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.\"
.IX Title "Math::PlanePath::PentSpiral 3pm"
.TH Math::PlanePath::PentSpiral 3pm "2014-06-14" "perl v5.20.1" "User Contributed Perl Documentation"
.\" For nroff, turn off justification.  Always turn off hyphenation; it makes
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.nh
.SH "NAME"
Math::PlanePath::PentSpiral \-\- integer points in a pentagonal shape
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
.Vb 3
\& use Math::PlanePath::PentSpiral;
\& my $path = Math::PlanePath::PentSpiral\->new;
\& my ($x, $y) = $path\->n_to_xy (123);
.Ve
.SH "DESCRIPTION"
.IX Header "DESCRIPTION"
This path makes a pentagonal (five-sided) spiral with points spread out to
fit on a square grid.
.PP
.Vb 1
\&                      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
.Ve
.PP
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.
.PP
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.
.SS "N Start"
.IX Subsection "N Start"
The default is to number points starting N=1 as shown above.  An optional
\&\f(CW\*(C`n_start\*(C'\fR can give a different start, in the same pattern.  For example to
start at 0,
.PP
.Vb 1
\&    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
.Ve
.SH "FUNCTIONS"
.IX Header "FUNCTIONS"
See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes.
.ie n .IP """$path = Math::PlanePath::PentSpiral\->new ()""" 4
.el .IP "\f(CW$path = Math::PlanePath::PentSpiral\->new ()\fR" 4
.IX Item "$path = Math::PlanePath::PentSpiral->new ()"
.PD 0
.ie n .IP """$path = Math::PlanePath::PentSpiral\->new (n_start => $n)""" 4
.el .IP "\f(CW$path = Math::PlanePath::PentSpiral\->new (n_start => $n)\fR" 4
.IX Item "$path = Math::PlanePath::PentSpiral->new (n_start => $n)"
.PD
Create and return a new pentagon spiral object.
.ie n .IP """$n = $path\->xy_to_n ($x,$y)""" 4
.el .IP "\f(CW$n = $path\->xy_to_n ($x,$y)\fR" 4
.IX Item "$n = $path->xy_to_n ($x,$y)"
Return the point number for coordinates \f(CW\*(C`$x,$y\*(C'\fR.  \f(CW$x\fR and \f(CW$y\fR are
each rounded to the nearest integer, which has the effect of treating each
point in the path as a square of side 1.
.SH "FORMULAS"
.IX Header "FORMULAS"
.SS "N to X,Y"
.IX Subsection "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,
.PP
.Vb 10
\&                      21                loop d=3
\&                   \-\-    \-\-
\&                22          20
\&             \-\-                \-\-
\&          23                      19
\&       \-\-                            \-\-
\&    24                 0                18
\&      \e                                /
\&       25          .                 17
\&         \e                          /
\&          26    13\-\-\-\-14\-\-\-\-15\-\-\-\-16
\&            \e
\&             .
.Ve
.PP
The \s-1SW\s0 diagonal is N=0,4,13,27,46,etc which is
.PP
.Vb 1
\&    N = (5d\-7)*d/2 + 1           # starting d=1 first loop
.Ve
.PP
This can be inverted to get d from N
.PP
.Vb 1
\&    d = floor( (sqrt(40*N + 9) + 7) / 10 )
.Ve
.PP
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.
.SH "OEIS"
.IX Header "OEIS"
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this
path include
.Sp
.RS 4
<http://oeis.org/A140066> (etc)
.RE
.PP
.Vb 6
\&    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
.Ve
.SH "SEE ALSO"
.IX Header "SEE ALSO"
Math::PlanePath,
Math::PlanePath::PentSpiralSkewed,
Math::PlanePath::HexSpiral
.SH "HOME PAGE"
.IX Header "HOME PAGE"
<http://user42.tuxfamily.org/math\-planepath/index.html>
.SH "LICENSE"
.IX Header "LICENSE"
Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde
.PP
This file is part of Math-PlanePath.
.PP
Math-PlanePath is free software; you can redistribute it and/or modify it
under the terms of the \s-1GNU\s0 General Public License as published by the Free
Software Foundation; either version 3, or (at your option) any later
version.
.PP
Math-PlanePath is distributed in the hope that it will be useful, but
\&\s-1WITHOUT ANY WARRANTY\s0; without even the implied warranty of \s-1MERCHANTABILITY\s0
or \s-1FITNESS FOR A PARTICULAR PURPOSE. \s0 See the \s-1GNU\s0 General Public License for
more details.
.PP
You should have received a copy of the \s-1GNU\s0 General Public License along with
Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.