NAME
Math::PlanePath::SierpinskiCurveStair -- Sierpinski curve with stair-step diagonals
SYNOPSIS
use Math::PlanePath::SierpinskiCurveStair;
my $path = Math::PlanePath::SierpinskiCurveStair->new (arms => 2);
my ($x, $y) = $path->n_to_xy (123);DESCRIPTION
This is a variation on the SierpinskiCurve with stair-step diagonal parts.
10 | 52-53
| | |
9 | 50-51 54-55
| | |
8 | 49-48 57-56
| | |
7 | 42-43 46-47 58-59 62-63
| | | | | | |
6 | 40-41 44-45 60-61 64-65
| | |
5 | 39-38 35-34 71-70 67-66
| | | | | | |
4 | 12-13 37-36 33-32 73-72 69-68 92-93
| | | | | | |
3 | 10-11 14-15 30-31 74-75 90-91 94-95
| | | | | | |
2 | 9--8 17-16 29-28 77-76 89-88 97-96
| | | | | | |
1 | 2--3 6--7 18-19 22-23 26-27 78-79 82-83 86-87 98-99
| | | | | | | | | | | | |
Y=0 | 0--1 4--5 20-21 24-25 80-81 84-85 ...
|
+-------------------------------------------------------------
^
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19The tiling is the same as the SierpinskiCurve, but each diagonal is a stair step horizontal and vertical. The correspondence is
SierpinskiCurve SierpinskiCurveStair
7-- 12--
/ |
6 10-11
| |
5 9--8
\ |
1--2 4 2--3 6--7
/ \ / | | |
0 3 0--1 4--5The SierpinskiCurve N=0 to N=3 corresponds to N=0 to N=5 here. N=7 to N=12 which is a copy of the N=0 to N=5 base. Point N=6 is an extra in between the parts. The next such extra is N=19.
Diagonal Length
The diagonal_length option can make longer diagonals, still in stair-step style. For example
diagonal_length => 4
10 | 36-37
| | |
9 | 34-35 38-39
| | |
8 | 32-33 40-41
| | |
7 | 30-31 42-43
| | |
6 | 28-29 44-45
| | |
5 | 27-26 47-46
| | |
4 | 8--9 25-24 49-48 ...
| | | | | |
3 | 6--7 10-11 23-22 51-50 62-63
| | | | | |
2 | 4--5 12-13 21-20 53-52 60-61
| | | | | |
1 | 2--3 14-15 18-19 54-55 58-59
| | | | | |
Y=0 | 0--1 16-17 56-57
|
+------------------------------------------------------
^
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17The length is reckoned from N=0 to the end of the first side N=8, which is X=1 to X=5 for length 4 units.
Arms
The optional arms parameter can give up to eight copies of the curve, each advancing successively. For example
arms => 8
98-90 66-58 57-65 89-97 5
| | | | | |
99 82-74 50-42 41-49 73-81 96 4
| | | |
91-83 26-34 33-25 80-88 3
| | | |
67-75 18-10 9-17 72-64 2
| | | |
59-51 27-19 2 1 16-24 48-56 1
| | | | | |
43-35 11--3 . 0--8 32-40 <- Y=0
44-36 12--4 7-15 39-47 -1
| | | | | |
60-52 28-20 5 6 23-31 55-63 -2
| | | |
68-76 21-13 14-22 79-71 -3
| | | |
92-84 29-37 38-30 87-95 -4
| |
85-77 53-45 46-54 78-86 -5
| | | | | |
93 69-61 62-70 94 -6
^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6The multiples of 8 (or however many arms) N=0,8,16,etc is the original curve, and the further mod 8 parts are the copies.
The middle "." shown is the origin X=0,Y=0. It would be more symmetrical to have the origin the middle of the eight arms, which would be X=-0.5,Y=-0.5 in the above, but that would give fractional X,Y values. Apply an offset X+0.5,Y+0.5 to centre if desired.
