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

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Module Version: 126

# 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

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

```    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.

Math::Fractal::Curve

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