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# Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde |
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# This file is part of Math-PlanePath. |
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# |
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# Math-PlanePath is free software; you can redistribute it and/or modify |
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# it under the terms of the GNU General Public License as published by the |
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# Free Software Foundation; either version 3, or (at your option) any later |
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# version. |
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# |
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# Math-PlanePath is distributed in the hope that it will be useful, but |
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# WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY |
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# or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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# for more details. |
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# |
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# You should have received a copy of the GNU General Public License along |
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# with Math-PlanePath. If not, see . |
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# http://mathworld.wolfram.com/ElementaryCellularAutomaton.html |
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# |
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# Loeschian numbers strips on the right ... |
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package Math::PlanePath::CellularRule190; |
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use 5.004; |
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use strict; |
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use vars '$VERSION', '@ISA'; |
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$VERSION = 128; |
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use Math::PlanePath; |
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*_sqrtint = \&Math::PlanePath::_sqrtint; |
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@ISA = ('Math::PlanePath'); |
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use Math::PlanePath::Base::Generic |
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'round_nearest'; |
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use Math::PlanePath::CellularRule54; |
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*_rect_for_V = \&Math::PlanePath::CellularRule54::_rect_for_V; |
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# uncomment this to run the ### lines |
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# use Smart::Comments; |
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use constant class_y_negative => 0; |
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use constant n_frac_discontinuity => .5; |
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use constant parameter_info_array => |
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[ { name => 'mirror', |
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display => 'Mirror', |
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type => 'boolean', |
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default => 0, |
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description => 'Mirror to "rule 246" instead.', |
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}, |
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Math::PlanePath::Base::Generic::parameter_info_nstart1(), |
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]; |
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57
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sub x_negative_at_n { |
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my ($self) = @_; |
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return $self->n_start + 1; |
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} |
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use constant sumxy_minimum => 0; # triangular X>=-Y so X+Y>=0 |
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use constant diffxy_maximum => 0; # triangular X<=Y so X-Y<=0 |
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use constant dx_maximum => 2; # across gap |
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use constant dy_minimum => 0; |
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use constant dy_maximum => 1; |
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use constant absdx_minimum => 1; |
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use constant dsumxy_maximum => 2; # straight East dX=+2 |
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use constant ddiffxy_maximum => 2; # straight East dX=+2 |
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use constant dir_maximum_dxdy => (-1,0); # supremum, West except dY=+1 |
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#------------------------------------------------------------------------------ |
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sub new { |
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1
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1827
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my $self = shift->SUPER::new (@_); |
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if (! defined $self->{'n_start'}) { |
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$self->{'n_start'} = $self->default_n_start; |
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} |
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return $self; |
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} |
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# 31 32 33 34 35 36 37 38 39 40 |
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# 22 23 24 25 26 27 28 29 30 |
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# 15 6 17 18 19 20 21 |
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# 9 10 11 12 13 14 |
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# 5 6 7 8 |
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# 2 3 4 |
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# 1 |
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# |
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# even y = [ 0, 2, 4, 6 ] |
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# N = [ 1, 5, 15, 31 ] |
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# Neven = (3/4 y^2 + 1/2 y + 1) |
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# = (3y + 2)*y/4 + 1 |
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# = ((3y + 2)*y + 4) /4 |
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# = (3 (y/2)^2 + (y/2) + 1) |
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# = (3*(y/2) + 1)*(y/2) + 1 |
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# |
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# odd y = [ 1, 3, 5,7 ] |
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# N = [ 2,9,22,41 ] |
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# Nodd = (3/4 y^2 + 1/2 y + 3/4) |
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# = ((3y+2)*y+ 3) / 4 |
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# |
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# pair even d = [0,1,2,3] |
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# N = [ 1, 5, 15, 31 ] |
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# Npair = (3 d^2 + d + 1) |
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# d = -1/6 + sqrt(1/3 * $n + -11/36) |
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# = [ -1 + sqrt(1/3 * $n + -11/36)*6 ] / 6 |
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# = [ -1 + sqrt(1/3 * $n*36 + -11/36*36) ] / 6 |
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# = [ -1 + sqrt(12n-11) ] / 6 |
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# |
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sub n_to_xy { |
112
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140
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140
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1
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10297
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my ($self, $n) = @_; |
113
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### CellularRule190 n_to_xy(): $n |
114
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115
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140
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261
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$n = $n - $self->{'n_start'} + 1; # to N=1 basis, and warn if $n undef |
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140
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210
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my $frac; |
117
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{ |
118
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140
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187
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my $int = int($n); |
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220
