Kevin Ryde > Math-PlanePath-116 > Math::PlanePath::GosperSide

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Module Version: 116   Source   Latest Release: Math-PlanePath-117

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.

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 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/>.

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