Provided by: libmath-planepath-perl_129-1_all bug

NAME

       Math::PlanePath::GosperSide -- one side of the Gosper island

SYNOPSIS

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

DESCRIPTION

       This path is a single side of the Gosper island, in integers ("Triangular Lattice" in
       Math::PlanePath).

                                               20-...        14
                                              /
                                      18----19               13
                                     /
                                   17                        12
                                     \
                                      16                     11
                                     /
                                   15                        10
                                     \
                                      14----13                9
                                              \
                                               12             8
                                              /
                                            11                7
                                              \
                                               10             6
                                              /
                                       8---- 9                5
                                     /
                              6---- 7                         4
                            /
                           5                                  3
                            \
                              4                               2
                            /
                     2---- 3                                  1
                   /
            0---- 1                                       <- Y=0

            ^
           X=0 1  2  3  4  5  6  7  8  9 10 11 12 13 ...

       The path slowly spirals around counter clockwise, with a lot of wiggling in between.  The
       N=3^level point is at

          N = 3^level
          angle = level * atan(sqrt(3)/5)
                = level * 19.106 degrees
          radius = sqrt(7) ^ level

       A full revolution for example takes roughly level=19 which is about N=1,162,000,000.

       Both ends of such levels are in fact sub-spirals, like an "S" shape.

       The path is both the sides and the radial spokes of the "GosperIslands" path, as described
       in "Side and Radial Lines" in Math::PlanePath::GosperIslands.  Each N=3^level point is the
       start of a "GosperIslands" ring.

       The path is the same as the "TerdragonCurve" except the turns here are by 60 degrees each,
       whereas "TerdragonCurve" is by 120 degrees.  See Math::PlanePath::TerdragonCurve for the
       turn sequence and total direction formulas etc.

FUNCTIONS

       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

       "$path = Math::PlanePath::GosperSide->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.

           Fractional $n gives a point on the straight line between integer N.

   Level Methods
       "($n_lo, $n_hi) = $path->level_to_n_range($level)"
           Return "(0, 3**$level)".

FORMULAS

   Level Endpoint
       The endpoint of each level N=3^k is at

           X + Y*i*sqrt(3) = b^k
           where b = 2 + w = 5/2 + sqrt(3)/2*i
                 where w=1/2 + sqrt(3)/2*i sixth root of unity

           X(k) = ( 5*X(k-1) - 3*Y(k-1) )/2        for k>=1
           Y(k) = (   X(k-1) + 5*Y(k-1) )/2
                  starting X(0)=2 Y(0)=0

           X(k) = 5*X(k-1) - 7*X(k-2)        for k>=2
                  starting X(0)=2 X(1)=5
                = 2, 5, 11, 20, 23, -25, -286, -1255, -4273, -12580, -32989,..

           Y(k) = 5*Y(k-1) - Y*X(k-2)        for k>=2
                  starting Y(0)=0 Y(1)=1
                = 0, 1,  5, 18, 55, 149,  360,   757,  1265, 1026, -3725, ...
                                                                   (A099450)

       The curve base figure is XY(k)=XY(k-1)+rot60(XY(k-1))+XY(k-1) giving XY(k) = (2+w)^k = b^k
       where w is the sixth root of unity giving the rotation by +60 degrees.

       The mutual recurrences are similar with the rotation done by (X-3Y)/2, (Y+X)/2 per
       "Triangular Lattice" in Math::PlanePath.  The separate recurrences are found by using the
       first to get Y(k-1) = -2/3*X(k) + 5/3*X(k-1) and substitute into the other to get X(k+1).
       Similar the other way around for Y(k+1).

OEIS

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

           <http://oeis.org/A099450> (etc)

           A229215   direction 1,2,3,-1,-2,-3 (clockwise)
           A099450   Y at N=3^k (for k>=1)

       Also the turn sequence is the same as the terdragon curve, see "OEIS" in
       Math::PlanePath::TerdragonCurve for the several turn forms, N positions of turns, etc.

SEE ALSO

       Math::PlanePath, Math::PlanePath::GosperIslands, Math::PlanePath::TerdragonCurve,
       Math::PlanePath::KochCurve

       Math::Fractal::Curve

HOME PAGE

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

LICENSE

       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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/>.