Math::PlanePath::ComplexPlus -- points in complex base i+r
use Math::PlanePath::ComplexPlus; my $path = Math::PlanePath::ComplexPlus->new; my ($x, $y) = $path->n_to_xy (123);
This path traverses points by a complex number base i+r for integer r>=1. The default is base i+1
30 31 14 15 5 28 29 12 13 4 26 27 22 23 10 11 6 7 3 24 25 20 21 8 9 4 5 2 62 63 46 47 18 19 2 3 1 60 61 44 45 16 17 0 1 <- Y=0 58 59 54 55 42 43 38 39 -1 56 57 52 53 40 41 36 37 -2 50 51 94 95 34 35 78 79 -3 48 49 92 93 32 33 76 77 -4 90 91 86 87 74 75 70 71 -5 88 89 84 85 72 73 68 69 -6 126 127 110 111 82 83 66 67 -7 124 125 108 109 80 81 64 65 -8 122 123 118 119 106 107 102 103 -9 120 121 116 117 104 105 100 101 -10 114 115 98 99 -11 112 113 96 97 -12 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1 2
The shape of these points N=0 to N=2^k-1 inclusive is equivalent to the twindragon turned 135 degrees. Each complex base point corresponds to a unit square inside the twindragon curve (two DragonCurve back-to-back).
Option realpart => $r selects another r for complex base b=i+r. For example
realpart => $r
realpart => 2 45 46 47 48 49 8 40 41 42 43 44 7 35 36 37 38 39 6 30 31 32 33 34 5 25 26 27 28 29 20 21 22 23 24 4 15 16 17 18 19 3 10 11 12 13 14 2 5 6 7 8 9 1 0 1 2 3 4 <- Y=0 ^ X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
N is broken into digits of a base norm=r*r+1, ie. digits 0 to r*r inclusive.
norm = r*r + 1 Nstart = 0 Nlevel = norm^level - 1
The low digit of N makes horizontal runs of r*r+1 many points, such as N=0 to N=4, then N=5 to N=9, etc shown above. In the default r=1 these runs are 2 long. For r=2 shown above they're 2*2+1=5 long, or r=3 would be 3*3+1=10, etc.
The offset for each successive run is i+r, ie. Y=1,X=r such as at N=5 shown above. Then the offset for the next level is (i+r)^2 = (2r*i + r^2-1) so N=25 begins at Y=2*r=4, X=r*r-1=3. In general each level adds an angle
angle = atan(1/r) Nlevel_angle = level * angle
So the points spiral around anti-clockwise. For r=1 the angle is atan(1/1)=45 degrees, so that for example level=4 is angle 4*45=180 degrees, putting N=2^4=16 on the negative X axis as shown in the first sample above.
As r becomes bigger the angle becomes smaller, making it spiral more slowly. The points never fill the plane, but the set of points N=0 to Nlevel are all touching.
For realpart => 1, an optional arms => 2 adds a second copy of the curve rotated 180 degrees and starting from X=0,Y=1. It meshes perfectly to fill the plane. Each arm advances successively so N=0,2,4,etc is the plain path and N=1,3,5,7,etc is the copy
realpart => 1
arms => 2
arms=>2 60 62 28 30 5 56 58 24 26 4 52 54 44 46 20 22 12 14 3 48 50 40 42 16 18 8 10 2 36 38 3 1 4 6 35 33 1 32 34 7 5 0 2 39 37 <- Y=0 11 9 19 17 43 41 51 49 -1 15 13 23 21 47 45 55 53 -2 27 25 59 57 -3 31 29 63 61 -4 ^ -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6
There's no arms parameter for other realpart values as yet, only for i+1. Is there a good rotated arrangement for others? Do "norm" many copies fill the plane in general?
arms
realpart
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::ComplexPlus->new ()
Create and return a new path object.
($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
$n < 0
($n_lo, $n_hi) = $path->level_to_n_range($level)
Return (0, 2**$level - 1), or for 2 arms return (0, 2 * 2**$level - 1). With the realpart option return (0, $norm**$level - 1) where norm=realpart^2+1.
(0, 2**$level - 1)
(0, 2 * 2**$level - 1)
(0, $norm**$level - 1)
Various formulas and pictures etc for the i+1 case can be found in the author's long mathematical write-up (section "Complex Base i+1")
http://user42.tuxfamily.org/dragon/index.html
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A290885 (etc)
realpart=1 (i+1, the default) A290885 -X A290884 Y A290886 norm X^2 + Y^2 A146559 dX at N=2^k-1 (step to next replication level) A077950,A077870 location of ComplexMinus origin in ComplexPlus (mirror horizontal even level, vertical odd level)
Math::PlanePath, Math::PlanePath::ComplexMinus, Math::PlanePath::ComplexRevolving, Math::PlanePath::DragonCurve
http://user42.tuxfamily.org/math-planepath/index.html
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.
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To install Math::PlanePath, copy and paste the appropriate command in to your terminal.
cpanm
cpanm Math::PlanePath
CPAN shell
perl -MCPAN -e shell install Math::PlanePath
For more information on module installation, please visit the detailed CPAN module installation guide.