Provided by: libmath-planepath-perl_122-1_all 
NAME
Math::PlanePath::AnvilSpiral -- integer points around an "anvil" shape
SYNOPSIS
use Math::PlanePath::AnvilSpiral;
my $path = Math::PlanePath::AnvilSpiral->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This path makes a spiral around an anvil style shape,
...-78-77-76-75-74 4
/
49-48-47-46-45-44-43-42-41-40-39-38 73 3
\ / /
50 21-20-19-18-17-16-15-14 37 72 2
\ \ / / /
51 22 5--4--3--2 13 36 71 1
\ \ \ / / / /
52 23 6 1 12 35 70 <- Y=0
/ / / \ \ \
53 24 7--8--9-10-11 34 69 -1
/ / \ \
54 25-26-27-28-29-30-31-32-33 68 -2
/ \
55-56-57-58-59-60-61-62-63-64-65-66-67 -3
^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7
The pentagonal numbers 1,5,12,22,etc, P(k) = (3k-1)*k/2 fall alternately on the X axis
X>0, and on the Y=1 horizontal X<0.
Those pentagonals are always composites, from the factorization shown, and as noted in
"Step 3 Pentagonals" in Math::PlanePath::PyramidRows, the immediately preceding P(k)-1 and
P(k)-2 are also composites. So plotting the primes on the spiral has a 3-high horizontal
blank line at Y=0,-1,-2 for positive X, and Y=1,2,3 for negative X (after the first few
values).
Each loop around the spiral is 12 longer than the preceding. This is 4* more than the
step=3 "PyramidRows" so straight lines on a "PyramidRows" like these pentagonals are also
straight lines here, but split into two parts.
The outward diagonal excursions are similar to the "OctagramSpiral", but there's just 4 of
them here where the "OctagramSpiral" has 8. This is reflected in the loop step. The
basic "SquareSpiral" is step 8, but by taking 4 excursions here increases that to 12, and
in the "OctagramSpiral" 8 excursions adds 8 to make step 16.
Wider
An optional "wider" parameter makes the path wider by starting with a horizontal section
of given width. For example
$path = Math::PlanePath::SquareSpiral->new (wider => 3);
gives
33-32-31-30-29-28-27-26-25-24-23 ... 2
\ / /
34 11-10--9--8--7--6--5 22 51 1
\ \ / / /
35 12 1--2--3--4 21 50 <- Y=0
/ / \ \
36 13-14-15-16-17-18-19-20 49 -1
/ \
37-38-39-40-41-42-43-44-45-46-47-48 -2
^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5
The starting point 1 is shifted to the left by ceil(wider/2) places to keep the spiral
centred on the origin X=0,Y=0. This is the same starting offset as the "SquareSpiral"
"wider".
Widening doesn't change the nature of the straight lines which arise, it just rotates them
around. Each loop is still 12 longer than the previous, since the widening is essentially
a constant amount in each loop.
N Start
The default is to number points starting N=1 as shown above. An optional "n_start" can
give a different start with the same shape. For example to start at 0,
n_start => 0
20-19-18-17-16-15-14-13 ...
\ / /
21 4--3--2--1 12 35
\ \ / / /
22 5 0 11 34
/ / \ \
23 6--7--8--9-10 33
/ \
24-25-26-27-28-29-30-31-32
The only effect is to push the N values around by a constant amount. It might help match
coordinates with something else zero-based.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::AnvilSpiral->new ()"
"$path = Math::PlanePath::AnvilSpiral->new (wider => $integer, n_start => $n)"
Create and return a new anvil spiral object. An optional "wider" parameter widens the
spiral path, it defaults to 0 which is no widening.
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
<http://oeis.org/A033581> (etc)
default wider=0, n_start=1
A033570 N on X axis, alternate pentagonals (2n+1)*(3n+1)
A126587 N on Y axis
A136392 N on Y negative (n=-Y+1)
A033568 N on X=Y diagonal, alternate second pents (2*n-1)*(3*n-1)
A085473 N on south-east diagonal
wider=0, n_start=0
A211014 N on X axis, 14-gonal numbers of the second kind
A139267 N on Y axis, 2*octagonal
A049452 N on X negative, alternate pentagonals
A033580 N on Y negative, 4*pentagonals
A051866 N on X=Y diagonal, 14-gonal numbers
A094159 N on north-west diagonal, 3*hexagonals
A049453 N on south-west diagonal, alternate second pentagonal
A195319 N on south-east diagonal, 3*second hexagonals
wider=1, n_start=0
A051866 N on X axis, 14-gonal numbers
A049453 N on Y negative, alternate second pentagonal
A033569 N on north-west diagonal
A085473 N on south-west diagonal
A080859 N on Y negative
A033570 N on south-east diagonal
alternate pentagonals (2n+1)*(3n+1)
wider=2, n_start=1
A033581 N on Y axis (6*n^2) except for initial N=2
SEE ALSO
Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::OctagramSpiral,
Math::PlanePath::HexSpiral
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015 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/>.