NAME
Math::PlanePath::ToothpickReplicate -- toothpick pattern by replication
SYNOPSIS
use Math::PlanePath::ToothpickReplicate;
my $path = Math::PlanePath::ToothpickReplicate->new;
my ($x, $y) = $path->n_to_xy (123);DESCRIPTION
This is the "toothpick" pattern of the ToothpickTree path arranged as a self-similar replicating pattern.
...
|
..-24-- --26-- --18-- --16--43 4
| | | | |
23--20--25 17--12--15 ... 3
| | | |
19---6-- ---4--11 2
| | | | | |
22--21- 5---1---3 -13--14 1
| | | | |
0 <- Y=0
| | | | |
30--29- 7---2---9 -37--38 -1
| | | | | |
27---8-- --10--35 -2
| | | |
31--28--33 41--36--39 -3
| | | |
..-32-- --34-- --42-- --40--.. -4
^
-3 -2 -1 X=0 1 2 3 4Replication
The points visited are the same as Math::PlanePath::ToothpickTree, but in a self-similar order. The pattern within each quarter repeats at 2^level size blocks.
+------------+------------+
| | |
| block 3 block 2 |
| mirror same |
| |
| --B-- |
| | |
+---------- A ---+
| | |
| block 0 block 1 |
| | rot +90 |
| | |
| | |
+------------+------------+In the parts=quarter above
N=1 to N=10 "0" block
N=11 "A" middle point
N=12 "B" middle point
N=13 to N=22 "1" block, rotated +90 degrees
N=23 to N=32 "2" block, same layout as the "0" block
N=33 to N=42 "3" block, mirror image of "0" blockThe very first points N=1 and N=2 are effectively the "A" and "B" middle toothpicks with no points at all in the 0,1,2,3 lower blocks.
The full parts=all form is four quarters, each advancing by a replication level each time.
The initial N=0,1,2 make the centre, and then each quadrant is extended in turn by blocks.
+------------+------------A
| | |
| block 3 block 2 | in each quadrant
| mirror same |
| ^ ^ |
| \ --B-- / |
| \ | / |
+---------- A ---+
| | |
| block 0 block 1 |
| ^ | \ rot +90 |
| / | \ |
| / | v |
+------------+------------+Block 0 is the existing part. Then toothpick A and B are counted, followed by replications of block 0 in blocks 1,2,3. For example in the first quadrant
N=11 toothpick "A"
N=12 toothpick "B"
N=13,14 block 1 \
N=15,16 block 2 | replicating block 0 N=3,N=4
N=17,18 block 3 /Each such replication doubles the size in a quadrant, so the "A" toothpick is on a power-of-2 X=2^k,Y=2^k. For example N=11 at X=2,Y=2 and N=43 at X=4,Y=4.
One Quadrant
Option parts => 1 selects a single quadrant of replications.
| ...
| |
8 | --39-- --41-- --31-- --29--42
| | | | | |
7 | 38--35--40 30--25--28 ...
| | | | |
6 | 34--33-- --23--24
| | | | | | |
5 | 37--36- 32--11--22 -26--27
| | | |
4 | ---9-- ---7--10
| | | | | |
3 | 8---3---6 -12--13 20--21
| | | | | |
2 | ---1---2 16--14--15
| | | | | | |
1 | 0 --4---5 17 --18--19
| | | |
Y=0 |
+-----------------------------------
X=0 1 2 3 4 5 6 7 8Half Plane
Option parts => 2 confines the pattern to the upper half plane Y>=1, giving two symmetric parts above the X axis. N=0 at X=0,Y=1 is the first toothpick of the full pattern which is wholly within this half plane.
... ... 5
| |
53--18-- --20-- --12-- --10--21 4
| | | | | |
... 17--14--19 11---6---9 ... 3
| | | |
13---4-- ---2---5 2
| | | | | |
16--15- 3---0---1 --7---8 1
| | | |
<- Y=0
------------------------------------
^
-4 -3 -2 -1 X=0 1 2 3 4Three Parts
Option parts => 3 is the three replications which occur from an X=2^k,Y=2^k point, but continued on indefinitely confined to the upper and right three quadrants.
..--30-- --32-- --24-- --22--.. 4
| | | |
29--26--31 23--18--21 3
| | | |
25---7-- ---5--17 2
| | | | | |
28--27- 6---1---4 -19--20 1
| | | | |
0 <- Y=0
| | |
--2---3 -16--17 -1
| | |
12---8---9 -2
| | |
--13-- --14--15 -3
|
... -4
^
-4 -3 -2 -1 X=0 1 2 3 4FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::ToothpickReplicate->new ()$path = Math::PlanePath::ToothpickTree->new (parts => $integer)-
Create and return a new path object.
partscan be 1, 2, 3 or 4.
SEE ALSO
Math::PlanePath, Math::PlanePath::ToothpickTree, Math::PlanePath::UlamWarburton
HOME PAGE
http://user42.tuxfamily.org/math-planepath/index.html
LICENSE
Copyright 2012 Kevin Ryde
This file is part of Math-PlanePath-Toothpick.
Math-PlanePath-Toothpick 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-Toothpick 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-Toothpick. If not, see <http://www.gnu.org/licenses/>.