NAME

Math::PlanePath::DekkingCentres -- 5x5 self-similar

SYNOPSIS

use Math::PlanePath::DekkingCentres;
my $path = Math::PlanePath::DekkingCentres->new;
my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This is a variation on a 5x5 self-similar curve from

F. M. Dekking, "Recurrent Sets", Advances in Mathematics, volume 44, 1982, pages 79-104, section 4.9 "Gosper-Type Curves"

and which is a horizontal mirror image of the E-curve of McKenna 1978.

The form visits the "centres" of the 5x5 self-similar unit squares of the pattern. The result is some diagonal steps, but replications wholly within 5x5 areas.

                          ...
    |                     /
  9 |  115-116 122-123-124  89--88  86--85--84
    |    |   |    \          |    \  |       |
  8 |  114 117-118 121-120  90  92  87  82--83
    |    |        \   /      |/   \      |
  7 |  113-112 106 119 102  91  94--93  81  77
    |     /   /  |    /  |    /       /   /  |
  6 |  111 107 105 103 101  95--96  80  78  76
    |    |    \   \  |   |        \   \  |   |
  5 |  110-109-108 104 100--99--98--97  79  75
    |                                         \
  4 |   10--11  13--14--15  35--36  38--39--40  74  70  66--65--64
    |    |    \  |       |   |    \  |       |   |   |\   \      |
  3 |    9   7  12  17--16  34  32  37  42--41  73  71  69  67  63
    |    |/   \      |       |/   \      |       |/      |/   /
  2 |    8   5-- 6  18  22  33  30--31  43  47  72  55  68  62--61
    |      /      /   /  |    /       /   /  |    /   \          |
  1 |    4-- 3  19  21  23  29--28  44  46  48  54--53  56--57  60
    |         \   \  |   |        \   \  |   |        \      |   |
Y=0 |    0-- 1-- 2  20  24--25--26--27  45  49--50--51--52  58--59
    +---------------------------------------------------------------
       X=0   1   2   3   4   5   6   7   8   9  10  11  12  13  14

The base pattern is the N=0 to N=24 section. It repeats with rotations or reversals which make the ends join. For example N=75 to N=99 is the base pattern in reverse. Or N=50 to N=74 is reverse and also rotate by -90.

FUNCTIONS

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

$path = Math::PlanePath::DekkingCentres->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.

Level Methods

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

Return (0, 25**$level - 1).

SEE ALSO

Math::PlanePath, Math::PlanePath::DekkingCurve, Math::PlanePath::CincoCurve, Math::PlanePath::PeanoCurve

HOME PAGE

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

LICENSE

Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde

This file is part of Math-PlanePath.

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/>.