# Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017 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/>.
#------------------------------------------------------------------------------
# cf
# Ulam/Warburton with cells turning off too
# A079315 cells OFF -> ON
# A079317 cells ON at stage n
# A079316 cells ON at stage n, in first quadrant
# A151921 net gain ON cells
#------------------------------------------------------------------------------
package Math::PlanePath::UlamWarburton;
use 5.004;
use strict;
use Carp 'croak';
use List::Util 'sum';
use vars '$VERSION', '@ISA';
$VERSION = 125;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
*_divrem = \&Math::PlanePath::_divrem;
*_divrem_mutate = \&Math::PlanePath::_divrem_mutate;
use Math::PlanePath::Base::Generic
'is_infinite',
'round_nearest';
use Math::PlanePath::Base::Digits
'round_up_pow',
'round_down_pow',
'digit_split_lowtohigh';
use Math::PlanePath::UlamWarburtonQuarter;
# uncomment this to run the ### lines
# use Smart::Comments;
use constant parameter_info_array =>
[
{ name => 'parts',
share_key => 'parts_ulamwarburton',
display => 'Parts',
type => 'enum',
default => '4',
choices => ['4','2','1','octant','octant_up' ],
choices_display => ['4','2','1','Octant','Octant Up' ],
description => 'Which parts of the plane to fill.',
},
Math::PlanePath::Base::Generic::parameter_info_nstart1(),
];
# octant_up goes up the Y axis spine, dX=0
# all others always have dX!=0
sub absdx_minimum {
my ($self) = @_;
return ($self->{'parts'} eq 'octant_up' ? 0 : 1);
}
# used also to validate $self->{'parts'}
my %x_negative = (4 => 1,
2 => 1,
1 => 0,
octant => 0,
octant_up => 0,
);
sub x_negative {
my ($self) = @_;
return $x_negative{$self->{'parts'}};
}
sub y_negative {
my ($self) = @_;
return $self->{'parts'} eq '4';
}
sub x_negative_at_n {
my ($self) = @_;
return ($x_negative{$self->{'parts'}} ? $self->n_start + 3 : undef);
}
sub y_negative_at_n {
my ($self) = @_;
return ($self->{'parts'} eq '4' ? $self->n_start + 4 : undef);
}
sub diffxy_minimum {
my ($self) = @_;
return ($self->{'parts'} eq 'octant' ? 0 : undef);
}
sub diffxy_maximum {
my ($self) = @_;
return ($self->{'parts'} eq 'octant_up' ? 0 : undef);
}
{
my %dir_maximum_dxdy = (4 => [1,-1], # N=4 South-East
2 => [1,-1], # N=44 South-East
1 => [2,-1], # N=3 ESE
octant => [10,-3], # N=51
octant_up => [2,-1], # N=8 ESE
);
sub dir_maximum_dxdy {
my ($self) = @_;
return @{$dir_maximum_dxdy{$self->{'parts'}}};
}
}
{
my %_UNDOCUMENTED__turn_any_right_at_n
= (
4 => 20,
2 => 35,
1 => 2,
octant => 4,
octant_up => 2,
);
sub _UNDOCUMENTED__turn_any_right_at_n {
my ($self) = @_;
return $self->n_start
+ $_UNDOCUMENTED__turn_any_right_at_n{$self->{'parts'}};
}
}
sub tree_num_children_list {
my ($self) = @_;
return ($self->{'parts'} eq '4'
? (0, 1, 3, 4)
: (0, 1, 2, 3 ));
}
#------------------------------------------------------------------------------
sub new {
my $self = shift->SUPER::new(@_);
if (! defined $self->{'n_start'}) {
$self->{'n_start'} = $self->default_n_start;
}
my $parts = ($self->{'parts'} ||= '4');
if (! exists $x_negative{$parts}) {
croak "Unrecognised parts option: ", $parts;
}
return $self;
}
sub n_to_xy {
my ($self, $n) = @_;
### UlamWarburton n_to_xy(): "$n parts=$self->{'parts'}"
if ($n < $self->{'n_start'}) { return; }
if (is_infinite($n)) { return ($n,$n); }
{
my $int = int($n);
### $int
### $n
if ($n != $int) {