Level Ranges
The N=0 point is reckoned as level=0, then N=0 to N=5 inclusive is level=1, etc. Each level is 4 copies of the previous and an extra 2 points between.
LevelPoints[k] = 4*LevelPoints[k-1] + 2 starting LevelPoints[0]=1
= 2 + 2*4 + 2*4^2 + ... + 2*4^(k-1) + 1*4^k
= (5*4^k - 2)/3
Nlevel[k] = LevelPoints[k] - 1 since starting at N=0
= 5*(4^k - 1)/3
= 0, 5, 25, 105, 425, 1705, 6825, 27305, ... (A146882)The width along the X axis of a level doubles each time, plus an extra distance 3 between.
LevelWidth[k] = 2*LevelWidth[k-1] + 3 starting LevelWidth[0]=0
= 3 + 3*2 + 3*2^2 + ... + 3*2^(k-1) + 0*2^k
= 3*(2^k - 1)
Xlevel[k] = 1 + LevelWidth[k]
= 3*2^k - 2
= 1, 4, 10, 22, 46, 94, 190, 382, ... (A033484)Level Ranges with Diagonal Length
With diagonal_length = L, level=0 is reckoned as having L many points instead of just 1.
LevelPoints[k] = 2 + 2*4 + 2*4^2 + ... + 2*4^(k-1) + L*4^k
= ( (3L+2)*4^k - 2 )/3
Nlevel[k] = LevelPoints[k] - 1
= ( (3L+2)*4^k - 5 )/3The width of level=0 becomes L-1 instead of 0.
LevelWidth[k] = 2*LevelWidth[k-1] + 3 starting LevelWidth[0]=L-1
= 3 + 3*2 + 3*2^2 + ... + 3*2^(k-1) + (L-1)*2^k
= (L+2)*2^k - 3
Xlevel[k] = 1 + LevelWidth[k]
= (L+2)*2^k - 2Level=0 as L many points can be thought of as a little block which is replicated in mirror image to make level=1. For example the diagonal 4 example above becomes
8 9 diagonal_length => 4
| |
6--7 10-11
| |
. 5 12 .
2--3 14-15
| |
0--1 16-17The spacing between the parts is had in the tiling by taking a margin of 1/2 at the base and 1 horizontally left and right.
Level Fill
The curve doesn't visit all the points in the eighth of the plane below the X=Y diagonal. In general Nlevel+1 many points of the triangular area Xlevel^2/4 are visited, for a filled fraction which approaches a constant
Nlevel 4*(3L+2)
FillFrac = ------------ -> ---------
Xlevel^2 / 4 3*(L+2)^2For example the default L=1 has FillFrac=20/27=0.74. Or L=2 FillFrac=2/3=0.66. As the diagonal length increases the fraction decreases due to the growing holes in the pattern.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.
$path = Math::PlanePath::SierpinskiCurveStair->new ()$path = Math::PlanePath::SierpinskiCurveStair->new (diagonal_length => $L, arms => $A)-
Create and return a new path object.
($x,$y) = $path->n_to_xy ($n)-
Return the X,Y coordinates of point number
$non the path. Points begin at 0 and if$n < 0then the return is an empty list.Fractional positions give an X,Y position along a straight line between the integer positions.
$n = $path->n_start()-
Return 0, the first N in the path.
Level Methods
($n_lo, $n_hi) = $path->level_to_n_range($level)-
Return
(0, ((3*$diagonal_length +2) * 4**$level - 5)/3as per "Level Ranges with Diagonal Length" above.
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A146882 (etc)
A146882 Nlevel, for level=1 up
A033484 Xmax and Ymax in level, being 3*2^n - 2SEE ALSO
Math::PlanePath, Math::PlanePath::SierpinskiCurve
HOME PAGE
http://user42.tuxfamily.org/math-planepath/index.html
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019 Kevin Ryde
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/>.