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.

Turn Sequence

The sequence of turns made by the curve is straightforward. In the base 3 representation of N, the lowest non-zero digit gives the turn

   low digit      turn
   ---------   -----------
      1        +60 degrees
      2        -60 degrees

When the least significant digit is non-zero it determines the turn, to make the base N=0 to N=3 shape. When the low digit is zero it's instead the next level up, the N=0,3,6,9 shape which is in control, applying a turn for the base which follows. So for example at N=6 = 20 base3 is a turn -60 degrees.

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for the 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.

SEE ALSO

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

Math::Fractal::Curve

HOME PAGE

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

LICENSE

Copyright 2011, 2012 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/>.