NAME

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

SYNOPSIS

 use Math::PlanePath::SierpinskiArrowheadCentres;
 my $path = Math::PlanePath::SierpinskiArrowheadCentres->new;
 my ($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 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.

FORMULAS

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.

SEE ALSO

Math::PlanePath, Math::PlanePath::SierpinskiArrowhead, Math::PlanePath::SierpinskiTriangle

HOME PAGE

http://user42.tuxfamily.org/math-planepath/index.html

LICENSE

Copyright 2011, 2012 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/>.

cpanm Math::PlanePath

perl -MCPAN -e shell
install Math::PlanePath