my ($x1,$y1) = $self->n_to_xy($int);
my ($x2,$y2) = $self->n_to_xy($int+1);
my $frac = $n - $int; # inherit possible BigFloat
my $dx = $x2-$x1;
my $dy = $y2-$y1;
return ($frac*$dx + $x1, $frac*$dy + $y1);
}
$n = $int; # BigFloat int() gives BigInt, use that
}
$n = $n - $self->{'n_start'}; # N=0 basis
if ($n == 0) { return (0,0); }
my $parts = $self->{'parts'};
my ($depthsum, $factor, $nrem) = _n0_to_depthsum_factor_rem($n, $parts)
or return $n; # N=nan or +inf
### depthsum: join(',',@$depthsum)
### $factor
### n rem within row: $nrem
if ($parts eq '4') {
$factor /= 4;
} elsif ($parts eq '2') {
$factor /= 2;
$nrem += ($factor-1)/2;
} elsif ($parts eq 'octant_up') {
$nrem += $factor;
} else {
$nrem += ($factor-1)/2;
}
(my $quad, $nrem) = _divrem ($nrem, $factor);
### factor modulus: $factor
### $quad
### n rem within quad: $nrem
### assert: $quad >= 0
### assert: $quad <= 3
my $dhigh = shift @$depthsum; # highest term
my @ndigits = digit_split_lowtohigh($nrem,3);
### $dhigh
### ndigits low to high: join(',',@ndigits)
my $x = 0;
my $y = 0;
foreach my $depthterm (reverse @$depthsum) { # depth terms low to high
my $ndigit = shift @ndigits; # N digits low to high
### $depthterm
### $ndigit
$x += $depthterm;
### bit to x: "$x,$y"
if ($ndigit) {
if ($ndigit == 2) {
($x,$y) = (-$y,$x); # rotate +90
}
} else {
# $ndigit==0 (or undef when @ndigits shorter than @$depthsum)
($x,$y) = ($y,-$x); # rotate -90
}
### rotate to: "$x,$y"
}
$x += $dhigh;
### xy before quad: "$x,$y"
if ($quad & 2) {
$x = -$x;
$y = -$y;
}
if ($quad & 1) {
($x,$y) = (-$y,$x); # rotate +90
}
### final: "$x,$y"
return $x,$y;
}
# no Smart::Comments;
sub xy_to_n {
my ($self, $x, $y) = @_;
### UlamWarburton xy_to_n(): "$x, $y"
$x = round_nearest ($x);
$y = round_nearest ($y);
if ($x == 0 && $y == 0) {
return $self->{'n_start'};
}
my $parts = $self->{'parts'};
if ($parts ne '4'
&& ($y < 0
|| ($parts ne '2' && $x < ($parts eq 'octant' ? $y : 0))
|| ($parts eq 'octant_up' && $x > $y))) {
return undef;
}
my $quad;
if ($y > $x) {
### quad above leading diagonal ...
# /
# above /
# /
if ($y > -$x) {
### quad above opposite diagonal, top quarter ...
# top
# \ /
# \/
$quad = 1;
($x,$y) = ($y,-$x); # rotate -90
} else {
### quad below opposite diagonal, left quarter ...
# \
# left \
# /
# /
$quad = 2;
$x = -$x; # rotate -180
$y = -$y;
}
} else {
### quad below leading diagonal ...
# /
# / below
# /
if ($y > -$x) {
### quad above opposite diagonal, right quarter ...
# /
# / right
# \
# \
$quad = 0;
} else {
### quad below opposite diagonal, bottom quarter ...
# /\
# / \
# bottom
$quad = 3;
($x,$y) = (-$y,$x); # rotate +90
}
}
### $quad
### quad rotated xy: "$x,$y"
### assert: ! ($y > $x)
### assert: ! ($y < -$x)
my ($len, $exp) = round_down_pow ($x + abs($y), 2);
if (is_infinite($exp)) { return ($exp); }
my $depth =
my $ndigits =
my $n = ($x * 0 * $y); # inherit bignum 0
while ($exp-- >= 0) {
### at: "$x,$y n=$n len=$len"
my $abs_y = abs($y);
if ($x && $x == $abs_y) {
return undef;
}
# right quarter diamond
### assert: $x >= 0
### assert: $x >= abs($y)
### assert: $x+abs($y) < 2*$len || $x==abs($y)
if ($x + $abs_y >= $len) {
# one of the three quarter diamonds away from the origin
$x -= $len;
### shift to: "$x,$y"
$depth += $len;
if ($x || $y) {
$n *= 3;
$ndigits++;
if ($y < -$x) {
### bottom, digit 0 ...
($x,$y) = (-$y,$x); # rotate +90
} elsif ($y > $x) {
### top, digit 2 ...
($x,$y) = ($y,-$x); # rotate -90
$n += 2;
