1Math::PlanePath::SacksSUpsierralC(o3n)tributed Perl DocuMmaetnht:a:tPiloannePath::SacksSpiral(3)
2
6 Math::PlanePath::SacksSpiral -- circular spiral squaring each
7 revolution
8
10 use Math::PlanePath::SacksSpiral;
11 my $path = Math::PlanePath::SacksSpiral->new;
12 my ($x, $y) = $path->n_to_xy (123);
13
15 The Sacks spiral by Robert Sacks is an Archimedean spiral with points N
16 placed on the spiral so the perfect squares fall on a line going to the
17 right. Read more at
18 <http://www.numberspiral.com>
20 An Archimedean spiral means radial distance a constant factor of the
22 angle, and so each loop a constant distance out from the preceding
23 loop, in this case 1 unit out. The polar coordinates are
24 R = sqrt(N)
26 theta = sqrt(N) * 2pi
27 which comes out roughly as
29 18
31 19 11 10 17
32 5
33 20 12 6 2
35 0 1 4 9 16 25
36 3
38 21 13 7 8
39 15 24
40 14
41 22 23
42 The X,Y positions returned are fractional, except for the perfect
44 squares on the positive X axis at X=0,1,2,3,etc. The perfect squares
45 are the closest points, at 1 unit apart. Other points are a little
46 further apart.
47 The arms going to the right like N=5,10,17,etc or N=8,15,24,etc are
49 constant offsets from the perfect squares, ie. d^2 + c for positive or
50 negative integer c. To the left the central arm N=2,6,12,20,etc is the
51 pronic numbers d^2 + d = d*(d+1), half way between the successive
52 perfect squares. Other arms going to the left are offsets from that,
53 ie. d*(d+1) + c for integer c.
54 Euler's quadratic d^2+d+41 is one such arm going left. Low values loop
56 around a few times before straightening out at about y=-127. This
57 quadratic has relatively many primes and in a plot of primes on the
58 spiral it can be seen standing out from its surrounds.
59 Plotting various quadratic sequences of points can form attractive
61 patterns. For example the triangular numbers k*(k+1)/2 come out as
62 spiral arcs going clockwise and anti-clockwise.
63 See examples/sacks-xpm.pl for a complete program plotting the spiral
65 points to an XPM image.
66
68 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
69 classes.
70 "$path = Math::PlanePath::SacksSpiral->new ()"
72 Create and return a new path object.
73 "($x,$y) = $path->n_to_xy ($n)"
75 Return the X,Y coordinates of point number $n on the path.
76 $n can be any value "$n >= 0" and fractions give positions on the
78 spiral in between the integer points.
79 For "$n < 0" the return is an empty list, it being considered there
81 are no negative points in the spiral.
82 "$rsquared = $path->n_to_rsquared ($n)"
84 Return the radial distance R^2 of point $n, or "undef" if there's
85 no point $n. This is simply $n itself, since R=sqrt(N).
86 "$n = $path->xy_to_n ($x,$y)"
88 Return an integer point number for coordinates "$x,$y". Each
89 integer N is considered the centre of a circle of diameter 1 and an
90 "$x,$y" within that circle returns N.
91 The unit spacing of the spiral means those circles don't overlap,
93 but they also don't cover the plane and if "$x,$y" is not within
94 one then the return is "undef".
95 Descriptive Methods
97 "$dx = $path->dx_minimum()"
98 "$dx = $path->dx_maximum()"
99 "$dy = $path->dy_minimum()"
100 "$dy = $path->dy_maximum()"
101 dX and dY have minimum -pi=-3.14159 and maximum pi=3.14159. The
102 loop beginning at N=2^k is approximately a polygon of 2k+1 many
103 sides and radius R=k. Each side is therefore
104 side = sin(2pi/(2k+1)) * k
106 -> 2pi/(2k+1) * k
107 -> pi
108 "$str = $path->figure ()"
110 Return "circle".
111
113 Rectangle to N Range
114 R=sqrt(N) here is the same as in the "TheodorusSpiral" and the code is
115 shared here. See "Rectangle to N Range" in
116 Math::PlanePath::TheodorusSpiral.
117 The accuracy could be improved here by taking into account the polar
119 angle of the corners which are candidates for the maximum radius. On
120 the X axis the stripes of N are from X-0.5 to X+0.5, but up on the Y
121 axis it's 0.25 further out at Y-0.25 to Y+0.75. The stripe the corner
122 falls in can thus be biased by theta expressed as a fraction 0 to 1
123 around the plane.
124 An exact theta 0 to 1 would require an arctan, but approximations 0,
126 0.25, 0.5, 0.75 from the quadrants, or eighths of the plane by X>Y etc
127 diagonals. As noted for the Theodorus spiral the over-estimate from
128 ignoring the angle is at worst R many points, which corresponds to a
129 full loop here. Using the angle would reduce that to 1/4 or 1/8 etc of
130 a loop.
131
133 Math::PlanePath, Math::PlanePath::PyramidRows,
134 Math::PlanePath::ArchimedeanChords, Math::PlanePath::TheodorusSpiral,
135 Math::PlanePath::VogelFloret
136
138 <http://user42.tuxfamily.org/math-planepath/index.html>
139
141 Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019,
142 2020 Kevin Ryde
143 This file is part of Math-PlanePath.
145 Math-PlanePath is free software; you can redistribute it and/or modify
147 it under the terms of the GNU General Public License as published by
148 the Free Software Foundation; either version 3, or (at your option) any
149 later version.
150 Math-PlanePath is distributed in the hope that it will be useful, but
152 WITHOUT ANY WARRANTY; without even the implied warranty of
153 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
154 General Public License for more details.
155 You should have received a copy of the GNU General Public License along
157 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
158perl v5.34.0 2021-07-22 Math::PlanePath::SacksSpiral(3)