and 1 contributors

# NAME

Math::PlanePath::ComplexPlus -- points in complex base i+r

# SYNOPSIS

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

# DESCRIPTION

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).

## Real Part

Option `realpart => \$r` selects another r for complex base b=i+r. For example

``````    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.

## Arms

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

``````    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?

# FUNCTIONS

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.

## Level Methods

`(\$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.

# FORMULAS

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")

# OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

``````    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)``````

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