Provided by: libmath-planepath-perl_129-1_all 
NAME
Math::PlanePath::SierpinskiCurve -- Sierpinski curve
SYNOPSIS
use Math::PlanePath::SierpinskiCurve;
my $path = Math::PlanePath::SierpinskiCurve->new (arms => 2);
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This is an integer version of the self-similar curve by Waclaw Sierpinski traversing the
plane by right triangles. The default is a single arm of the curve in an eighth of the
plane.
10 | 31-32
| / \
9 | 30 33
| | |
8 | 29 34
| \ /
7 | 25-26 28 35 37-38
| / \ / \ / \
6 | 24 27 36 39
| | |
5 | 23 20 43 40
| \ / \ / \ /
4 | 7--8 22-21 19 44 42-41 55-...
| / \ / \ /
3 | 6 9 18 45 54
| | | | | |
2 | 5 10 17 46 53
| \ / \ / \
1 | 1--2 4 11 13-14 16 47 49-50 52
| / \ / \ / \ / \ / \ /
Y=0 | . 0 3 12 15 48 51
|
+-----------------------------------------------------------
^
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
The tiling it represents is
/
/|\
/ | \
/ | \
/ 7| 8 \
/ \ | / \
/ \ | / \
/ 6 \|/ 9 \
/-------|-------\
/|\ 5 /|\ 10 /|\
/ | \ / | \ / | \
/ | \ / | \ / | \
/ 1| 2 X 4 |11 X 13|14 \
/ \ | / \ | / \ | / \ ...
/ \ | / \ | / \ | / \
/ 0 \|/ 3 \|/ 12 \|/ 15 \
----------------------------------
The points are on a square grid with integer X,Y. 4 points are used in each 3x3 block.
In general a point is used if
X%3==1 or Y%3==1 but not both
which means
((X%3)+(Y%3)) % 2 == 1
The X axis N=0,3,12,15,48,etc are all the integers which use only digits 0 and 3 in base
4. For example N=51 is 303 base4. Or equivalently the values all have doubled bits in
binary, for example N=48 is 110000 binary. (Compare the "CornerReplicate" which also has
these values along the X axis.)
Level Ranges
Counting the N=0 point as level=0, and with each level being 4 copies of the previous, the
levels end at
Nlevel = 4^level - 1 = 0, 3, 15, ...
Xlevel = 3*2^level - 2 = 1, 4, 10, ...
Ylevel = 0
For example level=2 is Nlevel = 2^(2*2)-1 = 15 at X=3*2^2-2 = 10.
Doubling a level is the middle of the next level and is the top of the triangle in that
next level.
Ntop = 2*4^level - 1 = 1, 7, 31, ...
Xtop = 3*2^level - 1 = 2, 5, 11, ...
Ytop = 3*2^level - 2 = Xlevel = 1, 4, 10, ...
For example doubling level=2 is Ntop = 2*4^2-1 = 31 at X=3*2^2-1 = 11 and Y=3*2^2-2 = 10.
The factor of 3 arises from the three steps which make up the N=0,1,2,3 section. The
Xlevel width grows as
Xlevel(1) = 3
Xlevel(level) = 2*Xwidth(level-1) + 3
which dividing out the factor of 3 is 2*w+1, giving 2^k-1 (in binary a left shift and
bring in a new 1 bit).
Notice too the Nlevel points as a fraction of the triangular area Xlevel*(Xlevel-1)/2
gives the 4 out of 9 points filled,
FillFrac = Nlevel / (Xlevel*(Xlevel-1)/2)
-> 4/9
Arms
The optional "arms" parameter can draw multiple curves, each advancing successively. For
example 2 arms,
arms => 2 ...
|
11 | 33 39 57 63
| / \ / \ / \ /
10 | 31 35-37 41 55 59-61 62-...
| \ / \ /
9 | 29 43 53 60
| | | | |
8 | 27 45 51 58
| / \ / \
7 | 25 21-19 47-49 50-52 56
| \ / \ / \ /
6 | 23 17 48 54
| | |
5 | 9 15 46 40
| / \ / \ / \
4 | 7 11-13 14-16 44-42 38
| \ / \ /
3 | 5 12 18 36
| | | | |
2 | 3 10 20 34
| / \ / \
1 | 1 2--4 8 22 26-28 32
| / \ / \ / \ /
Y=0 | 0 6 24 30
|
+-----------------------------------------
^
X=0 1 2 3 4 5 6 7 8 9 10 11
The N=0 point is at X=1,Y=0 (in all arms forms) so that the second arm is within the first
quadrant.
