Provided by: libmath-planepath-perl_113-1_all 
NAME
Math::PlanePath::ComplexMinus -- twindragon and other complex number base i-r
SYNOPSIS
use Math::PlanePath::ComplexMinus;
my $path = Math::PlanePath::ComplexMinus->new (realpart=>1);
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This path traverses points by a complex number base i-r for given integer r. The default is base i-1 as
per
Walter Penny, A "Binary" System for Complex Numbers, Journal of the ACM, volume 12, number 2, April
1965, pages 247-248.
When continued to a power-of-2 extent this has come to be called the "twindragon" shape.
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 7
A complex integer can be represented as a set of powers,
X+Yi = a[n]*b^n + ... + a[2]*b^2 + a[1]*b + a[0]
base b=i-1
digits a[n] to a[0] each = 0 or 1
N = a[n]*2^n + ... + a[2]*2^2 + a[1]*2 + a[0]
N is those a[i] digits as bits and X,Y is the resulting complex number. It can be shown that this is a
one-to-one mapping so every integer X,Y of the plane is visited.
The shape of points N=0 to N=2^level-1 repeats as N=2^level to N=2^(level+1)-1. For example N=0 to N=7
is repeated as N=8 to N=15, but starting at X=2,Y=2 instead of the origin. That position 2,2 is because
b^3 = 2+2i. There's no rotations or mirroring etc in this replication, just position offsets.
N=0 to N=7 N=8 to N=15 repeat shape
2 3 10 11
0 1 8 9
6 7 14 15
4 5 12 13
For b=i-1 each N=2^level point starts at X+Yi=b^level. The powering of that b means the start position
rotates around by +135 degrees each time and outward by a radius factor sqrt(2) each time. 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 integer 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 10
N is broken into digits of base=norm=r*r+1, ie. digits 0 to r*r inclusive. This makes horizontal runs of
r*r+1 many points, such as N=5 to N=9 etc above. In the default r=1 these runs 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 Y=-2*2=-4, X=2*2-1=3.
The successive replications tile the plane for any r, though the N values needed to rotate around and do
so become large if norm=r*r+1 is large.
X Axis Values
For base i-1, the X axis N=0,1,12,13,16,17,etc is integers using only digits 0,1,0xC,0xD in hexadecimal.
Those on the positive X axis have an odd number of digits and on the X negative axis an even number of
digits.
To be on the X axis the imaginary parts of the base powers b^k must cancel out to leave just a real part.
The powers repeat in an 8-long cycle
k b^k for b=i-1
0 +1
1 i -1
2 -2i +0 \ pair cancel
3 2i +2 /
4 -4
5 -4i +4
6 8i +0 \ pair cancel
7 -8i -8 /
The k=0 and k=4 bits are always reals and can always be included. Bits k=2 and k=3 have imaginary parts
-2i and 2i which cancel out, so they can be included together. Similarly k=6 and k=7 with 8i and -8i.
The two blocks k=0to3 and k=4to7 differ only in a negation so the bits can be reckoned in groups of 4,
which is hexadecimal. Bit 1 is digit value 1 and bits 2,3 together are digit value 0xC, so adding one or
both of those gives combinations are 0,1,0xC,0xD.
The high hex digit determines the sign, positive or negative, of the total real part. Bits k=0 or k=2,3
are positive. Bits k=4 or k=6,7 are negative, so
N for X>0 N for X<0
0x01.. 0x1_.. even number of hex 0,1,C,D following
0x0C.. 0xC_.. "_" digit any of 0,1,C,D
0x0D.. 0xD_..
which is equivalent to X>0 is an odd number of hex digits or X<0 is an even number. For example
N=28=0x1C is at X=-2 since that N is X<0 form "0x1_".
The order of the values on the positive X axis is obtained by taking the digits in reverse order on
alternate positions
0,1,C,D high digit
D,C,1,0
0,1,C,D
...
D,C,1,0
0,1,C,D low digit
For example in the following notice the first and third digit increases, but the middle digit decreases,
X=4to7 N=0x1D0,0x1D1,0x1DC,0x1DD
X=8to11 N=0x1C0,0x1C1,0x1CC,0x1CD
X=12to15 N=0x110,0x111,0x11C,0x11D
X=16to19 N=0x100,0x101,0x10C,0x10D
X=20to23 N=0xCD0,0xCD1,0xCDC,0xCDD
For the negative X axis it's the same if reading by increasing X, ie. upwards toward +infinity, or the
opposite way around if reading decreasing X, ie. more negative downwards toward -infinity.
Fractal
The i-1 twindragon is usually conceived as taking fractional N like 0.abcde in binary and giving
fractional complex X+iY. The twindragon is then all the points of the complex plane reached by such
fractional N. This set of points can be shown to be connected and to fill a certain radius around the
origin.
