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

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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)

    A099450   Y at N=3^k (for k>=1)

The turn sequence is the same as the terdragon curve, see "OEIS" in Math::PlanePath::TerdragonCurve for the numerous 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 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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