# Copyright 2011, 2012, 2013, 2014 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/>.
# math-image --path=GosperReplicate --lines --scale=10
# math-image --path=GosperReplicate --all --output=numbers_dash
#
package Math::PlanePath::GosperReplicate;
use 5.004;
use strict;
use List::Util qw(max);
use POSIX 'ceil';
use Math::Libm 'hypot';
use Math::PlanePath::SacksSpiral;
use vars '$VERSION', '@ISA';
$VERSION = 117;
use Math::PlanePath;
@ISA = ('Math::PlanePath');
use Math::PlanePath::Base::Generic
'is_infinite',
'round_nearest';
use Math::PlanePath::Base::Digits
'digit_split_lowtohigh',
'digit_join_lowtohigh';
# uncomment this to run the ### lines
#use Smart::Comments;
use constant n_start => 0;
*xy_is_visited = \&Math::PlanePath::Base::Generic::xy_is_even;
use constant x_negative_at_n => 3;
use constant y_negative_at_n => 5;
use constant absdx_minimum => 1;
use constant dir_maximum_dxdy => (3,-1);
#------------------------------------------------------------------------------
sub n_to_xy {
my ($self, $n) = @_;
### GosperReplicate n_to_xy(): $n
if ($n < 0) {
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
}
my $x = 0;
my $y = 0;
my $sx = 2;
my $sy = 0;
# digit
# 3 2
# \ /
# 4---0---1
# / \
# 5 6
foreach my $digit (digit_split_lowtohigh($n,7)) {
### digit: "$digit $x,$y side $sx,$sy"
if ($digit == 1) {
### right ...
# $x = -$x; # rotate 180
# $y = -$y;
$x += $sx;
$y += $sy;
} elsif ($digit == 2) {
### up right ...
# ($x,$y) = ((3*$y-$x)/2, # rotate -120
# ($x+$y)/-2);
$x += ($sx - 3*$sy)/2; # at +60
$y += ($sx + $sy)/2;
} elsif ($digit == 3) {
### up left ...
# ($x,$y) = (($x+3*$y)/2, # -60
# ($y-$x)/2);
$x += ($sx + 3*$sy)/-2; # at +120
$y += ($sx - $sy)/2;
} elsif ($digit == 4) {
### left
$x -= $sx; # at -180
$y -= $sy;
} elsif ($digit == 5) {
### down left
# ($x,$y) = (($x-3*$y)/2, # rotate +60
# ($x+$y)/2);
$x += (3*$sy - $sx)/2; # at -120
$y += ($sx + $sy)/-2;
} elsif ($digit == 6) {
### down right
# ($x,$y) = (($x+3*$y)/-2, # rotate +120
# ($x-$y)/2);
$x += ($sx + 3*$sy)/2; # at -60
$y += ($sy - $sx)/2;
}
# 2*(sx,sy) + rot+60(sx,sy)
($sx,$sy) = ((5*$sx - 3*$sy) / 2,
($sx + 5*$sy) / 2);
}
return ($x,$y);
}
# modulus
# 1 3
# \ /
# 5---0---2
# / \
# 4 6
# 0 1 2 3 4 5 6
my @modulus_to_x = (0,-1, 2, 1,-1,-2, 1);
my @modulus_to_y = (0, 1, 0, 1,-1, 0,-1);
my @modulus_to_digit = (0, 3, 1, 2, 5, 4, 6);
sub xy_to_n {
my ($self, $x, $y) = @_;
### GosperReplicate xy_to_n(): "$x, $y"
$x = round_nearest($x);
$y = round_nearest($y);
if (($x + $y) % 2) {
return undef;
}
my $level = _xy_to_level_ceil($x,$y);
if (is_infinite($level)) {
return $level;
}
my $zero = ($x * 0 * $y); # inherit bignum 0
my @n; # digits low to high
while ($level-- >= 0 && ($x || $y)) {
### at: "$x,$y m=".(($x + 2*$y) % 7)
my $m = ($x + 2*$y) % 7;
push @n, $modulus_to_digit[$m];
$x -= $modulus_to_x[$m];
$y -= $modulus_to_y[$m];
### digit: "to $x,$y"
### assert: (3 * $y + 5 * $x) % 14 == 0
### assert: (5 * $y - $x) % 14 == 0
# shrink
($x,$y) = ((3*$y + 5*$x) / 14,
(5*$y - $x) / 14);
}
return digit_join_lowtohigh (\@n, 7, $zero);
}
# not exact
sub rect_to_n_range {
my ($self, $x1,$y1, $x2,$y2) = @_;
$y1 *= sqrt(3);
$y2 *= sqrt(3);
my ($r_lo, $r_hi) = Math::PlanePath::SacksSpiral::_rect_to_radius_range
($x1,$y1, $x2,$y2);
$r_hi *= 2;
my $level_plus_1 = ceil( log(max(1,$r_hi/4)) / log(sqrt(7)) ) + 2;
return (0, 7**$level_plus_1 - 1);
}
sub _xy_to_level_ceil {
my ($x,$y) = @_;
my $r = hypot($x,$y);
$r *= 2;
return ceil( log(max(1,$r/4)) / log(sqrt(7)) ) + 1;
}
#------------------------------------------------------------------------------
# levels
sub level_to_n_range {
my ($self, $level) = @_;
return (0, 7**$level - 1);
}
sub n_to_level {
my ($self, $n) = @_;
if ($n < 0) { return undef; }
if (is_infinite($n)) { return $n; }
$n = round_nearest($n);
my ($pow, $exp) = round_down_pow ($n, 7);
return $exp + 1;
}
#------------------------------------------------------------------------------
1;
__END__
=for stopwords eg Ryde Gosper Math-PlanePath
=head1 NAME
Math::PlanePath::GosperReplicate -- self-similar hexagon replications
=head1 SYNOPSIS
use Math::PlanePath::GosperReplicate;
my $path = Math::PlanePath::GosperReplicate->new;
my ($x, $y) = $path->n_to_xy (123);
=head1 DESCRIPTION
This is a self-similar hexagonal tiling of the plane. At each level the
shape is the Gosper island.