} else {
### right, digit 1 ...
$n += 1;
}
}
}
$len /= 2;
}
### $n
### $depth
### $ndigits
### npower: 3**$ndigits
### $quad
### quad powered: $quad*3**$ndigits
my $npower = 3**$ndigits;
if ($parts eq 'octant_up') {
$n -= $npower;
} elsif ($parts ne '4') {
$n -= ($npower-1)/2;
}
return $n + $quad*$npower + $self->tree_depth_to_n($depth);
}
# not exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
### UlamWarburton rect_to_n_range(): "$x1,$y1 $x2,$y2"
my ($dlo, $dhi)
= _rect_to_diamond_range (round_nearest($x1), round_nearest($y1),
round_nearest($x2), round_nearest($y2));
### $dlo
### $dhi
if ($dlo) {
($dlo) = round_down_pow ($dlo,2);
}
($dhi) = round_down_pow ($dhi,2);
### rounded to pow2: "$dlo ".(2*$dhi)
return ($self->tree_depth_to_n($dlo),
$self->tree_depth_to_n(2*$dhi) - 1);
}
# x1 | x2
# +--------|-------+ y2 xzero true, yzero false
# | | | diamond min is y1
# +--------|-------+ y1
# |
# ----------O-------------
#
# | x1 x2
# | +--------+ y2 xzero false, yzero true
# | | | diamond min is x1
# -O--------------------
# | | |
# | +--------+ y1
# |
#
sub _rect_to_diamond_range {
my ($x1,$y1, $x2,$y2) = @_;
my $xzero = ($x1 < 0) != ($x2 < 0); # x range covers x=0
my $yzero = ($y1 < 0) != ($y2 < 0); # y range covers y=0
$x1 = abs($x1);
$y1 = abs($y1);
$x2 = abs($x2);
$y2 = abs($y2);
if ($x1 > $x2) { ($x1,$x2) = ($x2,$x1) }
if ($y1 > $y2) { ($y1,$y2) = ($y2,$y1) }
return (($yzero ? 0 : $y1) + ($xzero ? 0 : $x1),
$x2+$y2);
}
#------------------------------------------------------------------------------
use constant tree_num_roots => 1;
# ENHANCE-ME: step by the bits, not by X,Y
# ENHANCE-ME: tree_n_to_depth() by probe
sub tree_n_children {
my ($self, $n) = @_;
### UlamWarburton tree_n_children(): $n
if ($n < $self->{'n_start'}) {
return;
}
my ($x,$y) = $self->n_to_xy($n);
my @ret;
my $dx = 1;
my $dy = 0;
foreach (1 .. 4) {
if (defined (my $n_child = $self->xy_to_n($x+$dx,$y+$dy))) {
if ($n_child > $n) {
push @ret, $n_child;
}
}
($dx,$dy) = (-$dy,$dx); # rotate +90
}
return sort {$a<=>$b} @ret;
}
sub tree_n_parent {
my ($self, $n) = @_;
### UlamWarburton tree_n_parent(): $n
if ($n <= $self->{'n_start'}) {
return undef;
}
my ($x,$y) = $self->n_to_xy($n);
my $dx = 1;
my $dy = 0;
foreach (1 .. 4) {
if (defined (my $n_parent = $self->xy_to_n($x+$dx,$y+$dy))) {
if ($n_parent < $n) {
return $n_parent;
}
}
($dx,$dy) = (-$dy,$dx); # rotate +90
}
return undef;
