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 numbered 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 4One 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 8Replication
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=1 above ("One Quadrant"),
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 for the 0,1,2,3 sub-blocks.
The full parts=4 form (the default) 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.
Half 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.
..--29-- --31-- --23-- --21--.. 4
| | | |
28--25--30 22--17--20 3
| | | |
24---7-- ---5--16 2
| | | | | |
27--26- 6---1---4 -18--19 1
| | | | |
0 <- Y=0
| | |
--2---3 -14--15 -1
| | |
10---8---9 -2
| | |
--11-- --12--13 -3
|
... -4
^
-4 -3 -2 -1 X=0 1 2 3 4N=1,4,5,6,7,16,etc above the X axis have an odd number of bits when written in binary. For example N=6 is binary "110" which is 3 digits. Conversely N=0,2,3,8,etc below the X axis have an even number of digits. For example N=8 is "1000" which is 4 digits.
odd bit odd bit
length | length
|
"11" | "10" high two bits of N,
| at odd bit position
----------+----------
|
| "01"
|
| even bit lengthThis occurs because each quadrant contains (4^k-1)*2/3 many points on each doubling. Three of them plus A and B make 3*(4^k-1)*2/3+2 = 2*4^k at each doubling, so with the origin as N=0 each replication level starts
Nlevel_start = 2*4^k
Nlast_below = 2*4^k + 3*(4^k-1)*2/3+2 - 1
= 2*4^k + 2*4^k-1For example k=1 has Nlevel_start = 2*4^1 = 8 and runs through to Nlast_below = 2*4^1 + 2*4^1-1 = 15. In binary this is "1000" through "1111" which are all length 4. The first quadrant then runs 32 to 47 which is binary "10000" to 101111", and the second quadrant 48 to 63 "110000" to "111111".
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::ToothpickReplicate->new ()$path = Math::PlanePath::ToothpickTree->new (parts => $integer)-
Create and return a new path object.
partscan be4 whole plane (the default) 3 three quadrants 2 half plane 1 quadrant
Level Methods
($n_lo, $n_hi) = $path->level_to_n_range($level)-
Return
$n_lo = 0andparts $n_hi ----- ----- 4 (4*8 * 4**$level - 2) / 3 3 (3*8 * 4**$level - 3) / 3 2 (2*8 * 4**$level - 4) / 3 1 ( 8 * 4**$level - 5) / 3It can be noted that parts=3 finishes one N point sooner than the corresponding parts=3 pattern of the
ToothpickTreeform. This is because the Replicate form here finishes the upper quadrants before continuing the lower quadrant, whereasToothpickTreeis by rows so continues to grow the lower quadrant at the same time as the last row of the upper two quadrants. That lower quadrant growth is a single point.
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A053738 (etc)
parts=3
A053738 N of points with Y>0, being odd bit length
also N of parts=3 when taking X,Y points by parts=2 order
A053754 N of points with Y<=0, being even bit lengthSEE ALSO
Math::PlanePath, Math::PlanePath::ToothpickTree, Math::PlanePath::LCornerReplicate, Math::PlanePath::UlamWarburton
HOME PAGE
http://user42.tuxfamily.org/math-planepath/index.html
LICENSE
Copyright 2012, 2013, 2014, 2015 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/>.