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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::HexSpiral \-\- integer points around a hexagonal spiral .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::HexSpiral; \& my $path = Math::PlanePath::HexSpiral\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path makes a hexagonal spiral, with points spread out horizontally to fit on a square grid. .PP .Vb 10 \& 28 \-\- 27 \-\- 26 \-\- 25 3 \& / \e \& 29 13 \-\- 12 \-\- 11 24 2 \& / / \e \e \& 30 14 4 \-\-\- 3 10 23 1 \& / / / \e \e \e \& 31 15 5 1 \-\-\- 2 9 22 <\- Y=0 \& \e \e \e / / \& 32 16 6 \-\-\- 7 \-\-\- 8 21 \-1 \& \e \e / \& 33 17 \-\- 18 \-\- 19 \-\- 20 \-2 \& \e \& 34 \-\- 35 ... \-3 \& \& ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ \& \-6 \-5 \-4 \-3 \-2 \-1 X=0 1 2 3 4 5 6 .Ve .PP Each horizontal gap is 2, so for instance n=1 is at X=0,Y=0 then n=2 is at X=2,Y=0. The diagonals are just 1 across, so n=3 is at X=1,Y=1. Each alternate row is offset from the one above or below. The result is a triangular lattice per \*(L"Triangular Lattice\*(R" in Math::PlanePath. .PP The octagonal numbers 8,21,40,65, etc 3*k^2\-2*k fall on a horizontal straight line at Y=\-1. In general straight lines are 3*k^2 + b*k + c. A plain 3*k^2 goes diagonally up to the left, then b is a 1/6 turn anti-clockwise, or clockwise if negative. So b=1 goes horizontally to the left, b=2 diagonally down to the left, b=3 diagonally down to the right, etc. .SS "Wider" .IX Subsection "Wider" An optional \f(CW\*(C`wider\*(C'\fR parameter makes the path wider, stretched along the top and bottom horizontals. For example .PP .Vb 1 \& $path = Math::PlanePath::HexSpiral\->new (wider => 2); .Ve .PP gives .PP .Vb 11 \& ... 36\-\-\-\-35 3 \& \e \& 21\-\-\-\-20\-\-\-\-19\-\-\-\-18\-\-\-\-17 34 2 \& / \e \e \& 22 8\-\-\-\- 7\-\-\-\- 6\-\-\-\- 5 16 33 1 \& / / \e \e \e \& 23 9 1\-\-\-\- 2\-\-\-\- 3\-\-\-\- 4 15 32 <\- Y=0 \& \e \e / / \& 24 10\-\-\-\-11\-\-\-\-12\-\-\-\-13\-\-\-\-14 31 \-1 \& \e / \& 25\-\-\-\-26\-\-\-\-27\-\-\-\-28\-\-\-29\-\-\-\-30 \-2 \& \& ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ \& \-7 \-6 \-5 \-4 \-3 \-2 \-1 X=0 1 2 3 4 5 6 7 .Ve .PP The centre horizontal from N=1 is extended by \f(CW\*(C`wider\*(C'\fR many extra places, then the path loops around that shape. The starting point N=1 is shifted to the left by wider many places to keep the spiral centred on the origin X=0,Y=0. Each horizontal gap is still 2. .PP Each loop is still 6 longer than the previous, since the widening is basically a constant amount added into each loop. .SS "N Start" .IX Subsection "N Start" The default is to number points starting N=1 as shown above. An optional \&\f(CW\*(C`n_start\*(C'\fR can give a different start with the same shape etc. For example to start at 0, .PP .Vb 1 \& n_start => 0 \& \& 27 26 25 24 3 \& 28 12 11 10 23 2 \& 29 13 3 2 9 22 1 \& 30 14 4 0 1 8 21 <\- Y=0 \& 31 15 5 6 7 20 ... \-1 \& 32 16 17 18 19 38 \-2 \& 33 34 35 36 37 \-3 \& ^ \& \-6 \-5 \-4 \-3 \-2 \-1 X=0 1 2 3 4 5 6 .Ve .PP In this numbering the X axis N=0,1,8,21,etc is the octagonal numbers 3*X*(X+1). .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::HexSpiral\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::HexSpiral\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::HexSpiral->new ()" .PD 0 .ie n .IP """$path = Math::PlanePath::HexSpiral\->new (wider => $w)""" 4 .el .IP "\f(CW$path = Math::PlanePath::HexSpiral\->new (wider => $w)\fR" 4 .IX Item "$path = Math::PlanePath::HexSpiral->new (wider => $w)" .PD Create and return a new hex spiral object. An optional \f(CW\*(C`wider\*(C'\fR parameter widens the path, it defaults to 0 which is no widening. .ie n .IP """($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$n\fR on the path. .Sp For \f(CW\*(C`$n < 1\*(C'\fR the return is an empty list, it being considered the path starts at 1. .ie n .IP """$n = $path\->xy_to_n ($x,$y)""" 4 .el .IP "\f(CW$n = $path\->xy_to_n ($x,$y)\fR" 4 .IX Item "$n = $path->xy_to_n ($x,$y)" Return the point number for coordinates \f(CW\*(C`$x,$y\*(C'\fR. \f(CW$x\fR and \f(CW$y\fR are each rounded to the nearest integer, which has the effect of treating each \&\f(CW$n\fR in the path as a square of side 1. .Sp Only every second square in the plane has an N, being those where X,Y both odd or both even. If \f(CW\*(C`$x,$y\*(C'\fR is a position without an N, ie. one of X,Y odd the other even, then the return is \f(CW\*(C`undef\*(C'\fR. .SH "OEIS" .IX Header "OEIS" Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include .Sp .RS 4 (etc) .RE .PP .Vb 6 \& A056105 N on X axis \& A056106 N on X=Y diagonal \& A056107 N on North\-West diagonal \& A056108 N on negative X axis \& A056109 N on South\-West diagonal \& A003215 N on South\-East diagonal \& \& A063178 total sum N previous row or diagonal \& A135711 boundary length of N hexagons \& A135708 grid sticks of N hexagons \& \& n_start=0 \& A000567 N on X axis, octagonal numbers \& A049451 N on X negative axis \& A049450 N on X=Y diagonal north\-east \& A033428 N on north\-west diagonal, 3*k^2 \& A045944 N on south\-west diagonal, octagonal numbers second kind \& A063436 N on WSW slope dX=\-3,dY=\-1 \& A028896 N on south\-east diagonal .Ve .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::HexSpiralSkewed, Math::PlanePath::HexArms, Math::PlanePath::TriangleSpiral, Math::PlanePath::TriangularHypot .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2010, 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 .