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119
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183
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$frac = $n - $int; |
120
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140
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183
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$n = $int; # BigFloat int() gives BigInt, use that |
121
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140
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336
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if (2*$frac >= 1) { |
122
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0
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$frac -= 1; |
123
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$n += 1; |
124
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} |
125
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# now -0.5 <= $frac < 0.5 |
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### assert: 2*$frac >= -1 || $n!=$n || $n+1==$n |
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### assert: 2*$frac < 1 || $n!=$n || $n+1==$n |
128
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} |
129
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130
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140
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100
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258
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if ($n < 1) { |
131
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17
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return; |
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} |
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134
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# d is the two-row number, ie. d=2*y, where n belongs |
135
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# start of the two-row group is nbase = 3 d^2 + d + 1 |
136
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# |
137
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134
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343
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my $d = int ((_sqrtint(12*$n-11) - 1) / 6); |
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134
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247
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$n -= ((3*$d + 1)*$d + 1); # remainder within two-row |
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### $d |
140
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### remainder: $n |
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134
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100
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258
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if ($n <= 3*$d) { |
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# 3d+1 many points in the Y=0,2,4,6 etc even row |
143
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121
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$d *= 2; # y=2*d |
144
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79
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100
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263
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return ($frac + $n + int(($n + ($self->{'mirror'} ? 2 : 0))/3) - $d, |
145
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$d); |
146
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} else { |
147
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# 3*d many points in the Y=1,3,5,7 etc odd row, using 3 in 4 cells |
148
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55
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91
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$n -= 3*$d+1; # remainder 0 upwards into odd row |
149
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55
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83
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$d = 2*$d+1; # y=2*d+1 |
150
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55
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165
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return ($frac + $n + int($n/3) - $d, |
151
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$d); |
152
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} |
153
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} |
154
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155
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sub xy_to_n { |
156
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1628
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1628
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1
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17330
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my ($self, $x, $y) = @_; |
157
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1628
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3223
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$x = round_nearest ($x); |
158
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1628
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3077
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$y = round_nearest ($y); |
159
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### CellularRule190 xy_to_n(): "$x,$y" |
160
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161
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1628
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100
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66
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5004
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if ($y < 0 || $x > $y) { |
162
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240
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437
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return undef; |
163
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} |
164
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165
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1388
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1918
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$x += $y; # move to have x=0 the start of the row |
166
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1388
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100
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2426
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if ($x < 0) { |
167
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240
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494
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return undef; |
168
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} |
169
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170
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### x centred: $x |
171
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1148
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100
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1979
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if ($y % 2) { |
172
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### odd row, 3s from the start ... |
173
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508
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100
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878
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if (($x % 4) == 3) { |
174
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56
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120
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return undef; |
175
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} |
176
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# 3y^2+2y-1 = (3y-1)*(y+1) |
177
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return ($x |
178
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- int($x/4) |
179
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+ ((3*$y+2)*$y-1)/4 |
180
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452
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1475
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+ $self->{'n_start'}); |
181
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} else { |
182
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### even row, 3s then 1 sep, or mirror 1 sep start ... |
183
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640
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1005
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my $mirror = $self->{'mirror'}; |
184
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640
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100
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1321
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if (($x % 4) == ($mirror ? 1 : 3)) { |
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100
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185
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56
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return undef; |
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} |
187
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return ($x |
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- int(($x+($mirror ? 2 : 1))/4) |
189
