Math::...Spiral(3pm)

📦
version 129 · 3 versions
, in debian bookworm, debian trixie, ubuntu jammy, ubuntu noble, ubuntu resolute, ubuntu stonking

💖 Math::PlanePath::AnvilSpiral — integer points around an "anvil" shape 💖

okay so bestie, welcome to the Math::...Spiral page — that's the manual name, she's fancy like that — and today we're talking about Math::PlanePath::AnvilSpiral, which is literally just integer points spiralling around an anvil shape. iconic. no cap. 🔨✨

🌸 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, and honestly the way it does that? it's giving geometry queen 💅

                              ...-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). 💀 the primes really said "not here bestie"

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. 🧚 math is so satisfying when it lines up like that fr fr

🦋 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. 🔄 lowkey elegant

🎀 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. zero-based besties rise up 🙌

💅 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. ✨ fresh object, fresh vibes

🔥 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

the way these sequence numbers just keep stacking up? obsessed 😭👑

🌈 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, 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/>.

bestie, real talk: that warranty disclaimer is not a joke, it's the whole point — no warranty, no promises, that's the deal 💀

and that's the whole page, you ate it up 🫶 this was Math::...Spiral(3pm) from the User Contributed Perl Documentation collection, running on perl v5.32.0 and dated 2021-01-23 — slay, go forth and spiral 💖🎀🦋

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