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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::TriangleSpiral \-\- integer points drawn around an equilateral triangle .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::TriangleSpiral; \& my $path = Math::PlanePath::TriangleSpiral\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path makes a spiral shaped as an equilateral triangle (each side the same length). .PP .Vb 10 \& 16 4 \& / \e \& 17 15 3 \& / \e \& 18 4 14 ... 2 \& / / \e \e \e \& 19 5 3 13 32 1 \& / / \e \e \e \& 20 6 1\-\-\-\-\-2 12 31 <\- Y=0 \& / / \e \e \& 21 7\-\-\-\-\-8\-\-\-\-\-9\-\-\-\-10\-\-\-\-11 30 \-1 \& / \e \& 22\-\-\-\-23\-\-\-\-24\-\-\-\-25\-\-\-\-26\-\-\-\-27\-\-\-\-28\-\-\-\-29 \-2 \& \& ^ \& \-6 \-5 \-4 \-3 \-2 \-1 X=0 1 2 3 4 5 6 7 8 .Ve .PP Cells are spread horizontally to fit on a square grid as per \&\*(L"Triangular Lattice\*(R" in Math::PlanePath. The horizontal gaps are 2, so for instance n=1 is at x=0,y=0 then n=2 is at x=2,y=0. The diagonals are 1 across and 1 up or down, so n=3 is at x=1,y=1. Each alternate row is offset from the one above or below. .PP This grid is the same as the \f(CW\*(C`HexSpiral\*(C'\fR and the path is like that spiral except instead of a flat top and \s-1SE,SW\s0 sides it extends to triangular peaks. The result is a longer loop and each successive loop is step=9 longer than the previous (whereas the \f(CW\*(C`HexSpiral\*(C'\fR is step=6 more). .PP The triangular numbers 1, 3, 6, 10, 15, 21, 28, 36 etc, k*(k+1)/2, fall one before the successive corners of the triangle, so when plotted make three lines going vertically and angled down left and right. .IX Xref "Triangular numbers" .PP The 11\-gonal \*(L"hendecagonal\*(R" numbers 11, 30, 58, etc, k*(9k\-7)/2 fall on a straight line horizontally to the right. (As per the general rule that a step \*(L"s\*(R" lines up the (s+2)\-gonal numbers.) .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 10 \& n_start => 0 15 \& / \e \& 16 14 \& / \e \& 17 3 13 \& / / \e \e \& 18 4 2 12 ... \& / / \e \e \e \& 19 5 0\-\-\-\-\-1 11 30 \& / / \e \e \& 20 6\-\-\-\-\-7\-\-\-\-\-8\-\-\-\-\-9\-\-\-\-10 29 \& / \e \& 21\-\-\-\-22\-\-\-\-23\-\-\-\-24\-\-\-\-25\-\-\-\-26\-\-\-\-27\-\-\-\-28 .Ve .PP With this adjustment the X axis N=0,1,11,30,etc is the hendecagonal numbers (9k\-7)*k/2. And N=0,8,25,etc diagonally South-East is the hendecagonals of the second kind which is (9k\-7)*k/2 for k negative. .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::TriangleSpiral\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::TriangleSpiral\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::TriangleSpiral->new ()" .PD 0 .ie n .IP """$path = Math::PlanePath::TriangleSpiral\->new (n_start => $n)""" 4 .el .IP "\f(CW$path = Math::PlanePath::TriangleSpiral\->new (n_start => $n)\fR" 4 .IX Item "$path = Math::PlanePath::TriangleSpiral->new (n_start => $n)" .PD Create and return a new triangle spiral object. .ie n .IP """($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$n\fR on the path. .Sp For \f(CW\*(C`$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. If \f(CW\*(C`$x,$y\*(C'\fR is a position without an N 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 2 \& n_start=1 (default) \& A010054 turn 1=left,0=straight, extra initial 1 \& \& A117625 N on X axis \& A081272 N on Y axis \& A006137 N on X negative axis \& A064226 N on X=Y leading diagonal, but without initial value=1 \& A064225 N on X=Y negative South\-West diagonal \& A081267 N on X=\-Y negative South\-East diagonal \& A081589 N on ENE slope dX=3,dY=1 \& A038764 N on WSW slope dX=\-3,dY=\-1 \& A060544 N on ESE slope dX=3,dY=\-1 diagonal \& \& A063177 total sum previous row or diagonal \& \& n_start=0 \& A051682 N on X axis (11\-gonal numbers) \& A062741 N on Y axis \& A062708 N on X=Y leading diagonal \& A081268 N on X=Y+2 diagonal (right of leading diagonal) \& A062728 N on South\-East diagonal (11\-gonal second kind) \& A062725 N on South\-West diagonal \& A081275 N on ENE slope from X=2,Y=0 then dX=+3,dY=+1 \& A081266 N on WSW slope dX=\-3,dY=\-1 \& A081271 N on X=2 vertical \& \& n_start=\-1 \& A023531 N position of turns (to the left) \& 1 at N=k*(k+3)/2 .Ve .PP A023531 is \f(CW\*(C`n_start=\-1\*(C'\fR to match its \*(L"offset=0\*(R" for the first turn, being the second point of the path. A010054 which is 1 at triangular numbers k*(k+1)/2 is the same except for an extra initial 1. .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::TriangleSpiralSkewed, Math::PlanePath::HexSpiral .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 .