17----16 4
/ \
24----23 18 14----15 3
/ \ \
25 21----22 19----20 10---- 9 2
\ / \
26----27 3---- 2 11 7---- 8 1
/ \ \
31----30 4 0---- 1 12----13 <- Y=0
/ \ \
32 28----29 5---- 6 45----44 -1
\ / \
33----34 38----37 46 42----43 -2
/ \ \
39 35----36 47----48 -3
\
40----41 -4
^
-7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7
The points are spread out on every second X coordinate to make a a
triangular lattice in integer coordinates (see L<Math::PlanePath/Triangular
Lattice>).
The base pattern is the inner N=0 to N=6, then six copies of that shape are
arranged around as the blocks N=7,14,21,28,35,42. Then six copies of the
resulting N=0 to N=48 shape are replicated around, etc.
Each point represents a little hexagon, thus tiling the plane with hexagons.
The innermost N=0 to N=6 are for instance,
* *
/ \ / \
/ \ / \
* * *
| 3 | 2 |
* * *
/ \ / \ / \
/ \ / \ / \
* * * *
| 4 | 0 | 1 |
* * * *
\ / \ / \ /
\ / \ / \ /
* * *
| 5 | 6 |
* * *
\ / \ /
\ / \ /
* *
The further replications are the same arrangement, but the sides become ever
wigglier and the centres rotate around. The rotation can be seen at N=7
X=5,Y=1 which is up from the X axis.
The C<FlowsnakeCentres> path is this same replicating shape, but starting
from a side instead of the middle and traversing in such as way as to make
each N adjacent. The C<Flowsnake> curve itself is this replication too, but
following edges.
=head2 Complex Base
The path corresponds to expressing complex integers X+i*Y in a base
b = 5/2 + i*sqrt(3)/2
with some scaling to put equilateral triangles on a square
grid. So for integer X,Y with X and Y either both odd or both even,
X/2 + i*Y*sqrt(3)/2 = a[n]*b^n + ... + a[2]*b^2 + a[1]*b + a[0]
where each digit a[i] is either 0 or a sixth root of unity encoded into N as
base 7 digits,
r = e^(i*pi/3)
= 1/2 + i*sqrt(3)/2 sixth root of unity
N digit a[i] complex number
------- -------------------
0 0
1 r^0 = 1
2 r^2 = 1/2 + i*sqrt(3)/2
3 r^3 = -1/2 + i*sqrt(3)/2
4 r^4 = -1
5 r^5 = -1/2 - i*sqrt(3)/2
6 r^6 = 1/2 - i*sqrt(3)/2
7 digits suffice because
norm(b) = (5/2)^2 + (sqrt(3)/2)^2 = 7
=head1 FUNCTIONS
See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.
=over 4
=item C<$path = Math::PlanePath::GosperReplicate-E<gt>new ()>
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<$n> on the path. Points begin
at 0 and if C<$n E<lt> 0> then the return is an empty list.
=back
=head2 Level Methods
=over
=item C<($n_lo, $n_hi) = $path-E<gt>level_to_n_range($level)>
Return C<(0, 7**$level - 1)>.
=back
=head1 SEE ALSO
L<Math::PlanePath>,
L<Math::PlanePath::GosperIslands>,
L<Math::PlanePath::Flowsnake>,
L<Math::PlanePath::FlowsnakeCentres>,
L<Math::PlanePath::QuintetReplicate>,
L<Math::PlanePath::ComplexPlus>
=head1 HOME PAGE
L<http://user42.tuxfamily.org/math-planepath/index.html>
=head1 LICENSE
Copyright 2011, 2012, 2013, 2014 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