Math::PlanePath::PentSpiral(3pm)

1Math::PlanePath::PentSpUisrearl(C3o)ntributed Perl DocumMeanttha:t:iPolnanePath::PentSpiral(3)
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NAME

6       Math::PlanePath::PentSpiral -- integer points in a pentagonal shape
7

SYNOPSIS

9        use Math::PlanePath::PentSpiral;
10        my $path = Math::PlanePath::PentSpiral->new;
11        my ($x, $y) = $path->n_to_xy (123);
12

DESCRIPTION

14       This path makes a pentagonal (five-sided) spiral with points spread out
15       to fit on a square grid.
16
17                             22                              3
18
19                       23    10    21                        2
20
21                 24    11     3     9    20                  1
22
23           25    12     4     1     2     8    19       <- Y=0
24
25              26    13     5     6     7    18    ...       -1
26
27                 27    14    15    16    17    33           -2
28
29                    28    29    30    31    32              -2
30
31
32            ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^  ^
33           -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7
34
35       Each horizontal gap is 2, so for instance n=1 is at x=0,y=0 then n=2 is
36       at x=2,y=0.  The lower diagonals are 1 across and 1 down, so n=17 is at
37       x=4,y=-2 and n=18 is x=5,y=-1.  But the upper angles go 2 across and 1
38       up, so n=20 is x=4,y=1 then n=21 is x=2,y=2.
39
40       The effect is to make the sides equal length, except for a kink at the
41       lower right corner.  Only every second square in the plane is used.  In
42       the top half (y>=0) those points line up, in the lower half (y<0)
43       they're offset on alternate rows.
44
45   N Start
46       The default is to number points starting N=1 as shown above.  An
47       optional "n_start" can give a different start, in the same pattern.
48       For example to start at 0,
49
50           n_start => 0            38
51
52                             39    21    37
53                                                  ...
54                       40    22     9    20    36    57
55
56                 41    23    10     2     8    19    35    56
57
58           42    24    11     3     0     1     7    18    34    55
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60              43    25    12     4     5     6    17    33    54
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62                 44    26    13    14    15    16    32    53
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64                    45    27    28    29    30    31    52
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66                       46    47    48    49    50    51
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FUNCTIONS

69       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
70       classes.
71
72       "$path = Math::PlanePath::PentSpiral->new ()"
73       "$path = Math::PlanePath::PentSpiral->new (n_start => $n)"
74           Create and return a new pentagon spiral object.
75
76       "$n = $path->xy_to_n ($x,$y)"
77           Return the point number for coordinates "$x,$y".  $x and $y are
78           each rounded to the nearest integer, which has the effect of
79           treating each point in the path as a square of side 1.
80

FORMULAS

82   N to X,Y
83       It's convenient to work in terms of Nstart=0 and to take each loop as
84       beginning on the South-West diagonal,
85
86                             21                loop d=3
87                          --    --
88                       22          20
89                    --                --
90                 23                      19
91              --                            --
92           24                 0                18
93             \                                /
94              25          .                 17
95                \                          /
96                 26    13----14----15----16
97                   \
98                    .
99
100       The SW diagonal is N=0,4,13,27,46,etc which is
101
102           N = (5d-7)*d/2 + 1           # starting d=1 first loop
103
104       This can be inverted to get d from N
105
106           d = floor( (sqrt(40*N + 9) + 7) / 10 )
107
108       Each side is length d, except the lower right diagonal slope which is
109       d-1.  For the very first loop that lower right is length 0.
110

OEIS

112       Entries in Sloane's Online Encyclopedia of Integer Sequences related to
113       this path include
114
115           <http://oeis.org/A140066> (etc)
116
117           n_start=1 (the default)
118             A192136    N on X axis, (5*n^2 - 3*n + 2)/2
119             A140066    N on Y axis
120             A116668    N on X negative axis
121             A005891    N on South-East diagonal, centred pentagonals
122             A134238    N on South-West diagonal
123
124           n_start=0
125             A000566    N on X axis, heptagonal numbers
126             A005476    N on Y axis
127             A028895    N on South-East diagonal
128

HOME PAGE

134       <http://user42.tuxfamily.org/math-planepath/index.html>
135

LICENSE

137       Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019,
138       2020 Kevin Ryde
139
140       This file is part of Math-PlanePath.
141
142       Math-PlanePath is free software; you can redistribute it and/or modify
143       it under the terms of the GNU General Public License as published by
144       the Free Software Foundation; either version 3, or (at your option) any
145       later version.
146
147       Math-PlanePath is distributed in the hope that it will be useful, but
148       WITHOUT ANY WARRANTY; without even the implied warranty of
149       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
150       General Public License for more details.
151
152       You should have received a copy of the GNU General Public License along
153       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
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157perl v5.34.0                      2022-01-21    Math::PlanePath::PentSpiral(3)