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# NAME

Math::PlanePath::SierpinskiArrowheadCentres -- self-similar triangular path traversal

# SYNOPSIS

`````` use Math::PlanePath::SierpinskiArrowheadCentres;
my (\$x, \$y) = \$path->n_to_xy (123);``````

# DESCRIPTION

This is a version of the Sierpinski arrowhead path taking the centres of each triangle represented by the arrowhead segments. The effect is to traverse the Sierpinski triangle.

``````              ...                                 ...
/                                   /
.    30     .     .     .     .     .    65     .   ...
/                                      \        /
28----29     .     .     .     .     .     .    66    68     9
\                                               \  /
27     .     .     .     .     .     .     .    67        8
\
26----25    19----18----17    15----14----13           7
/        \           \  /           /
24     .    20     .    16     .    12              6
\        /                       /
23    21     .     .    10----11                 5
\  /                    \
22     .     .     .     9                    4
/
4---- 5---- 6     8                       3
\           \  /
3     .     7                          2
\
2---- 1                             1
/
0                            <- Y=0

-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7``````

The base figure is the N=0 to N=2 shape. It's repeated up in mirror image as N=3 to N=6 then across rotated as N=6 to N=9. At the next level the same is done with the N=0 to N=8 shape, up mirrored as N=9 to N=17 and across rotated as N=18 to N=26, etc.

The X,Y coordinates are on a triangular lattice using every second integer X, per "Triangular Lattice" in Math::PlanePath.

The base pattern is a triangle like

``````      .-------.-------.
\     / \     /
\ 2 / m \ 1 /
\ /     \ /
.- - - -.
\     /
\ 0 /
\ /
``````

Higher levels replicate this within the triangles 0,1,2 but the middle "m" is not traversed. The result is the familiar Sierpinski triangle by connected steps 2 across or 1 diagonal.

``````    * * * * * * * * * * * * * * * *
*   *   *   *   *   *   *   *
* *     * *     * *     * *
*       *       *       *
* * * *         * * * *
*   *           *   *
* *             * *
*               *
* * * * * * * *
*   *   *   *
* *     * *
*       *
* * * *
*   *
* *
*                                       ``````

See the SierpinskiTriangle path to traverse by rows instead.

## Level Ranges

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

``````    Nstart = 0
Nlevel = 3^level - 1     inclusive``````

For example level 2 from N=0 to N=3^2-1=9. Each level doubles in size,

``````                 0  <= Y <= 2^level - 1
- (2^level - 1) <= X <= 2^level - 1``````

The Nlevel position is alternately on the right or left,

``````    Xlevel = /  2^level - 1      if level even
\  - 2^level + 1    if level odd``````

The Y axis ie. X=0, is crossed just after N=1,5,17,etc which is is 2/3 through the level, which is after two replications of the previous level,

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

# FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.

`\$path = Math::PlanePath::SierpinskiArrowheadCentres->new ()`

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.

If `\$n` is not an integer then the return is on a straight line between the integer points.

## Rectangle to N Range

An easy over-estimate of the range can be had from inverting the Nlevel formulas in "Level Ranges" above.

``````    level = floor(log2(Ymax)) + 1
Nmax = 3^level - 1``````

For example Y=5, level=floor(log2(11))+1=3, so Nmax=3^3-1=26, which is the left end of the Y=7 row, ie. rounded up to the end of the Y=4 to Y=7 replication.