1 to 8 arms can be done this way. For example 8 arms are
arms => 8
... ... 6
| |
58 34 33 57 5
\ / \ / \ /
...-59 50-42 26 25 41-49 56-... 4
\ / \ /
51 18 17 48 3
| | | |
43 10 9 40 2
/ \ / \
35 19-11 2 1 8-16 32 1
\ / \ / \ /
27 3 . 0 24 <- Y=0
28 4 7 31 -1
/ \ / \ / \
36 20-12 5 6 15-23 39 -2
\ / \ /
44 13 14 47 -3
| | | |
52 21 22 55 -4
/ \ / \
...-60 53-45 29 30 46-54 63-... -5
/ \ / \ / \
61 37 38 62 -6
| |
... ... -7
^
-7 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6
The middle "." is the origin X=0,Y=0. It would be more symmetrical to make the origin the
middle of the eight arms, at 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 it if desired.
Spacing
The optional "diagonal_spacing" and "straight_spacing" can increase the space between
points diagonally or vertically+horizontally. The default for each is 1.
straight_spacing => 2
diagonal_spacing => 1
7 ----- 8
/ \
6 9
| |
| |
| |
5 10 ...
\ / \
1 ----- 2 4 11 13 ---- 14 16
/ \ / \ / \ /
0 3 12 15
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 ...
The effect is only to spread the points. The straight lines are both horizontal and
vertical so when they're stretched the curve remains on a 45 degree angle in an eighth of
the plane.
In the level formulas above the "3" factor becomes 2*d+s, effectively being the N=0 to N=3
section sized as d+s+d.
d = diagonal_spacing
s = straight_spacing
Xlevel = (2d+s)*(2^level - 1) + 1
Xtop = (2d+s)*2^(level-1) - d - s + 1
Ytop = (2d+s)*2^(level-1) - d - s
Closed Curve
Sierpinski's original conception was a closed curve filling a unit square by ever greater
self-similar detail,
/\_/\ /\_/\ /\_/\ /\_/\
\ / \ / \ / \ /
| | | | | | | |
/ _ \_/ _ \ / _ \_/ _ \
\/ \ / \/ \/ \ / \/
| | | |
/\_/ _ \_/\ /\_/ _ \_/\
\ / \ / \ / \ /
| | | | | | | |
/ _ \ / _ \_/ _ \ / _ \
\/ \/ \/ \ / \/ \/ \/
| |
/\_/\ /\_/ _ \_/\ /\_/\
\ / \ / \ / \ /
| | | | | | | |
/ _ \_/ _ \ / _ \_/ _ \
\/ \ / \/ \/ \ / \/
| | | |
/\_/ _ \_/\ /\_/ _ \_/\
\ / \ / \ / \ /
| | | | | | | |
/ _ \ / _ \ / _ \ / _ \
\/ \/ \/ \/ \/ \/ \/ \/
The code here might be pressed into use for this by drawing a mirror image of the curve
N=0 through Nlevel. Or using the "arms=>2" form N=0 to N=4^level - 1, inclusive, and
joining up the ends.
The curve is also usually conceived as scaling down by quarters. This can be had with
"straight_spacing => 2" and then an offset to X+1,Y+1 to centre in a 4*2^level square
Koch Curve Midpoints
The replicating structure is the same as the Koch curve (Math::PlanePath::KochCurve) in
that the curve repeats four times to make the next level.
The Sierpinski curve points are midpoints of a Koch curve of 90 degree angles with a unit
gap between verticals.
Koch Curve Koch Curve
90 degree angles, unit gap
/\ | |
/ \ | |
/ \ | |
----- ----- ------ ------
Sierpinski curve points "*" as midpoints
| |
7 8
| |
---6--- ---9---
---5--- --10---
| | | | | |
1 2 4 11 13 14
| | | | | |
---0--- ---3--- --12--- --15---
Koch Curve Rounded
The Sierpinski curve in mirror image across the X=Y diagonal and rotated -45 degrees is
pairs of points on the lines of the Koch curve 90 degree angles unit gap from above.
Sierpinski curve mirror image and turn -45 degrees
two points on each Koch line segment
15 16
| |
14 17
12--13 . . 18--19
11--10 . . 21--20
3 4 9 22 27 28
| | | | | |
2 5 8 23 26 29
0---1 . . 6---7 . . 24--25 . . 30--31
This is a kind of "rounded" form of the 90-degree Koch, similar what "DragonRounded" does
for the "DragonCurve". Each 90 turn of the Koch curve is done by two turns of 45 degrees
in the Sierpinski curve here, and each 180 degree turn in the Koch is two 90 degree turns
here. So the Sierpinski turn sequence is pairs of the Koch turn sequence, as follows.
The mirroring means a swap left<->right between the two.
