NAME
Math::PlanePath::SquareArms  four spiral arms
SYNOPSIS
use Math::PlanePath::SquareArms;
my $path = Math::PlanePath::SquareArms>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
This path follows four spiral arms, each advancing successively,
...3329 3

2622181410 25 2
  
30 11 7 3 6 21 1
   
... 15 4 1 2 17 ... < Y=0
    
19 8 5 913 32 1
  
23 1216202428 2

2731... 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.
FUNCTIONS
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.
Descriptive Methods
FORMULAS
Rectangle N Range
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 dlo1 square,
Nlo = / 1 if dlo=0
\ (2*(dlo1)+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 offdiagonal 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.
SEE ALSO
Math::PlanePath, Math::PlanePath::DiamondArms, Math::PlanePath::HexArms, Math::PlanePath::SquareSpiral
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
This file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.