Math::PlanePath::PentSpiralSkewed -- integer points in a pentagonal shape
use Math::PlanePath::PentSpiralSkewed; my $path = Math::PlanePath::PentSpiralSkewed->new; my ($x, $y) = $path->n_to_xy (123);
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
).
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
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.
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A192136 (etc)
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
Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::DiamondSpiral, Math::PlanePath::HexSpiralSkewed
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017 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.
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