Math::PlanePath::SierpinskiCurve -- Sierpinski curve
use Math::PlanePath::SierpinskiCurve; my $path = Math::PlanePath::SierpinskiCurve->new (arms => 2); my ($x, $y) = $path->n_to_xy (123);
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.)
Counting the N=0 to N=3 as level=1, N=0 to N=15 as level 2, etc, the end of each level, back at the X axis, is
Nlevel = 4^level - 1 Xlevel = 3*2^level - 2 Ylevel = 0
For example level=2 is Nend = 2^(2*2)-1 = 15 at X=3*2^2-2 = 10.
The top of each level is half way along,
Ntop = (4^level)/2 - 1 Xtop = 3*2^(level-1) - 1 Ytop = 3*2^(level-1) - 2
For example level=3 is Ntop = 2^(2*3-1)-1 = 31 at X=3*2^(3-1)-1 = 11 and Y=3*2^(3-1)-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+1) = 2*Xwidth(level) + 3
which dividing out the factor of 3 is 2*w+1, given 2^k-1 (in binary a left shift and bring in a new 1 bit, giving 2^k-1).
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
The optional arms
parameter can draw multiple curves, each advancing successively. For example arms => 2
,
... | 33 39 57 63 11 / \ / \ / \ / 31 35-37 41 55 59-61 62-.. 10 \ / \ / 29 43 53 60 9 | | | | 27 45 51 58 8 / \ / \ 25 21-19 47-49 50-52 56 7 \ / \ / \ / 23 17 48 54 6 | | 9 15 46 40 5 / \ / \ / \ 7 11-13 14-16 44-42 38 4 \ / \ / 5 12 18 36 3 | | | | 3 10 20 34 2 / \ / \ 1 2--4 8 22 26-28 32 1 / \ / \ / \ / 0 6 24 30 <- Y=0 ^ 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. arms=>8
is as follows.
... ... 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 with X+0.5,Y+0.5 to centre it if desired.
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. 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
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 (above). Or using the arms=>2
form N=0 to N=4^level, 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
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,
Koch Curve Sierpinski Curve (mirror image) | | / \ | | / \ | | --- --- --- ---
The turns in the Sierpinski curve are by 90 degrees and 180 degrees, done in two steps 45+45=90 when turning right or 90+90=180 when turning left.
The turn sequence left or right is the same as the Koch curve ("N to Turn" in Math::PlanePath::KochCurve) except the Sierpinski curve makes each turn in two steps, and mirrored to swap L<->R. For example the Koch curve starts with Left at N=1 which for the Sierpinski curve becomes two turns Right,Right at N=1,N=2.
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 ...
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.
The curve direction at an even N 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 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 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.
N=2k or 2k+1 direction = 1 + sum direction[base-4 digits of k] + if N odd then nextturn[low-non-3 of k] 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 for on the low base-4 digit, and the next turn calculation for fractional N becomes hairier. A little state table could no doubt 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.
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.
Math::PlanePath, Math::PlanePath::SierpinskiCurveStair, Math::PlanePath::SierpinskiArrowhead, Math::PlanePath::KochCurve
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014 Kevin Ryde
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