Provided by: libmath-planepath-perl_129-1_all 
NAME
Math::PlanePath::TriangleSpiral -- integer points drawn around an equilateral triangle
SYNOPSIS
use Math::PlanePath::TriangleSpiral;
my $path = Math::PlanePath::TriangleSpiral->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This path makes a spiral shaped as an equilateral triangle (each side the same length).
16 4
/ \
17 15 3
/ \
18 4 14 ... 2
/ / \ \ \
19 5 3 13 32 1
/ / \ \ \
20 6 1-----2 12 31 <- Y=0
/ / \ \
21 7-----8-----9----10----11 30 -1
/ \
22----23----24----25----26----27----28----29 -2
^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8
Cells are spread horizontally to fit on a square grid as per "Triangular Lattice" in
Math::PlanePath. The horizontal gaps are 2, so for instance n=1 is at x=0,y=0 then n=2 is
at x=2,y=0. The diagonals are 1 across and 1 up or down, so n=3 is at x=1,y=1. Each
alternate row is offset from the one above or below.
This grid is the same as the "HexSpiral" and the path is like that spiral except instead
of a flat top and SE,SW sides it extends to triangular peaks. The result is a longer loop
and each successive loop is step=9 longer than the previous (whereas the "HexSpiral" is
step=6 more).
The triangular numbers 1, 3, 6, 10, 15, 21, 28, 36 etc, k*(k+1)/2, fall one before the
successive corners of the triangle, so when plotted make three lines going vertically and
angled down left and right.
The 11-gonal "hendecagonal" numbers 11, 30, 58, etc, k*(9k-7)/2 fall on a straight line
horizontally to the right. (As per the general rule that a step "s" lines up the
(s+2)-gonal numbers.)
N Start
The default is to number points starting N=1 as shown above. An optional "n_start" can
give a different start with the same shape etc. For example to start at 0,
n_start => 0 15
/ \
16 14
/ \
17 3 13
/ / \ \
18 4 2 12 ...
/ / \ \ \
19 5 0-----1 11 30
/ / \ \
20 6-----7-----8-----9----10 29
/ \
21----22----23----24----25----26----27----28
With this adjustment the X axis N=0,1,11,30,etc is the hendecagonal numbers (9k-7)*k/2.
And N=0,8,25,etc diagonally South-East is the hendecagonals of the second kind which is
(9k-7)*k/2 for k negative.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::TriangleSpiral->new ()"
"$path = Math::PlanePath::TriangleSpiral->new (n_start => $n)"
Create and return a new triangle spiral object.
"($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, it being considered the path starts at 1.
"$n = $path->xy_to_n ($x,$y)"
Return the point number for coordinates "$x,$y". $x and $y are each rounded to the
nearest integer, which has the effect of treating each $n in the path as a square of
side 1.
Only every second square in the plane has an N. If "$x,$y" is a position without an N
then the return is "undef".
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
<http://oeis.org/A010054> (etc)
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 turn 1=left,0=straight, being 1 at N=k*(k+3)/2
A023532 turn 1=straight,0=left
A023531 is "n_start=-1" to match its "offset=0" 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.
SEE ALSO
Math::PlanePath, Math::PlanePath::TriangleSpiralSkewed, Math::PlanePath::HexSpiral
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2010, 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 <http://www.gnu.org/licenses/>.