NAME
Math::PlanePath::MathImageComplexMinus -- complex number base i-r, including "twindragon"
SYNOPSIS
use Math::PlanePath::MathImageComplexMinus;
my $path = Math::PlanePath::MathImageComplexMinus->new (realpart=>1);
my ($x, $y) = $path->n_to_xy (123);DESCRIPTION
In progress.
This path traverses points by complex number base i-r. The default is base i-1 giving an integer version of the "twindragon".
26 27 10 11 3
24 25 8 9 2
18 19 30 31 2 3 14 15 1
16 17 28 29 0 1 12 13 <- Y=0
22 23 6 7 58 59 42 43 -1
20 21 4 5 56 57 40 41 -2
50 51 62 63 34 35 46 47 -3
48 49 60 61 32 33 44 45 -4
54 55 38 39 -5
52 53 36 37 -6
^
-5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7With base b=i-1, a complex integer can be represented
X+Yi = a[n]*b^n + ... + a[2]*b^2 + a[1]*b + a[0]The digits a[n] to a[0] are all either 0 or 1. N is those a[i] as bits, the X,Y is the resulting complex number. It can be shown that this is a one-to-one transformation so every integer point of the plane is visited.
The shape of a given N=0 to N=2^level-1 range is repeated in the next N=2^level to N=(2*2^level)-1. For example the N=0 to N=7 is repeated as N=8 to N=15, starting at X=2,Y=2 instead of the origin. That 2,2 is because b^3 = 2+2i. There's no rotations or mirroring etc in this replication, just the position.
In general each N=2^level point starts at b^level. The powering of that b means the start rotates around by +135 degrees each time, and outward by a factor sqrt(2) on the radius. So for example b^3 = 2+2i is followed by b^4 = -4, which is 135 degrees around, and radius |b^3|=sqrt(8) becomes |b^4|=sqrt(16).
Real Part
The realpart => $r option gives a complex base b=i-r for a given r>=1. For example realpart => 2 is
20 21 22 23 24 4
15 16 17 18 19 3
10 11 12 13 14 2
5 6 7 8 9 1
45 46 47 48 49 0 1 2 3 4 <- Y=0
40 41 42 43 44 -1
35 36 37 38 39 -2
30 31 32 33 34 -3
70 71 72 73 74 25 26 27 28 29 -4
65 66 67 68 69 -5
60 61 62 63 64 -6
55 56 57 58 59 -7
50 51 52 53 54 -8
^
-8 -7 -6 -5-4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9 10N is broken into digits 0 to r*r, inclusive. This makes horizontal runs of norm=r*r+1 many points, for example N=0 to N=4, then N=5 to N=9, etc. In the default r=1 above these are 2 long, whereas for r=2 they're 2*2+1=5 long, or r=3 would be 3*3+1=10, etc.
The offset back for each run like N=5 shown is the r in i-r, then the next level is (i-r)^2 = (-2r*i + r^2-1) so N=25 begins at X=-2*2=-4,Y=2*2-1=3.
The successive replications end up tiling the plane for any r, though the N values needed to come around and do so may become large if the norm=r*r+1 is large.
Radius Range
In general for i-1 after the first few innermost levels each N=2^level increases the covered radius around by a factor sqrt(2), ie.
N = 0 to 2^level-1
Xmin,Ymin closest to origin
Xmin^2+Ymin^2 approx 2^(level-7)The "level-7" is since the innermost few levels take a while to cover the points surrounding the origin. Notice for example X=1,Y=-1 is not reached until N=58. But after that it grows like N approx = pi*R^2.
Fractal
The i-1 twindragon is generally conceived as taking fractional N like 0.abcde in binary and giving fractional complex components X,Y. The twindragon is then all the points of the complex plane reached by such N. Those points can be shown to be connected and cover a certain radius around the origin which is completely covered.
The code here might be pressed into use for that for finite bits of N by taking a suitable power N*256^k to get an integer then use X/16^k, Y/16^k for fractional coordinates. 256 is a good base because b^8=16 so there's no rotations to apply to the X,Y, just a division. b^4=-4 for multiplier 16^k and divisor (-4)^k would be almost as easy too.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.
$path = Math::PlanePath::MathImageComplexMinus->new ()$path = Math::PlanePath::MathImageComplexMinus->new (realpart => $r)-
Create and return a new path object.
($x,$y) = $path->n_to_xy ($n)-
Return the X,Y coordinates of point number
$non the path. Points begin at 0 and if$n < 0then the return is an empty list.$nshould be an integer, it's unspecified yet what will be done for a fraction.
FORMULAS
X,Y to N
A given X,Y representing X+Yi can be broken into digits of N by complex division by i-1, with remainder 0 or 1 being each digit.
The remainder can be determined simply from (X+Y) mod 2. If it's 1 then X-1,Y gives a point which is an exact multiple of i-1 and that base can be divided out
X <- -(X-Y)/2
Y <- -(X+Y)/2This can also be thought of as a rotate by -135 degrees and divide by sqrt(2).
The binary bits of N from low to high are generated this way.