Math::PlanePath::GosperSide -- one side of the Gosper island
use Math::PlanePath::GosperSide; my $path = Math::PlanePath::GosperSide->new; my ($x, $y) = $path->n_to_xy (123);
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.
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 $n
gives a point on the straight line between integer N.
($n_lo, $n_hi) = $path->level_to_n_range($level)
Return (0, 3**$level)
.
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).
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A099450 (etc)
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::PlanePath, Math::PlanePath::GosperIslands, Math::PlanePath::TerdragonCurve, Math::PlanePath::KochCurve
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014, 2015, 2016 Kevin Ryde
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