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=head2 Arms

For C<realpart =E<gt> 1>, an optional C<arms =E<gt> 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 C<arms> parameter for other C<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?

=head1 FUNCTIONS

See L<Math::PlanePath/FUNCTIONS> for behaviour common to all path classes.

=over 4

=item C<$path = Math::PlanePath::ComplexPlus-E<gt>new ()>

Create and return a new path object.

=item C<($x,$y) = $path-E<gt>n_to_xy ($n)>

Return the X,Y coordinates of point number C<$n> on the path.  Points begin
at 0 and if C<$n E<lt> 0> then the return is an empty list.

=back

=head2 Level Methods

=over

=item C<($n_lo, $n_hi) = $path-E<gt>level_to_n_range($level)>

Return C<(0, 2**$level - 1)>, or for 2 arms return C<(0, 2 * 2**$level -
1)>.  With the C<realpart> option return C<(0, $norm**$level - 1)> where
norm=realpart^2+1.

=back

=head1 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")

=over

L<http://user42.tuxfamily.org/dragon/index.html>

=back

=head1 OEIS

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

=over

L<http://oeis.org/A290885> (etc)

=back

    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)

=head1 SEE ALSO

L<Math::PlanePath>,
L<Math::PlanePath::ComplexMinus>,
L<Math::PlanePath::ComplexRevolving>,
L<Math::PlanePath::DragonCurve>

=over

L<http://user42.tuxfamily.org/dragon/index.html>

=back

=head1 HOME PAGE

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

=head1 LICENSE

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.

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.