N=1 2 3 4 5 6 7 8
Koch L R L L L R L R ...
N=1,2 3,4 5,6 7,8 9,10 11,12 13,14 15,16
Sierp R R L L R R R R R R L L R R L L ...
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::SierpinskiCurve->new ()"
"$path = Math::PlanePath::SierpinskiCurve->new (arms => $integer, diagonal_spacing =>
$integer, straight_spacing => $integer)"
Create and return a new path object.
"($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 = $path->n_start()"
Return 0, the first N in the path.
Level Methods
"($n_lo, $n_hi) = $path->level_to_n_range($level)"
Return "(0, 4**$level - 1)", or for multiple arms return "(0, $arms * 4**$level - 1)".
There are 4^level points in a level, or arms*4^level when multiple arms, numbered
starting from 0.
FORMULAS
N to dX,dY
The curve direction at N even can be calculated from the base-4 digits of N/2 in a fashion
similar to the Koch curve ("N to Direction" in Math::PlanePath::KochCurve). Counting
direction in eighths so 0=East, 1=North-East, 2=North, etc,
digit direction
----- ---------
0 0
1 -2
2 2
3 0
direction = 1 + sum direction[base-4 digits of N/2]
for N even
For example the direction at N=10 has N/2=5 which is "11" in base-4, so direction =
1+(-2)+(-2) = -3 = south-west.
The 1 in 1+sum is direction north-east for N=0, then -2 or +2 for the digits follow the
curve. For an odd arm the curve is mirrored and the sign of each digit direction is
flipped, so a subtract instead of add,
direction
mirrored = 1 - sum direction[base-4 digits of N/2]
for N even
For odd N=2k+1 the direction at N=2k is calculated and then also the turn which is made
from N=2k to N=2(k+1). This is similar to the Koch curve next turn ("N to Next Turn" in
Math::PlanePath::KochCurve).
lowest non-3 next turn
digit of N/2 (at N=2k+1,N=2k+2)
------------ ----------------
0 -1 (right)
1 +2 (left)
2 -1 (right)
Again the turn is in eighths, so -1 means -45 degrees (to the right). For example at N=14
has N/2=7 which is "13" in base-4 so lowest non-3 is "1" which is turn +2, so at N=15 and
N=16 turn by 90 degrees left.
direction = 1 + sum direction[base-4 digits of k]
+ if N odd then nextturn[low-non-3 of k]
for N=2k or 2k+1
dX,dY = direction to 1,0 1,1 0,1 etc
For fractional N the same nextturn is applied to calculate the direction of the next
segment, and combined with the integer dX,dY as per "N to dX,dY -- Fractional" in
Math::PlanePath.
N=2k or 2k+1 + frac
direction = 1 + sum direction[base-4 digits of k]
if (frac != 0 or N odd)
turn = nextturn[low-non-3 of k]
if N odd then direction += turn
dX,dY = direction to 1,0 1,1 0,1 etc
if frac!=0 then
direction += turn
next_dX,next_dY = direction to 1,0 1,1 0,1 etc
dX += frac*(next_dX - dX)
dY += frac*(next_dY - dY)
For the "straight_spacing" and "diagonal_spacing" options the dX,dY values are not units
like dX=1,dY=0 but instead are the spacing amount, either straight or diagonal so
direction delta with spacing
--------- -------------------------
0 dX=straight_spacing, dY=0
1 dX=diagonal_spacing, dY=diagonal_spacing
2 dX=0, dY=straight_spacing
3 dX=-diagonal_spacing, dY=diagonal_spacing
etc
As an alternative, it's possible to take just base-4 digits of N, without separate
handling for the low-bit of N, but it requires an adjustment on the low base-4 digit, and
the next turn calculation for fractional N becomes hairier. A little state table could
encode the cumulative and lowest whatever if desired, to take N by base-4 digits high to
low, or equivalently by bits high to low with an initial state based on high bit at an odd
or even bit position.
OEIS
The Sierpinski curve is in Sloane's Online Encyclopedia of Integer Sequences as,
<http://oeis.org/A039963> (etc)
A039963 turn 1=right,0=left, doubling the KochCurve turns
A081706 N-1 of left turn positions
(first values 2,3 whereas N=3,4 here)
A127254 abs(dY), so 0=horizontal, 1=vertical or diagonal,
except extra initial 1
A081026 X at N=2^k, being successively 3*2^j-1, 3*2^j
A039963 is numbered starting n=0 for the first turn, which is at the point N=1 in the path
here.
SEE ALSO
Math::PlanePath, Math::PlanePath::SierpinskiCurveStair,
Math::PlanePath::SierpinskiArrowhead, Math::PlanePath::KochCurve
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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/>.