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+ (3*$y+2)*$y/4 |
190
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584
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100
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2113
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+ $self->{'n_start'}); |
191
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} |
192
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} |
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194
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# exact |
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sub rect_to_n_range { |
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222
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222
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1
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8715
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my ($self, $x1,$y1, $x2,$y2) = @_; |
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### CellularRule190 rect_to_n_range(): "$x1,$y1, $x2,$y2" |
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199
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222
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50
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($x1,$y1, $x2,$y2) = _rect_for_V ($x1,$y1, $x2,$y2) |
200
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or return (1,0); # rect outside pyramid |
201
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202
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# # inherit bignum (before collapsing some y1 to x1 etc) |
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# my $zero = ($x1 * 0 * $y1 * $x2 * $y2); |
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205
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222
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my $mirror = $self->{'mirror'}; |
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222
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my $unincremented_x1 = $x1; |
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208
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# \+------+ |
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# | | / |
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# |\ | / |
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# | \ | / |
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# | \ | / |
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# y1 +------+ / |
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# x1 \ / |
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# \/ |
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# |
217
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673
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if ($x1 < (my $neg_y1 = -$y1)) { |
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100
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50
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218
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### bottom-left outside, move across to: "$neg_y1,$y1" |
219
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11
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18
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$x1 = $neg_y1; |
220
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221
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# For the following blank checks a blank doesn't occur at the ends of a |
222
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# row, so when on a blank it's always possible to increment or decrement |
223
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# X to go to a non-blank -- as long as that adjacent space is within the |
224
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# rectangle. |
225
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# |
226
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} elsif ((($mirror ? $y1-$x1 : $x1+$y1) % 4) == 3) { |
227
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### x1,y1 bottom left is on a blank: "x1+y1=".($x1+$y1) |
228
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0
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0
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0
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if ($x1 < $x2) { |
229
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### rect width >= 2, so increase x1 ... |
230
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0
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0
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$x1 += 1; |
231
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} else { |
232
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### rect is a single column width==1, increase y1 ... |
233
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0
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0
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0
|
if (($y1 += 1) > $y2) { |
234
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### rect was a single blank square, contains no N ... |
235
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0
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0
|
return (1,0); |
236
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} |
237
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} |
238
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} |
239
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240
|
222
|
100
|
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534
|
if ((($mirror ? $y2-$x2 : $x2+$y2) % 4) == 3) { |
|
|
100
|
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241
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|
|
### x2,y2 top right is on a blank, decrement ... |
242
|
12
|
50
|
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|
25
|
if ($x2 > $unincremented_x1) { |
243
|
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|
|
### rect width >= 2, so decrease x2 ... |
244
|
12
|
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|
28
|
$x2 -= 1; |
245
|
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|
|
} else { |
246
|
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|
|
|
### rect is a single column width==1, decrease y2 ... |
247
|
0
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0
|
$y2 -= 1; |
248
|
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|
249
|
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|
|
|
# Can decrement without checking whether the rect is a single square. |
250
|
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|
|
# If the rect was a single blank square then the x1+y1 bottom left |
251
|
|
|
|
|
|
|
# above detects and returns. And the bottom left blank check |
252
|
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|
|
# incremented y1 to leave a single square then that's a non-blank |
253
|
|
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|
|
|
# because there's no vertical blank pairs (they go on the diagonal). |
254
|
|
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|
|
### assert $y2 >= $y1 |
255
|
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|
|
|
} |
256
|
|
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|
|
|
|
} |
257
|
|
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|
|
258
|
|
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|
|
|
|
# At this point $x1,$y1 is a non-blank bottom left corner, and $x2,$y2 |
259
|
|
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|
|
|
|
# is a non-blank top right corner, being the N lo to hi range. |
260
|
|
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|
|
|
|
261
|
|
|
|
|
|
|
### range: "bottom-right $x1,$y1 top-left $x2,$y2" |
262
|
222
|
|
|
|
|
481
|
return ($self->xy_to_n ($x1,$y1), |
263
|
|
|
|
|
|
|
$self->xy_to_n ($x2,$y2)); |
264
|
|
|
|
|
|
|
} |
265
|
|
|
|
|
|
|
|
266
|
|
|
|
|
|
|
|
267
|
|
|
|
|
|
|
# old rect_to_n_range() of row endpoints |
268
|
|
|
|
|
|
|
# |
269
|
|
|
|
|
|
|
# # even right y = [ 0, 2, 4, 6 ] |
270
|
|
|
|
|
|
|
# # N = [ 1,8,21,40 ] |
271
|
|
|
|
|
|
|
# # Nright = (3/4 y^2 + 2 y + 1) |
272
|
|
|
|
|
|
|
# # = (3 y^2 + 8 y + 4) / 4 |
273
|
|
|
|
|
|
|
# # = ((3y + 8)y + 4) / 4 |
274
|
|
|
|
|
|
|
# # |
275
|
|
|
|
|
|
|
# # odd right y = [ 1, 3, 5, 7 ] |
276
|
|
|
|
|
|
|
# # N = [ 4,14,30, 52 ] |
277
|
|
|
|
|
|
|
# # Nright = (3/4 y^2 + 2 y + 5/4) |
278
|
|
|
|
|
|
|
# # = (3 y^2 + 8 y + 5) / 4 |
279
|
|
|
|
|
|
|
# # = ((3y + 8)y + 5) / 4 |
280
|
|
|
|
|
|
|
# # |
281
|
|
|
|
|
|
|
# # Nleft y even ((3y+2)*y + 4)/4 |
282
|
|
|
|
|
|
|
# # Nleft y odd ((3y+2)*y + 3)/4 |
283
|
|
|
|
|
|
|
# # Nright even ((3(y+1)+2)*(y+1) + 3)/4 - 1 |
284
|
|
|
|
|
|
|
# # = ((3y+3+2)*(y+1) + 3 - 4)/4 |
285
|
|
|
|
|
|
|
# # = ((3y+5)*(y+1) - 1)/4 |
286
|
|
|
|
|
|
|
# # = ((3y^2 + 8y + 5 - 1)/4 |
287
|
|
|
|
|
|
|
# # = ((3y^2 + 8y + 4)/4 |
288
|
|
|
|
|
|
|
# # = ((3y+8)y + 4)/4 |
289
|
|
|
|
|
|
|
# # = ((3y+2)(y+2)/4 |
290
|
|
|
|
|
|
|
# # |
291
|
|
|
|
|
|
|
# ### $y1 |
292
|
|
|
|
|
|
|
# ### $y2 |
293
|
|
|
|
|
|
|
# $y2 += $zero; |
294
|
|
|
|
|
|
|
# $y1 += $zero; |
295
|
|
|
|
|
|
|
# return (((3*$y1 + 2)*$y1 + 4 - ($y1%2)) / 4, # even/odd Nleft |
296
|
|
|
|
|
|
|
# ((3*$y2 + 8)*$y2 + 4 + ($y2%2)) / 4); # even/odd Nright |
297
|
|
|
|
|
|
|
|
298
|
|
|
|
|
|
|
1; |
299
|
|
|
|
|
|
|
__END__ |