}
# sub tree_n_children {
# my ($self, $n) = @_;
# my ($power, $exp) = _round_down_pow (3*$n-2, 4);
# $exp -= 1;
# $power /= 4;
#
# ### $power
# ### $exp
# ### pow base: 2 + 4*(4**$exp - 1)/3
#
# $n -= ($power - 1)/3 * 4 + 2;
# ### n less pow base: $n
#
# my @$depthsum = (2**$exp);
# $power = 3**$exp;
#
# # find the cumulative levelpoints total <= $n, being the start of the
# # level containing $n
# #
# my $factor = 4;
# while (--$exp >= 0) {
# $power /= 3;
# my $sub = 4**$exp * $factor;
# ### $sub
# # $power*$factor;
# my $rem = $n - $sub;
#
# ### $n
# ### $power
# ### $factor
# ### consider subtract: $sub
# ### $rem
#
# if ($rem >= 0) {
# $n = $rem;
# push @$depthsum, 2**$exp;
# $factor *= 3;
# }
# }
#
# $n += $factor;
# if (1) {
# return ($n,$n+1,$n+2);
# } else {
# return $n,$n+1,$n+2;
# }
# }
# Converting quarter ...
# (N-start)*4+1+start = 4*N-4*start+1+start
# = 4*N-3*start+1
#
sub tree_depth_to_n {
my ($self, $depth) = @_;
### UlamWarburton tree_depth_to_n(): $depth
if ($depth == 0) {
return $self->{'n_start'};
}
my $n = $self->Math::PlanePath::UlamWarburtonQuarter::tree_depth_to_n($depth-1);
if (! defined $n) {
return undef;
}
my $parts = $self->{'parts'};
if ($parts eq '2') {
return 2*$n - $self->{'n_start'} + $depth;
}
if ($parts eq '1') {
return $n + $depth;
}
if ($parts eq 'octant' || $parts eq 'octant_up') {
return ($n + 1);
}
### assert: $parts eq '4'
return 4*$n - 3*$self->{'n_start'} + 1;
}
# sub _NOTWORKING__tree_depth_to_n_range {
# my ($self, $depth) = @_;
# my ($nstart, $nend) = $self->Math::PlanePath::UlamWarburtonQuarter::tree_depth_to_n_range($self, $depth)
# or return;
# return (4*$nstart-3 + $self->{'n_start'}-1,
# 4*$nend-3 + $self->{'n_start'}-1);
# }
sub tree_n_to_depth {
my ($self, $n) = @_;
### UlamWarburton tree_n_to_depth(): $n
$n = $n - $self->{'n_start'}; # N=0 basis
if ($n < 0) {
return undef;
}
$n = int($n);
if ($n == 0) {
return 0;
}
my ($depthsum) = _n0_to_depthsum_factor_rem($n, $self->{'parts'})
or return $n; # N=nan or +inf
return sum(@$depthsum);
