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Always turn off hyphenation; it makes .\" way too many mistakes in technical documents. .if n .ad l .nh .SH "NAME" Math::PlanePath::SquareArms \-\- four spiral arms .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::SquareArms; \& my $path = Math::PlanePath::SquareArms\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path follows four spiral arms, each advancing successively, .PP .Vb 10 \& ...\-\-33\-\-29 3 \& | \& 26\-\-22\-\-18\-\-14\-\-10 25 2 \& | | | \& 30 11\-\- 7\-\- 3 6 21 1 \& | | | | \& ... 15 4 1 2 17 ... <\- Y=0 \& | | | | | \& 19 8 5\-\- 9\-\-13 32 \-1 \& | | | \& 23 12\-\-16\-\-20\-\-24\-\-28 \-2 \& | \& 27\-\-31\-\-... \-3 \& \& ^ ^ ^ ^ ^ ^ ^ \& \-3 \-2 \-1 X=0 1 2 3 ... .Ve .PP Each arm is quadratic, with each loop 128 longer than the preceding. The perfect squares fall in eight straight lines 4, with the even squares on the X and Y axes and the odd squares on the diagonals X=Y and X=\-Y. .IX Xref "Square numbers" .PP Some novel straight lines arise from numbers which are a repdigit in one or more bases (Sloane's A167782). \*(L"111\*(R" in various bases falls on straight lines. Numbers \*(L"[16][16][16]\*(R" in bases 17,19,21,etc are a horizontal at Y=3 because they're perfect squares, and \*(L"[64][64][64]\*(R" in base 65,66,etc go a vertically downwards from X=12,Y=\-266 similarly because they're squares. .PP Each arm is N=4*k+rem for a remainder rem=0,1,2,3, so sequences related to multiples of 4 or with a modulo 4 pattern may fall on particular arms. .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::SquareArms\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::SquareArms\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::SquareArms->new ()" Create and return a new path object. .ie n .IP """($x,$y) = $path\->n_to_xy ($n)""" 4 .el .IP "\f(CW($x,$y) = $path\->n_to_xy ($n)\fR" 4 .IX Item "($x,$y) = $path->n_to_xy ($n)" Return the X,Y coordinates of point number \f(CW$n\fR on the path. For \f(CW\*(C`$n < 1\*(C'\fR the return is an empty list, as the path starts at 1. .Sp Fractional \f(CW$n\fR gives a point on the line between \f(CW$n\fR and \f(CW\*(C`$n+4\*(C'\fR, that \&\f(CW\*(C`$n+4\*(C'\fR being the next point on the same spiralling arm. This is probably of limited use, but arises fairly naturally from the calculation. .SS "Descriptive Methods" .IX Subsection "Descriptive Methods" .ie n .IP """$arms = $path\->arms_count()""" 4 .el .IP "\f(CW$arms = $path\->arms_count()\fR" 4 .IX Item "$arms = $path->arms_count()" Return 4. .SH "FORMULAS" .IX Header "FORMULAS" .SS "Rectangle N Range" .IX Subsection "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, .PP .Vb 1 \& d = max(abs(x1),abs(y1),abs(x2),abs(y2)) .Ve .PP from which .PP .Vb 2 \& Nmax = (2*d+1)^2 \& = (4*d + 4)*d + 1 .Ve .PP This can be used for a minimum too by finding the smallest d covered by the rectangle. .PP .Vb 4 \& dlo = max (0, \& min(abs(y1),abs(y2)) if x=0 not covered \& min(abs(x1),abs(x2)) if y=0 not covered \& ) .Ve .PP from which the maximum of the preceding dlo\-1 square, .PP .Vb 4 \& Nlo = / 1 if dlo=0 \& \e (2*(dlo\-1)+1)^2 +1 if dlo!=0 \& = (2*dlo \- 1)^2 \& = (4*dlo \- 4)*dlo + 1 .Ve .PP 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, .PP .Vb 4 \& Nhi = max (xy_to_n (x1,y1), \& xy_to_n (x1,y2), \& xy_to_n (x2,y1), \& xy_to_n (x2,y2)) .Ve .PP The current code uses a dlo for Nlo and the corners for Nhi, which means the high is exact but the low is not. .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::DiamondArms, Math::PlanePath::HexArms, Math::PlanePath::SquareSpiral .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2011, 2012, 2013, 2014 Kevin Ryde .PP This file is part of Math-PlanePath. .PP Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the \s-1GNU\s0 General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. .PP Math-PlanePath is distributed in the hope that it will be useful, but \&\s-1WITHOUT ANY WARRANTY\s0; without even the implied warranty of \s-1MERCHANTABILITY\s0 or \s-1FITNESS FOR A PARTICULAR PURPOSE. \s0 See the \s-1GNU\s0 General Public License for more details. .PP You should have received a copy of the \s-1GNU\s0 General Public License along with Math-PlanePath. If not, see .