Math::PlanePath::SierpinskiCurveStair -- Sierpinski curve with stair-step diagonals
use Math::PlanePath::SierpinskiCurveStair; my $path = Math::PlanePath::SierpinskiCurveStair->new (arms => 2); my ($x, $y) = $path->n_to_xy (123);
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 19
The 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--5
So the SierpinskiCurve
N=0 to N=1 diagonal corresponds to N=0 to N=2 here, and N=2 to N=3 corresponds to N=3 to N=5. The join section N=3 to N=4 gets an extra point at N=6 here, and later similar N=19, etc.
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 17
The 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.
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 6
The multiplies 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.
For diagonal_length
= L and reckoning the first diagonal side N=0 to N=2L as level 0, a level extends out to a triangle
Nlevel = ((6L+4)*4^level - 4) / 3 Xlevel = (L+2)*2^level - 1
For example level 2 in the default L=1 goes to N=((6*1+4)*4^2-4)/3=52 and Xlevel=(1+2)*2^2-1=11. Or in the L=4 sample above level 1 is N=((6*4+4)*4^1-4)/3=36 and Xlevel=(4+2)*2^1-1=11.
The power-of-4 in Nlevel is per the plain SierpinskiCurve
, with factor 2L+1 for the points making the diagonal stair. The "/3" arises from the extra points between replications. They become a power-of-4 series
Nextras = 1+4+4^2+...+4^(level-1) = (4^level-1)/3
For example level 1 is Nextras=(4^1-1)/3=1, being point N=6 in the default L=1. Or for level 2 Nextras=(4^2-1)/3=5 at N=6 and N=19,26,33,46.
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*(Xlevel-1)/2 are visited, for a filled fraction which approaches a constant
FillFrac = Nlevel / (Xlevel*(Xlevel-1)/2) -> 4/3 * (3L+2)/(L+2)^2
For 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.
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 $n
on the path. Points begin at 0 and if $n < 0
then 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.
Math::PlanePath, Math::PlanePath::SierpinskiCurve
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013 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.
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