Kevin Ryde > Math-PlanePath > Math::PlanePath::AnvilSpiral

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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 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/>.

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