Kevin Ryde > Math-PlanePath-111 > Math::PlanePath::GosperSide

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Module Version: 111   Source   Latest Release: Math-PlanePath-114

# 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::Fractal::Curve

# HOME PAGE

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

# LICENSE

Copyright 2011, 2012, 2013 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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