/ 
Math-PlanePath-129
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::PentSpiralSkewed

NAME

Math::PlanePath::PentSpiralSkewed -- integer points in a pentagonal shape

SYNOPSIS

use Math::PlanePath::PentSpiralSkewed;
my $path = Math::PlanePath::PentSpiralSkewed->new;
my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This path makes a pentagonal (five-sided) spiral with points skewed so as to fit a square grid and fully cover the plane.

      10 ...             2
     /  \  \
   11  3  9 20           1
  /  /  \  \  \
12  4  1--2  8 19    <- Y=0
  \  \       |  |
   13  5--6--7 18       -1
     \          |
      14-15-16-17       -2

 ^  ^  ^  ^  ^  ^
-2 -1 X=0 1  2  3 ...

The pattern is similar to the SquareSpiral but cuts three corners which makes each cycle is faster. Each cycle is just 5 steps longer than the previous (where it's 8 for a SquareSpiral).

N Start

The default is to number points starting N=1 as shown above. An optional n_start can give a different start, in the same pattern. For example to start at 0,

            38             n_start => 0
         39 21 37  ...
      40 22  9 20 36 57
   41 23 10  2  8 19 35 56
42 24 11  3  0  1  7 18 34 55
   43 25 12  4  5  6 17 33 54
      44 26 13 14 15 16 32 53
         45 27 28 29 30 31 52
            46 47 48 49 50 51

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

$path = Math::PlanePath::PentSpiral->new ()
$path = Math::PlanePath::PentSpiral->new (n_start => $n)

Create and return a new path object.

$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 point in the path as a square of side 1.

OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

n_start=1 (the default)
  A192136    N on X axis, (5*n^2 - 3*n + 2)/2
  A140066    N on Y axis
  A116668    N on X negative axis, (5n^2 + n + 2)/2
  A134238    N on Y negative axis
  A158187    N on North-West diagonal, 10*n^2 + 1
  A005891    N on South-East diagonal, centred pentagonals

n_start=0
  A000566    N on X axis, heptagonal numbers
  A005476    N on Y axis
  A005475    N on X negative axis
  A147875    N on Y negative axis, second heptagonals
  A033583    N on North-West diagonal, 10*n^2
  A028895    N on South-East diagonal, 5*triangular

SEE ALSO

Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::DiamondSpiral, Math::PlanePath::HexSpiralSkewed

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/>.