/2)
# left N = (2*floor((y-1)/2) + 1)*floor((y-1)/2) + 2
# = (yodd + 1)*yodd/2 + 2
# not exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
### CellularRule57 rect_to_n_range(): "$x1,$y1, $x2,$y2"
($x1,$y1, $x2,$y2) = _rect_for_V ($x1,$y1, $x2,$y2)
or return (1,0); # rect outside pyramid
my $zero = ($x1 * 0 * $y1 * $x2 * $y2); # inherit bignum
$y1 -= ! ($y1 % 2);
$y2 -= ! ($y2 % 2);
return ($zero + ($y1 < 1
? $self->{'n_start'}
: ($y1-1)*$y1/2 + 1 + $self->{'n_start'}),
$zero + ($y2+2)*($y2+1)/2 + $self->{'n_start'});
}
1;
__END__
=for stopwords straight-ish Ryde Math-PlanePath ie hexagonals 18-gonal Xmax-Xmin Nleft Nright OEIS
=head1 NAME
Math::PlanePath::CellularRule57 -- cellular automaton 57 and 99 points
=head1 SYNOPSIS
use Math::PlanePath::CellularRule57;
my $path = Math::PlanePath::CellularRule57->new;
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
X<Wolfram, Stephen>This is the pattern of Stephen Wolfram's "rule 57"
cellular automaton
=over
L<http://mathworld.wolfram.com/ElementaryCellularAutomaton.html>
=back
arranged as rows
=cut
# math-image --path=CellularRule57 --all --output=numbers --size=132x50
=pod
51 52 53 54 55 56 10
38 39 40 41 42 43 44 45 46 47 48 49 50 9
33 34 35 36 37 8
23 24 25 26 27 28 29 30 31 32 7
19 20 21 22 6
12 13 14 15 16 17 18 5
9 10 11 4
5 6 7 8 3
3 4 2
2 1
1 <- Y=0
-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9
X<Triangular numbers>The triangular numbers N=10,15,21,28,etc, k*(k+1)/2,
make a 1/2 sloping diagonal upwards.
On rows with odd Y there's a solid block at either end then 1 of 3 cells to
the left and 2 of 3 to the right of the centre. On even Y rows there's
similar 1 of 3 and 2 of 3 middle parts, but without the solid ends. Those 1
of 3 and 2 of 3 are successively offset so as to make lines going up towards
the centre as can be seen in the following plot.
=cut
# math-image --text --path=CellularRule57 --all
=pod
*********** * * * * * ** ** ** ************
* * * * ** ** ** **
********** * * * * ** ** ** ***********
* * * * * ** ** **
********* * * * ** ** ** **********
* * * * ** ** **
******** * * * * ** ** *********
* * * ** ** **
******* * * * ** ** ********
* * * * ** **
****** * * ** ** *******
* * * ** **
***** * * * ** ******
* * ** **
**** * * ** *****
* * * **
*** * ** ****
* * **
** * * ***
* **
* * **
* *
*
*
=head2 Mirror
The C<mirror =E<gt> 1> option gives the mirror image pattern which is "rule
99". The point numbering shifts but the total points on each row is the
same.
=cut
# math-image --path=CellularRule57,mirror=1 --all --output=numbers --size=132x50
=pod
51 52 53 54 55 56 10
38 39 40 41 42 43 44 45 46 47 48 49 50 9
33 34 35 36 37 8
23 24 25 26 27 28 29 30 31 32 7
19 20 21 22 6
12 13 14 15 16 17 18 5
9 10 11 4
5 6 7 8 3
3 4 2
2 1
1 <- Y=0
-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9
=head2 N Start
The default is to number points starting N=1 as shown above. An optional
C<n_start> can give a different start, in the same pattern. For example to
start at 0,
=cut
# math-image --path=CellularRule57,n_start=0 --all --output=numbers --size=75x8
# math-image --path=CellularRule57,n_start=0,mirror=1 --all --output=numbers --size=75x8
=pod
n_start => 0
22 23 24 25 26 27 28 29 30 31
18 19 20 21
11 12 13 14 15 16 17
8 9 10
4 5 6 7
2 3
1
0
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::CellularRule57-E<gt>new ()>
=item C<$path = Math::PlanePath::CellularRule57-E<gt>new (mirror =E<gt> $bool, n_start =E<gt> $n)>
Create and return a new path object.
=item C<($x,$y) = $path-E<gt>n_to_xy ($n)>
Return the X,Y coordinates of point number C<$n> on the path.
=item C<$n = $path-E<gt>xy_to_n ($x,$y)>
Return the point number for coordinates C<$x,$y>. C<$x> and C<$y> are each
rounded to the nearest integer, which has the effect of treating each cell
as a square of side 1. If C<$x,$y> is outside the pyramid or on a skipped
cell the return is C<undef>.
=back
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::CellularRule>,
L<Math::PlanePath::CellularRule54>,
L<Math::PlanePath::CellularRule190>,
L<Math::PlanePath::PyramidRows>
L<http://mathworld.wolfram.com/ElementaryCellularAutomaton.html>
=head1 HOME PAGE
L<http://user42.tuxfamily.org/math-planepath/index.html>
=head1 LICENSE
Copyright 2011, 2012, 2013 Kevin Ryde
This file is part of Math-PlanePath.
Math-PlanePath is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by the Free
Software Foundation; either version 3, or (at your option) any later
version.
Math-PlanePath is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for
more details.
You should have received a copy of the GNU General Public License along with
Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
=cut
|