Math::PlanePath::DekkingCentres -- 5x5 self-similar
use Math::PlanePath::DekkingCentres; my $path = Math::PlanePath::DekkingCentres->new; my ($x, $y) = $path->n_to_xy (123);
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.
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 $n < 0
then the return is an empty list.
($n_lo, $n_hi) = $path->level_to_n_range($level)
Return (0, 25**$level - 1)
.
Math::PlanePath, Math::PlanePath::DekkingCurve, Math::PlanePath::CincoCurve, Math::PlanePath::PeanoCurve
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017 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.
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