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)
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.
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016 Kevin RydeMath-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/>.