}
# 1+3+3+9=16
#
# 0 +1
# 1 +4 <- 0
# 5 +4 <- 1
# 9 +12
# 21 +4 <- 5 + 4+12 = 21 = 5 + 4*(1+3)
# 25 +12
# 37 +12
# 49 +36
# 85 +4 <- 21 + 4+12+12+36 = 21 + 4*(1+3+3+9)
# 89 +12 <- 8 +64
# 101 +12
# 113 +36
# 149
# 161
# 197
# 233
# 341
# 345 <- 16 +256
# 357
# 369
# 1+3 = 4 power 2
# 1+3+3+9 = 16 power 3
# 1+3+3+9+3+9+9+27 = 64 power 4
#
# 4*(1+4+...+4^(l-1)) = 4*(4^l-1)/3
# l=1 total=4*(4-1)/3 = 4
# l=2 total=4*(16-1)/3=4*5 = 20
# l=3 total=4*(64-1)/3=4*63/3 = 4*21 = 84
#
# n = 2 + 4*(4^l-1)/3
# (n-2) = 4*(4^l-1)/3
# 3*(n-2) = 4*(4^l-1)
# 3n-6 = 4^(l+1)-4
# 3n-2 = 4^(l+1)
#
# 3^0+3^1+3^1+3^2 = 1+3+3+9=16
# x+3x+3x+9x = 16x = 256
# 4 quads is 4*16=64
#
# 1+1+3 = 5
# 1+1+3 +1+1+3 +3+3+9 = 25
# 1+4 = 5
# 1+4+4+12 = 21 = 1 + 4*(1+1+3)
# 2 +1
# 3 +3
# 6 +1
# 7 +1
# 10 +3
# 13
# parts=1
# 1+4+...+4^(l-1) + 2^l
# = (4^l-1)/3 + 2^l
# = (4^l-1 + 3*2^l)/3
# = (2^l*(2^l + 3) - 1)/3
# l=1 total= 3
# l=2 total= 9
# l=3 total= 29
# l=4 total= 101
#
# N = (4^l-1)/3 + 2^l
# 3*(N-2^l)+1 = 4^l
# 12*(N-2^l)+1 = 4 * 4^l
#
# parts=2
# N = 2*(4^l-1)/3 + 2^l
# 3/2*(N-2^l)+1 = 4^l
# 6*(N-2^l)+1 = 4 * 4^l
#
# parts=4
# N = (4^l-1)/3
# 3*N+1 = 4 * 4^l
# use Smart::Comments;
# Return ($aref, $factor, $remaining_n).
# sum(@$aref) = depth starting depth=1
#
sub _n0_to_depthsum_factor_rem {
my ($n, $parts) = @_;
### _n0_to_depthsum_factor_rem(): "$n parts=$parts"
my $factor = ($parts eq '4' ? 4 : $parts eq '2' ? 2 : 1);
if ($n == 0) {
return ([], $factor, 0);
}
my $n3 = 3*$n + 1;
my $ndepth = 0;
my $power = $n3;
my $exp;
if ($parts eq '4') {
$power /= 4;
} elsif ($parts eq '2') {
$power /= 2;
$ndepth = -1;
} elsif ($parts =~ /octant/) {
$power *= 2;
$ndepth = 2;
}
($power, $exp) = round_down_pow ($power, 4);
### $n3
### $power
### $exp
if (is_infinite($exp)) {
return;
}
# ### pow base: ($power - 1)/3 * $factor + 1 + ($parts ne '4' && $exp)
# $n -= ($power - 1)/3 * $factor + 1;
# if ($parts ne '4') { $n -= $exp; }
# ### n less pow base: $n
my $twopow = 2**$exp;
my @depthsum;
for (;
$exp-- >= 0;
$power /= 4, $twopow /= 2) {
### at: "power=$power twopow=$twopow factor=$factor n3=$n3 ndepth=$ndepth depthsum=".join(',',@depthsum)
my $nmore = $power * $factor;
if ($parts ne '4') { $nmore += 3*$twopow; }
if ($parts =~ /octant/) {
### assert: $nmore % 2 == 0
$nmore = $nmore/2;
}
my $ncmp = $ndepth + $nmore;
### $nmore
### $ncmp
if ($n3 >= $ncmp) {
### go to ncmp, remainder: $n3-$ncmp
$factor *= 3;
$ndepth = $ncmp;
push @depthsum, $twopow;
}
}
if ($parts eq '2') {
$n3 += 1;
}
# ### assert: ($n3 - $ndepth)%3 == 0
$n = ($n3 - $ndepth) / 3;
$factor /= 3;
### $ndepth
### @depthsum
### remaining n: $n
### assert: $n >= 0
### assert: $n < $factor + ($parts ne '4')
return \@depthsum, $factor, $n;
}
sub tree_n_to_subheight {
my ($self, $n) = @_;
### tree_n_to_subheight(): $n
$n = int($n - $self->{'n_start'}); # N=0 basis
if ($n < 0) {
return undef;
}
my ($depthsum, $factor, $nrem) = _n0_to_depthsum_factor_rem($n, $self->{'parts'})
or return $n; # N=nan or +inf
### $depthsum
### $factor
### $nrem
my $parts = $self->{'parts'};
if ($parts eq '4') {
$factor /= 4;
} elsif ($parts eq '2') {
$factor /= 2;
$nrem += ($factor-1)/2;
} elsif ($parts eq 'octant_up') {
} else {
$nrem += ($factor-1)/2;
}
(my $quad, $nrem) = _divrem ($nrem, $factor);
my $sub = pop @$depthsum;
while (_divrem_mutate($nrem,3) == 1) { # low "1" ternary digits of Nrem
$sub += pop @$depthsum;
}
if (@$depthsum) {
return $depthsum->[-1] - 1 - $sub;
} else {
return undef; # N all 1-digits, on central infinite spine
}
}
#------------------------------------------------------------------------------
# levels
sub level_to_n_range {
my ($self, $level) = @_;
return ($self->{'n_start'},
$self->tree_depth_to_n_end(2**($level+1)-1));
}
sub n_to_level {
my ($self, $n) = @_;
my $depth = $self->tree_n_to_depth($n);
if (! defined $depth) { return undef; }
my ($pow, $exp) = round_down_pow ($depth, 2);
return $exp;
}
# parts=4
# Ndepth(2^a) = 2 + 4*(4^a-1)/3
# Nend(2^a-1) = 1 + 4*(4^a-1)/3 = (4^(a+1)-1)/3
# parts=2
#
# {
# my %factor = (4 => 16,
# 2 => 8,
# 1 => 4,
# octant => 2,
# octant_up => 2,
# );
# my %constant = (4 => -4,
# 2 => -5,
# 1 => -4,
# octant => 0,
# octant_up => 0,
# );
# my %spine = (4 => 0,
# 2 => 2,
# 1 => 2,
# octant => 1,
# octant_up => 1,
# );
# sub level_to_n_range {
# my ($self, $level) = @_;
# my $parts = $self->{'parts'};
# return ($self->{'n_start'},
# $self->{'n_start'}
# + (4**$level * $factor{$parts} + $constant{$parts}) / 3
# + 2**$level * $spine{$parts});
# }
# }
#------------------------------------------------------------------------------
1;
__END__
=for stopwords eg Ryde Math-PlanePath Ulam Warburton Ndepth OEIS ie
=head1 NAME
Math::PlanePath::UlamWarburton -- growth of a 2-D cellular automaton
=head1 SYNOPSIS
use Math::PlanePath::UlamWarburton;
my $path = Math::PlanePath::UlamWarburton->new;
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
X<Ulam, Stanislaw>X<Warburton>This is the pattern of a cellular automaton
studied by Ulam and Warburton, numbering cells by growth tree row and
anti-clockwise within the rows.
