/ 
Math-PlanePath-128
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::AnvilSpiral

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

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, 2016, 2017, 2018, 2019 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/>.