--- name: "Math::PlanePath::SquareArms" section: "3pm" platform: "linux" package: "libmath-planepath-perl" version: "129-1" release: "Debian 13 (Trixie)" platforms: ["linux"] url: "https://new.carta.tech/Math::PlanePath::SquareArms" --- # Math::PlanePath::SquareArms(3pm) ## 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, ``` ...--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. ## 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 - "$arms = $path-\>arms\_count()" Return 4. ## 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 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. ## SEE ALSO Math::PlanePath, Math::PlanePath::DiamondArms, Math::PlanePath::HexArms, Math::PlanePath::SquareSpiral ## HOME PAGE \ ## LICENSE Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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. You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see \.