=cut
# math-image --path=UlamWarburton --expression='i<100?i:0' --output=numbers
# and add N=100,N=101 manually
=pod
94 9
95 87 93 8
63 7
64 42 62 6
65 30 61 5
66 43 31 23 29 41 60 4
69 67 14 59 57 3
70 44 68 15 7 13 58 40 56 2
96 71 32 16 3 12 28 55 92 1
97 88 72 45 33 24 17 8 4 1 2 6 11 22 27 39 54 86 91 <- Y=0
98 73 34 18 5 10 26 53 90 -1
74 46 76 19 9 21 50 38 52 ... -2
75 77 20 85 51 -3
78 47 35 25 37 49 84 -4
79 36 83 -5
80 48 82 -6
81 -7
99 89 101 -8
100 -9
^
-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9
The growth rule is that a given cell grows up, down, left and right, but
only if the new cell has no neighbours (up, down, left or right). So the
initial cell "a" is N=1,
a initial depth=0 cell
The next row "b" cells are numbered N=2 to N=5 anti-clockwise from the
right,
b
b a b depth=1
b
Likewise the next row "c" cells N=6 to N=9. The "b" cells only grow
outwards as 4 "c"s since the other positions would have neighbours in the
existing "b"s.
c
b
c b a b c depth=2
b
c
The "d" cells are then N=10 to N=21, numbered following the previous row "c"
cell order and then anti-clockwise around each.
d
d c d
d b d
d c b a b c d depth=3
d b d
d c d
d
There's only 4 "e" cells since among the "d"s only the X,Y axes won't have
existing neighbours (the "b"s and "d"s).
e
d
d c d
d b d
e d c b a b c d e depth=4
d b d
d c d
d
e
In general the pattern always grows by 1 outward along the X and Y axes and
travels into the quarter planes between with a diamond shaped tree pattern
which fills 11 of 16 cells in each 4x4 square block.
=head2 Tree Row Ranges
Counting depth=0 as the N=1 at the origin and depth=1 as the next N=2,3,4,5
generation, the number of cells in a row is
rowwidth(0) = 1
then
rowwidth(depth) = 4 * 3^((count_1_bits(depth) - 1)
So depth=1 has 4*3^0=4 cells, as does depth=2 at N=6,7,8,9. Then depth=3
has 4*3^1=12 cells N=10 to N=21 because depth=3=0b11 has two 1-bits in
binary. The N start and end for a row is the cumulative total of those
before it,
Ndepth(depth) = 1 + (rowwidth(0) + ... + rowwidth(depth-1))
Nend(depth) = rowwidth(0) + ... + rowwidth(depth)
For example depth 3 ends at N=(1+4+4)=9.
depth Ndepth rowwidth Nend
0 1 1 1
1 2 4 5
2 6 4 9
3 10 12 21
4 22 4 25
5 26 12 37
6 38 12 49
7 50 36 85
8 86 4 89
9 90 12 101
For a power-of-2 depth the Ndepth is
Ndepth(2^a) = 2 + 4*(4^a-1)/3
For example depth=4=2^2 starts at N=2+4*(4^2-1)/3=22, or depth=8=2^3 starts
N=2+4*(4^3-1)/3=86.
Further bits in the depth value contribute powers-of-4 with a tripling for
each bit above. So if the depth number has bits a,b,c,d,etc in descending
order,
depth = 2^a + 2^b + 2^c + 2^d ... a>b>c>d...
Ndepth = 2 + 4*(-1
+ 4^a
+ 3 * 4^b
+ 3^2 * 4^c
+ 3^3 * 4^d + ... ) / 3
For example depth=6 = 2^2+2^1 is Ndepth = 2 + (1+4*(4^2-1)/3) + 4^(1+1) =
38. Or depth=7 = 2^2+2^1+2^0 is Ndepth = 1 + (1+4*(4^2-1)/3) + 4^(1+1) +
3*4^(0+1) = 50.
