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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::PentSpiralSkewed \-\- integer points in a pentagonal shape .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::PentSpiralSkewed; \& my $path = Math::PlanePath::PentSpiralSkewed\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path makes a pentagonal (five-sided) spiral with points skewed so as to fit a square grid and fully cover the plane. .PP .Vb 9 \& 10 ... 2 \& / \e \e \& 11 3 9 20 1 \& / / \e \e \e \& 12 4 1\-\-2 8 19 <\- Y=0 \& \e \e | | \& 13 5\-\-6\-\-7 18 \-1 \& \e | \& 14\-15\-16\-17 \-2 \& \& ^ ^ ^ ^ ^ ^ \& \-2 \-1 X=0 1 2 3 ... .Ve .PP The pattern is similar to the \f(CW\*(C`SquareSpiral\*(C'\fR but cuts three corners which makes each cycle is faster. Each cycle is just 5 steps longer than the previous (where it's 8 for a \f(CW\*(C`SquareSpiral\*(C'\fR). .SS "N Start" .IX Subsection "N Start" The default is to number points starting N=1 as shown above. An optional \&\f(CW\*(C`n_start\*(C'\fR can give a different start, in the same pattern. For example to start at 0, .PP .Vb 9 \& 38 n_start => 0 \& 39 21 37 ... \& 40 22 9 20 36 57 \& 41 23 10 2 8 19 35 56 \& 42 24 11 3 0 1 7 18 34 55 \& 43 25 12 4 5 6 17 33 54 \& 44 26 13 14 15 16 32 53 \& 45 27 28 29 30 31 52 \& 46 47 48 49 50 51 .Ve .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::PentSpiral\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::PentSpiral\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::PentSpiral->new ()" .PD 0 .ie n .IP """$path = Math::PlanePath::PentSpiral\->new (n_start => $n)""" 4 .el .IP "\f(CW$path = Math::PlanePath::PentSpiral\->new (n_start => $n)\fR" 4 .IX Item "$path = Math::PlanePath::PentSpiral->new (n_start => $n)" .PD Create and return a new path object. .ie n .IP """$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 point in the path as a square of side 1. .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 7 \& n_start=1 (the default) \& A192136 N on X axis, (5*n^2 \- 3*n + 2)/2 \& A140066 N on Y axis \& A116668 N on X negative axis, (5n^2 + n + 2)/2 \& A134238 N on Y negative axis \& A158187 N on North\-West diagonal, 10*n^2 + 1 \& A005891 N on South\-East diagonal, centred pentagonals \& \& n_start=0 \& A000566 N on X axis, heptagonal numbers \& A005476 N on Y axis \& A005475 N on X negative axis \& A147875 N on Y negative axis, second heptagonals \& A033583 N on North\-West diagonal, 10*n^2 \& A028895 N on South\-East diagonal, 5*triangular .Ve .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::DiamondSpiral, Math::PlanePath::HexSpiralSkewed .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 .