Kevin Ryde > Math-PlanePath > Math::PlanePath::SierpinskiCurve

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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 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/>.

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