SYNOPSIS
use Math::PlanePath::PentSpiral;
my $path = Math::PlanePath::PentSpiral->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This path makes a pentagonal (five-sided) spiral with points spread out to fit on a square grid.
22 3
23 10 21 2
24 11 3 9 20 1
25 12 4 1 2 8 19 <- Y=0
26 13 5 6 7 18 ... -1
27 14 15 16 17 33 -2
28 29 30 31 32 -2
^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
-6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7
Each horizontal gap is 2, so for instance n=1 is at x=0,y=0 then n=2 is at x=2,y=0. The lower diagonals are 1 across and 1 down, so n=17 is at x=4,y=-2 and n=18 is x=5,y=-1. But the upper angles go 2 across and 1 up, so n=20 is x=4,y=1 then n=21 is x=2,y=2.
The effect is to make the sides equal length, except for a kink at the lower right corner. Only every second square in the plane is used. In the top half (y>=0) those points line up, in the lower half (y<0) they're offset on alternate rows.
N Start
The default is to number points starting N=1 as shown above. An optional "n_start" can give a different start, in the same pattern. For example to start at 0,
n_start => 0 38
39 21 37
...
40 22 9 20 36 57
41 23 10 2 8 19 35 56
42 24 11 3 0 1 7 18 34 55
43 25 12 4 5 6 17 33 54
44 26 13 14 15 16 32 53
45 27 28 29 30 31 52
46 47 48 49 50 51
FUNCTIONS
See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes.- "$path = Math::PlanePath::PentSpiral->new ()"
- "$path = Math::PlanePath::PentSpiral->new (n_start => $n)"
- Create and return a new pentagon spiral object.
- "$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.
FORMULAS
N to X,Y
It's convenient to work in terms of Nstart=0 and to take each loop as beginning on the South-West diagonal,
21 loop d=3
-- --
22 20
-- --
23 19
-- --
24 0 18
\ /
25 . 17
\ /
26 13----14----15----16
\
.
The SW diagonal is N=0,4,13,27,46,etc which is
N = (5d-7)*d/2 + 1 # starting d=1 first loop
This can be inverted to get d from N
d = floor( (sqrt(40*N + 9) + 7) / 10 )
Each side is length d, except the lower right diagonal slope which is d-1. For the very first loop that lower right is length 0.
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
- <http://oeis.org/A140066> (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
A005891 N on South-East diagonal, centred pentagonals
A134238 N on South-West diagonal
n_start=0
A000566 N on X axis, heptagonal numbers
A005476 N on Y axis
A028895 N on South-East diagonal
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016 Kevin RydeThis 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/>.