Math::PlanePath::SquareArms -- four spiral arms
use Math::PlanePath::SquareArms; my $path = Math::PlanePath::SquareArms->new; my ($x, $y) = $path->n_to_xy (123);
This path follows four spiral arms, each advancing successively,
...--33--29 3 | 26--22--18--14--10 25 2 | | | 30 11-- 7-- 3 6 21 1 | | | | ... 15 4 1 2 17 ... <- Y=0 | | | | | 19 8 5-- 9--13 32 -1 | | | 23 12--16--20--24--28 -2 | 27--31--... -3 ^ ^ ^ ^ ^ ^ ^ -3 -2 -1 X=0 1 2 3 ...
Each arm is quadratic, with each loop 128 longer than the preceding. The perfect squares fall in eight straight lines 4, with the even squares on the X and Y axes and the odd squares on the diagonals X=Y and X=-Y.
Some novel straight lines arise from numbers which are a repdigit in one or more bases (Sloane's A167782). "111" in various bases falls on straight lines. Numbers "[16][16][16]" in bases 17,19,21,etc are a horizontal at Y=3 because they're perfect squares, and "[64][64][64]" in base 65,66,etc go a vertically downwards from X=12,Y=-266 similarly because they're squares.
Each arm is N=4*k+rem for a remainder rem=0,1,2,3, so sequences related to multiples of 4 or with a modulo 4 pattern may fall on particular arms.
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::SquareArms->new ()
Create and return a new path object.
($x,$y) = $path->n_to_xy ($n)
Return the X,Y coordinates of point number $n
on the path. For $n < 1
the return is an empty list, as the path starts at 1.
Fractional $n
gives a point on the line between $n
and $n+4
, that $n+4
being the next point on the same spiralling arm. This is probably of limited use, but arises fairly naturally from the calculation.
$arms = $path->arms_count()
Return 4.
Within a square X=-d...+d, and Y=-d...+d the biggest N is the end of the N=5 arm in that square, which is N=9, 25, 49, 81, etc, (2d+1)^2, in successive corners of the square. So for a rectangle find a surrounding d square,
d = max(abs(x1),abs(y1),abs(x2),abs(y2))
from which
Nmax = (2*d+1)^2 = (4*d + 4)*d + 1
This can be used for a minimum too by finding the smallest d covered by the rectangle.
dlo = max (0, min(abs(y1),abs(y2)) if x=0 not covered min(abs(x1),abs(x2)) if y=0 not covered )
from which the maximum of the preceding dlo-1 square,
Nlo = / 1 if dlo=0 \ (2*(dlo-1)+1)^2 +1 if dlo!=0 = (2*dlo - 1)^2 = (4*dlo - 4)*dlo + 1
For a tighter maximum, horizontally N increases to the left or right of the diagonal X=Y line (or X=Y+/-1 line), which means one end or the other is the maximum. Similar vertically N increases above or below the off-diagonal X=-Y so the top or bottom is the maximum. This means for a rectangle the biggest N is at one of the four corners,
Nhi = max (xy_to_n (x1,y1), xy_to_n (x1,y2), xy_to_n (x2,y1), xy_to_n (x2,y2))
The current code uses a dlo for Nlo and the corners for Nhi, which means the high is exact but the low is not.
Math::PlanePath, Math::PlanePath::DiamondArms, Math::PlanePath::HexArms, Math::PlanePath::SquareSpiral
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014, 2015, 2016 Kevin Ryde
This file is part of Math-PlanePath.
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.
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