Sierpinski Triangle¶

The path is related to the Sierpinski triangle or "gasket" by treating each line segment as the side of a little triangle. The N=0 to N=1 segment has a triangle on the left, N=1 to N=2 on the right, and N=2 to N=3 underneath, which are per the A,B,C parts shown above. Notice there's no middle little triangle "D" in the triplets of line segments. In general a segment N to N+1 has its little triangle to the left if N even or to the right if N odd.

This pattern of little triangles is why the N=4 to N=5 looks like it hasn't visited the vertex of the triangular N=0 to N=9 -- the 4 to 5 segment is standing in for a little triangle to the left of that segment. Similarly N=13 to N=14 and each alternate side midway through replication levels.

There's easier ways to generate the Sierpinski triangle though. One of the simplest is to take X,Y coordinates which have no 1 bit on common, ie. a bitwise-AND,

    ($x & $y) == 0

which gives the shape in the first quadrant X>=0,Y>=0. The same can be had with the "ZOrderCurve" path by plotting all numbers N which have no digit 3 in their base-4 representation (see "Power of 2 Values" in Math::PlanePath::ZOrderCurve), since digit 3s in that case are X,Y points with a 1 bit in common.

The attraction of this Arrowhead path is that it makes a connected traversal through the Sierpinski triangle pattern.

Level Sizes¶

Counting the N=0,1,2,3 part as level 1, each level goes from

    Nstart = 0
    Nlevel = 3^level

inclusive of the final triangle corner position. For example level 2 is from N=0 to N=3^2=9. Each level doubles in size,

           0  <= Y <= 2^level
    - 2^level <= X <= 2^level

The final Nlevel position is alternately on the right or left,

    Xlevel = /  2^level      if level even
             \  - 2^level    if level odd

The Y axis is crossed, ie. X=0, at N=2,6,18,etc which is is 2/3 through the level, ie. after two replications of the previous level,

    Ncross = 2/3 * 3^level
           = 2 * 3^(level-1)

Align Parameter¶

An optional "align" parameter controls how the points are arranged relative to the Y axis. The default shown above is "triangular". The choices are the same as for the "SierpinskiTriangle" path.

"right" means points to the right of the axis, packed next to each other and so using an eighth of the plane.

    align => "right"
        |   |
     8  |  27-26    19-18    15-14
        |      |   /    |   /    |
     7  |     25 20    17-16    13
        |    /    |            /
     6  |  24    21       11-12
        |   |   /        /
     5  |  23-22       10
        |               |
     4  |      5--6     9
        |    /    |   /
     3  |   4     7--8
        |   |
     2  |   3--2
        |      |
     1  |      1
        |    /
    Y=0 |   0
        +--------------------------
           X=0 1  2  3  4  5  6  7

"left" is similar but skewed to the left of the Y axis, ie. into negative X.

    align => "left"
    \
     27-26    19-18    15-14     |  8
          \    |   \    |   \    |
           25 20    17-16    13  |  7
            |   \             |  |
           24    21       11-12  |  6
             \    |        |     |
              23-22       10     |  5
                            \    |
                     5--6     9  |  4
                     |   \    |  |
                     4     7--8  |  3
                      \          |
                        3--2     |  2
                            \    |
                              1  |  1
                              |  |
                              0  | Y=0
    -----------------------------+
     -8 -7 -6 -5 -4 -3 -2 -1 X=0

"diagonal" put rows on diagonals down from the Y axis to the X axis. This uses the whole of the first quadrant (with gaps).

    align => "diagonal"
        |   |
     8  |  27
        |    \
     7  |     26
        |      |
     6  |  24-25
        |   |
     5  |  23    20-19
        |    \    |   \
     4  |     22-21    18
        |               |
     3  |   4--5       17
        |   |   \        \
     2  |   3     6       16-15
        |    \    |            \
     1  |      2  7    10-11    14
        |      |   \    |   \    |
    Y=0 |   0--1     8--9    12-13
        +--------------------------
           X=0 1  2  3  4  5  6  7

Sideways¶

Sierpinski presents the curve with a base along the X axis. That can be had here with a -60 degree rotation (see "Triangular Lattice" in Math::PlanePath),

    (3Y+X)/2, (Y-X)/2       rotate -60

The first point N=1 is then along the X axis at X=2,Y=0. Or to have it diagonally upwards first then apply a mirroring -X before rotating

    (3Y-X)/2, (Y+X)/2       mirror X and rotate -60

The plain -60 rotate puts the Nlevel=3^level point on the X axis for even number level, and at the top peak for odd level. With the extra mirroring it's the other way around. If drawing successive levels then the two ways can be alternated to have the endpoint on the X axis each time.

Level Methods¶

"($n_lo, $n_hi) = $path->level_to_n_range($level)"

Return "(0, 3**$level)".

Turn¶

The turn at N is given by ternary

    turn(N)    N + LowestNonZero(N) + CountLowZeros(N)
    -------    ---------------------------------------
     left                      even
     right                     odd

In the replications, turns N=1 and N=2 are both left. A low 0 digit expansion is mirror image to maintain initial segment direction. Parts "B" digit=1 above are each mirror images too so turns flip.

    [flip for each 1 digit]  [1 or 2]  [flip for each low 0 digit]

N is odd or even according as the number of ternary 1 digits is odd or even (all 2 digits being even of course), so N parity accounts for the "B" mirrorings. On a binary computer this is just the low bit rather than examining the high digits of N. In any case if the ternary lowest non-0 is a 1 then it is not such a mirror so adding LowestNonZero cancels that.

This turn rule is noted by Alexis Monnerot-Dumaine in OEIS A156595. That sequence is LowestNonZero(N) + CountLowZeros(N) mod 2 and flipping according as N odd or even is the arrowhead turns.