Math::PlanePath::PyramidSpiral(3pm)

1Math::PlanePath::PyramiUdsSeprirCaoln(t3r)ibuted Perl DoMcautmhe:n:tPaltainoenPath::PyramidSpiral(3)
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NAME

6       Math::PlanePath::PyramidSpiral -- integer points drawn around a pyramid
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SYNOPSIS

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

14       This path makes a pyramid shaped spiral,
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16                             31                          3
17                            /  \
18                          32 13 30                       2
19                         /  /  \  \
20                       33 14  3 12 29                    1
21                      /  /  /  \  \  \
22                    34 15  4  1--2 11 28 ...         <- Y=0
23                   /  /  /           \  \  \
24                 35 16  5--6--7--8--9-10 27 52          -1
25                /  /                       \  \
26              36 17-18-19-20-21-22-23-24-25-26 51       -2
27             /                                   \
28           37-38-39-40-41-42-43-44-45-46-47-48-49-50    -3
29
30                              ^
31           -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7
32
33       The perfect squares 1,4,9,16 fall one before the bottom left corner of
34       each loop, and the pronic numbers 2,6,12,20,30,etc are the vertical
35       upwards from X=1,Y=0.
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37   Square Spiral
38       This spiral goes around at the same rate as the "SquareSpiral".  It's
39       as if two corners are cut off (like the "DiamondSpiral") and two others
40       extended (like the "OctagramSpiral").  The net effect is the same
41       looping rate but the points pushed around a bit.
42
43       Taking points up to a perfect square shows the similarity.  The two
44       triangular cut-off corners marked by "."s are matched by the two
45       triangular extensions.
46
47                   +--------------------+   7x7 square
48                   | .  .  . 31  .  .  .|
49                   | .  . 32 13 30  .  .|
50                   | . 33 14  3 12 29  .|
51                   |34 15  4  1  2 11 28|
52                 35|16  5  6  7  8  9 10|27
53              36 17|18 19 20 21 22 23 24|25 26
54           37 38 39|40 41 42 43 44 45 46|47 48 49
55                   +--------------------+
56
57   N Start
58       The default is to number points starting N=1 as shown above.  An
59       optional "n_start" can give a different start, with the same shape etc.
60       For example to start at 0,
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62                       12         n_start => 0
63                      /  \
64                    13  2 11
65                   /  /  \  \
66                 14  3  0--1 10
67                /  /           \
68              15  4--5--6--7--8--9
69             /
70           16-17-18-19-20-21-22-...
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FUNCTIONS

73       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
74       classes.
75
76       "$path = Math::PlanePath::PyramidSpiral->new ()"
77       "$path = Math::PlanePath::PyramidSpiral->new (n_start => $n)"
78           Create and return a new pyramid spiral object.
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80       "$n = $path->xy_to_n ($x,$y)"
81           Return the point number for coordinates "$x,$y".  $x and $y are
82           each rounded to the nearest integer, which has the effect of
83           treating each N in the path as centred in a square of side 1, so
84           the entire plane is covered.
85

OEIS

87       This path is in Sloane's Online Encyclopedia of Integer Sequences as
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89           <http://oeis.org/A053615> (etc)
90
91           n_start=1 (the default)
92             A053615    abs(X), distance to next pronic, but starts n=0
93             A054552    N on X axis, 4n^2 - 3n + 1
94             A033951    N on South-East diagonal, 4n^2 + 3n + 1
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96             A214250    sum N of eight surrounding cells
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98             A217013    permutation N of points in SquareSpiral order
99                          rotated +90 degrees
100             A217294    inverse
101
102       In the two permutations the pyramid spiral is conceived as starting to
103       the left and the square spiral starting upwards.  The paths here start
104       in the same direction (both to the right), hence rotate 90 to adjust
105       the orientation.
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107           n_start=0
108             A329116    X coordinate
109             A329972    Y coordinate
110             A053615    abs(X)
111             A339265    dX-dY increments (runs +1,-1)
112             A001107    N on X axis, decagonal numbers
113             A002939    N on Y axis
114             A033991    N on X negative axis
115             A002943    N on Y negative axis
116             A007742    N on diagonal South-West
117             A033954    N on diagonal South-East, decagonal second kind
118
119           n_start=2
120             A185669    N on diagonal South-East
121

HOME PAGE

128       <http://user42.tuxfamily.org/math-planepath/index.html>
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LICENSE

131       Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019,
132       2020, 2021 Kevin Ryde
133
134       This file is part of Math-PlanePath.
135
136       Math-PlanePath is free software; you can redistribute it and/or modify
137       it under the terms of the GNU General Public License as published by
138       the Free Software Foundation; either version 3, or (at your option) any
139       later version.
140
141       Math-PlanePath is distributed in the hope that it will be useful, but
142       WITHOUT ANY WARRANTY; without even the implied warranty of
143       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
144       General Public License for more details.
145
146       You should have received a copy of the GNU General Public License along
147       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
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151perl v5.36.0                      2022-07-22 Math::PlanePath::PyramidSpiral(3)