The code here might be pressed into use for that to some finite number of bits by multiplying up to make
an integer N
Nint = Nfrac * 256^k
Xfrac = Xint / 16^k
Yfrac = Yint / 16^k
256 is a good power because b^8=16 is a positive real and so there's no rotations to apply to the
resulting X,Y, only a power-of-16 division (b^8)^k=16^k each. Using b^4=-4 for a multiplier 16^k and
divisor (-4)^k would be almost as easy too, requiring just sign changes if k odd.
Boundary Length
The length of the boundary of the first norm^k many points, ie. N=0 to N=norm^k-1 inclusive, is
calculated in
William J. Gilbert, "The Fractal Dimension of Sets Derived From Complex Bases", Canadian Math
Bulletin, volume 29(4), 1986. <http://www.math.uwaterloo.ca/~wgilbert/Research/GilbertFracDim.pdf>
The boundary formula is a 3rd-order recurrence. For the twindragon case,
realpart=1
boundary[k] = boundary[k-1] + 2*boundary[k-3]
4, 6, 10, 18, 30, 50, 86, 146, 246, 418, 710, ...
The first three boundaries are as follows. Then the recurrence gives the next boundary[3] = 10+2*4 = 18.
k area boundary[k]
--- ---- -----------
+---+
0 2^k = 1 4 | 0 |
+---+
+---+---+
1 2^k = 2 6 | 0 1 |
+---+---+
+---+---+
| 2 3 |
2 2^k = 4 10 +---+ +---+
| 0 1 |
+---+---+
Gilbert calculates the boundary of any i-r by taking it in three parts A,B,C and showing how in the next
replication level those boundary parts transform into multiple copies of the preceding level parts. The
replication is easier to visualize for a bigger "r" than for the twindragon because in bigger r it's
clearer how the A, B and C parts differ. The replications are
A -> A * (2*realpart-1) + C * 2*realpart
B -> A * (realpart^2-2*realpart+2) + C * (realpart-1)^2
C -> B
starting from
A = 2*realpart
B = 2
C = 2 - 2*realpart
total boundary = A+B+C
For the twindragon case realpart=1 these A,B,C are already in the form of a recurrence A->A+2*C, B->A,
C->B, per the formula above. For other real parts a little matrix rearrangement gives the recurrence
boundary[k] = boundary[k-1] * (2*realpart - 1)
+ boundary[k-2] * (norm - 2*realpart)
+ boundary[k-3] * norm
starting from
boundary[0] = 4 (ie. a single square cell)
boundary[1] = 2*norm + 2
boundary[2] = 2*(norm-1)*(realpart+2) + 4
For example
realpart=2
boundary[k] = 3*boundary[k-1] + 1*boundary[k-2] + 5*boundary[k-1]
4, 12, 36, 140, 516, 1868, 6820, 24908, ...
If calculating for large k values then the matrix form can be powered up rather than repeated additions.
(As usual for all such recurrences.)
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::ComplexMinus->new ()"
"$path = Math::PlanePath::ComplexMinus->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 $n on the path. Points begin at 0 and if "$n < 0" then
the return is an empty list.
$n should 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 turned into digits of N by successive complex divisions by i-r.
Each digit of N is a real remainder 0 to r*r inclusive from that division.
The base formula above is
X+Yi = a[n]*b^n + ... + a[2]*b^2 + a[1]*b + a[0]
and we want the a[0]=digit to be a real 0 to r*r inclusive. Subtracting a[0] and dividing by b will give
(X+Yi - digit) / (i-r)
= - (X-digit + Y*i) * (i+r) / norm
= (Y - (X-digit)*r)/norm
+ i * - ((X-digit) + Y*r)/norm
which is
Xnew = Y - (X-digit)*r)/norm
Ynew = -((X-digit) + Y*r)/norm
The a[0] digit must make both Xnew and Ynew parts integers. The easiest one to calculate from is the
imaginary part, from which require
- ((X-digit) + Y*r) == 0 mod norm
so
digit = X + Y*r mod norm
This digit value makes the real part a multiple of norm too, as can be seen from
Xnew = Y - (X-digit)*r
= Y - X*r - (X+Y*r)*r
= Y - X*r - X*r + Y*r*r
= Y*(r*r+1)
= Y*norm
Notice Ynew is the quotient from (X+Y*r)/norm rounded towards negative infinity. Ie. in the division
"X+Y*r mod norm" which calculates the digit, the quotient is Ynew and the remainder is the digit.
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
<http://oeis.org/A066321> (etc)
realpart=1 (the default)
A066321 N on X axis, being the base i-1 positive reals
A066323 N on X axis, in binary
A066322 diffs (N at X=16k+4) - (N at X=16k+3)
A003476 boundary length / 2
recurrence a(n) = a(n-1) + 2*a(n-3)
A203175 boundary length, starting from 4
(believe its conjectured recurrence is true)
A052537 boundary length part A, B or C, per Gilbert's paper
SEE ALSO
Math::PlanePath, Math::PlanePath::DragonCurve, Math::PlanePath::ComplexPlus
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2011, 2012, 2013 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/>.