=head2 Self-Similar Replication
The diamond shape depth=1 to depth=2^level-1 repeats three times. For
example an "a" part going to the right of the origin "O",
d
d d d
| a d c
--O a a a * c c c ...
| a b c
b b b
b
The 2x2 diamond shaped "a" repeats pointing up, down and right as "b", "c"
and "d". This resulting 4x4 diamond then likewise repeats up, down and
right. The same happens in the other quarters of the plane.
The points in the path here are numbered by tree rows rather than in this
sort of replication, but the replication helps to see the structure of the
pattern.
=head2 Half Plane
Option C<parts =E<gt> '2'> confines the pattern to the upper half plane
C<YE<gt>=0>,
=cut
# math-image --path=UlamWarburton,parts=2 --expression='i<32?i:0' --output=numbers --size=99x16
=pod
parts => "2"
28 6
21 5
29 22 16 20 27 4
11 3
30 12 6 10 26 2
23 13 3 9 19 1
31 24 17 14 7 4 1 2 5 8 15 18 25 <- Y=0
--------------------------------------
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6
Points are still numbered anti-clockwise around so X axis N=1,2,5,8,15,etc
is the first of row depth=X. X negative axis N=1,4,7,14,etc is the last of
row depth=-X. For depth=0 point N=1 is both the first and last of that row.
Within a row a line from point N to N+1 is always a 45-degree angle. This
is true of each 3 direct children, but also across groups of children by
symmetry. For this parts=2 the lines from the last of one row to the first
of the next are horizontal, making an attractive pattern of diagonals and
then across to the next row horizontally. For parts=4 or parts=1 the last
to first lines are at various different slopes and so upsets the pattern.
=head2 One Quadrant
Option C<parts =E<gt> '1'> confines the pattern to the first quadrant,
=cut
# math-image --path=UlamWarburton,parts=1 --expression='i<=73?i:0' --output=numbers --size=99x16
=pod
parts => "1" to depth=14
14 | 73
13 | 63
12 | 53 62 72
11 | 49
10 | 39 48 71
9 | 35 47 61
8 | 31 34 38 46 52 60 70
7 | 29 45 59
6 | 19 28 69 67
5 | 15 27 57
4 | 11 14 18 26 68 58 51 56 66
3 | 9 25 23 43
2 | 5 8 24 17 22 44 37 42 65
1 | 3 7 13 21 33 41 55
Y=0 | 1 2 4 6 10 12 16 20 30 32 36 40 50 54 64
+-----------------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
X axis N=1,2,4,6,10,etc is the first of each row X=depth. Y axis
N=1,3,5,9,11,etc is the last similarly Y=depth.
In this arrangement horizontal arms have even N and vertical arms have
odd N. For example the vertical at X=8 N=30,33,37,etc has N odd from N=33
up and when it turns to horizontal at N=42 or N=56 it switches to N even.
The children of N=66 are not shown but the verticals from there are N=79
below and N=81 above and so switch to odd again.
This odd/even pattern is true of N=2 horizontal and N=3 vertical and
thereafter is true due to each row having an even number of points and the
self-similar replications in the pattern,
|\ replication
| \ block 0 to 1 and 3
|3 \ and block 0 block 2 less sides
|----
|\ 2|\
| \ | \
|0 \|1 \
---------
Block 0 is the base and is replicated as block 1 and in reverse as block 3.
Block 2 is a further copy of block 0, but the two halves of block 0 rotated
inward 90 degrees, so the X axis of block 0 becomes the vertical of block 2,
and the Y axis of block 0 the horizontal of block 2. Those axis parts are
dropped since they're already covered by block 1 and 3 and dropping them
flips the odd/even parity to match the vertical/horizontal flip due to the
90-degree rotation.
=head2 Octant
Option C<parts =E<gt> 'octant'> confines the pattern to the first eighth of
the plane 0E<lt>=YE<lt>=X.
=cut
# math-image --path=UlamWarburton,parts=octant --expression='i<=51?i:0' --output=numbers --size=75x15
=pod
parts => "octant"
7 | 47 ...
6 | 48 36 46
5 | 49 31 45
4 | 50 37 32 27 30 35 44
3 | 14 51 24 43 41
2 | 15 10 13 25 20 23 42 34 40
1 | 5 8 12 18 22 29 39
Y=0 | 1 2 3 4 6 7 9 11 16 17 19 21 26 28 33 38
+-------------------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
In this arrangement N=1,2,3,4,6,7,etc on the X axis is the first N of each
row (C<tree_depth_to_n()>).
