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 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/>.