=head2 Upper Octant
Option C<parts =E<gt> 'octant_up'> confines the pattern to the upper octant
0E<lt>=XE<lt>=Y of the first quadrant.
=cut
# math-image --path=UlamWarburton,parts=octant_up --expression='i<=51?i:0' --output=numbers --size=75x15
=pod
parts => "octant_up"
8 | 16 17 19 22 26 29 34 42
7 | 15 21 28 41
6 | 10 14 38 33 40
5 | 8 13 39
4 | 6 7 9 12
3 | 5 11
2 | 3 4
1 | 2
Y=0 | 1
+--------------------------
X=0 1 2 3 4 5 6 7
In this arrangement N=1,2,3,5,6,8,etc on the Y axis the last N of each row
(C<tree_depth_to_n_end()>).
=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=UlamWarburton,n_start=0 --expression='i<38?i:0' --output=numbers
=pod
n_start => 0
29 5
30 22 28 4
13 3
14 6 12 2
31 15 2 11 27 1
32 23 16 7 3 0 1 5 10 21 26 <- Y=0
33 17 4 9 25 -1
18 8 20 37 -2
19 -3
34 24 36 -4
35 -5
^
-5 -4 -3 -2 -1 X=0 1 2 3 4 5
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::UlamWarburton-E<gt>new ()>
=item C<$path = Math::PlanePath::UlamWarburton-E<gt>new (parts =E<gt> $str, n_start =E<gt> $n)>
Create and return a new path object. The C<parts> option (a string) can be
"4" the default
"2"
"1"
=back
=head2 Tree Methods
=over
=item C<@n_children = $path-E<gt>tree_n_children($n)>
Return the children of C<$n>, or an empty list if C<$n> has no children
(including when C<$n E<lt> 1>, ie. before the start of the path).
The children are the cells turned on adjacent to C<$n> at the next row. The
way points are numbered means that when there's multiple children they're
consecutive N values, for example at N=6 the children are 10,11,12.
=back
=head2 Tree Descriptive Methods
=over
=item C<@nums = $path-E<gt>tree_num_children_list()>
Return a list of the possible number of children in C<$path>. This is the
set of possible return values from C<tree_n_num_children()>. The possible
children varies with the C<parts>,
parts tree_num_children_list()
----- ------------------------
4 0, 1, 3, 4 (the default)
2 0, 1, 2, 3
1 0, 1, 2, 3
parts=4 has 4 children at the origin N=0 and thereafter either 0, 1 or 3.
parts=2 and parts=1 can have 2 children on the boundaries where the 3rd
child is chopped off, otherwise 0, 1 or 3.
=item C<$n_parent = $path-E<gt>tree_n_parent($n)>
Return the parent node of C<$n>, or C<undef> if C<$n E<lt>= 1> (the start of
the path).
=back
=head2 Level Methods
=over
=item C<($n_lo, $n_hi) = $path-E<gt>level_to_n_range($level)>
Return C<$n_lo = $n_start> and
parts $n_hi
----- -----
4 $n_start + (16*4**$level - 4) / 3
2 $n_start + ( 8*4**$level - 5) / 3 + 2*2**$level
1 $n_start + ( 4*4**$level - 4) / 3 + 2*2**$level
C<$n_hi> is C<tree_depth_to_n_end(2**($level+1) - 1>.
=back
=head1 OEIS
This cellular automaton is in Sloane's Online Encyclopedia of Integer
Sequences as
=over
L<http://oeis.org/A147582> (etc)
=back
parts=4
A147562 total cells to depth, being tree_depth_to_n() n_start=0
A147582 added cells at depth
parts=2
A183060 total cells to depth=n in half plane
A183061 added cells at depth=n
parts=1
A151922 total cells to depth=n in quadrant
A079314 added cells at depth=n
The A147582 new cells sequence starts from n=1, so takes the innermost N=1
single cell as row n=1, then N=2,3,4,5 as row n=2 with 5 cells, etc. This
makes the formula a binary 1-bits count on n-1 rather than on N the way
rowwidth() above is expressed.
The 1-bits-count power 3^(count_1_bits(depth)) part of the rowwidth() is
also separately in A048883, and as n-1 in A147610.
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::UlamWarburtonQuarter>,
L<Math::PlanePath::LCornerTree>,
L<Math::PlanePath::CellularRule>
L<Math::PlanePath::SierpinskiTriangle> (a similar binary 1s-count related
calculation)
=head1 HOME PAGE
L<http://user42.tuxfamily.org/math-planepath/index.html>
=head1 LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017 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