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) = $path->n_to_xy (123);
13
15 The Sacks spiral by Robert Sacks is an Archimedean spiral with points N
16 placed on the spiral so the perfect squares fall on a line going to the
17 right. Read more at
18 <http://www.numberspiral.com>
20 An Archimedean spiral means each loop is a constant distance from the
22 preceding, in this case 1 unit. The polar coordinates are
23 R = sqrt(N)
25 theta = sqrt(N) * 2pi
26 which comes out roughly as
28 18
30 19 11 10 17
31 5
32 20 12 6 2
34 0 1 4 9 16 25
35 3
37 21 13 7 8
38 15 24
39 14
40 22 23
41 The X,Y positions returned are fractional, except for the perfect
43 squares on the positive X axis at X=0,1,2,3,etc. The perfect squares
44 are the closest points, at 1 unit apart. Other points are a little
45 further apart.
46 The arms going to the right like N=5,10,17,etc or N=8,15,24,etc are
48 constant offsets from the perfect squares, ie. d^2 + c for positive or
49 negative integer c. To the left the central arm N=2,6,12,20,etc is the
50 pronic numbers d^2 + d = d*(d+1), half way between the successive
51 perfect squares. Other arms going to the left are offsets from that,
52 ie. d*(d+1) + c for integer c.
53 Euler's quadratic d^2+d+41 is one such arm going left. Low values loop
55 around a few times before straightening out at about y=-127. This
56 quadratic has relatively many primes and in a plot of primes on the
57 spiral it can be seen standing out from its surrounds.
58 Plotting various quadratic sequences of points can form attractive
60 patterns. For example the triangular numbers k*(k+1)/2 come out as
61 spiral arcs going clockwise and anti-clockwise.
62 See examples/sacks-xpm.pl for a complete program plotting the spiral
64 points to an XPM image.
65
67 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
68 classes.
69 "$path = Math::PlanePath::SacksSpiral->new ()"
71 Create and return a new path object.
72 "($x,$y) = $path->n_to_xy ($n)"
74 Return the X,Y coordinates of point number $n on the path.
75 $n can be any value "$n >= 0" and fractions give positions on the
77 spiral in between the integer points.
78 For "$n < 0" the return is an empty list, it being considered there
80 are no negative points in the spiral.
81 "$rsquared = $path->n_to_rsquared ($n)"
83 Return the radial distance R^2 of point $n, or "undef" if there's
84 no point $n. This is simply $n itself, since R=sqrt(N).
85 "$n = $path->xy_to_n ($x,$y)"
87 Return an integer point number for coordinates "$x,$y". Each
88 integer N is considered the centre of a circle of diameter 1 and an
89 "$x,$y" within that circle returns N.
90 The unit spacing of the spiral means those circles don't overlap,
92 but they also don't cover the plane and if "$x,$y" is not within
93 one then the return is "undef".
94 Descriptive Methods
96 "$dx = $path->dx_minimum()"
97 "$dx = $path->dx_maximum()"
98 "$dy = $path->dy_minimum()"
99 "$dy = $path->dy_maximum()"
100 dX and dY have minimum -pi=-3.14159 and maximum pi=3.14159. The
101 loop beginning at N=2^k is approximately a polygon of 2k+1 many
102 sides and radius R=k. Each side is therefore
103 side = sin(2pi/(2k+1)) * k
105 -> 2pi/(2k+1) * k
106 -> pi
107 "$str = $path->figure ()"
109 Return "circle".
110
112 Rectangle to N Range
113 R=sqrt(N) here is the same as in the "TheodorusSpiral" and the code is
114 shared here. See "Rectangle to N Range" in
115 Math::PlanePath::TheodorusSpiral.
116 The accuracy could be improved here by taking into account the polar
118 angle of the corners which are candidates for the maximum radius. On
119 the X axis the stripes of N are from X-0.5 to X+0.5, but up on the Y
120 axis it's 0.25 further out at Y-0.25 to Y+0.75. The stripe the corner
121 falls in can thus be biased by theta expressed as a fraction 0 to 1
122 around the plane.
123 An exact theta 0 to 1 would require an arctan, but approximations 0,
125 0.25, 0.5, 0.75 from the quadrants, or eighths of the plane by X>Y etc
126 diagonals. As noted for the Theodorus spiral the over-estimate from
127 ignoring the angle is at worst R many points, which corresponds to a
128 full loop here. Using the angle would reduce that to 1/4 or 1/8 etc of
129 a loop.
130
132 Math::PlanePath, Math::PlanePath::PyramidRows,
133 Math::PlanePath::ArchimedeanChords, Math::PlanePath::TheodorusSpiral,
134 Math::PlanePath::VogelFloret
135
137 <http://user42.tuxfamily.org/math-planepath/index.html>
138
140 Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017 Kevin Ryde
141 This file is part of Math-PlanePath.
143 Math-PlanePath is free software; you can redistribute it and/or modify
145 it under the terms of the GNU General Public License as published by
146 the Free Software Foundation; either version 3, or (at your option) any
147 later version.
148 Math-PlanePath is distributed in the hope that it will be useful, but
150 WITHOUT ANY WARRANTY; without even the implied warranty of
151 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
152 General Public License for more details.
153 You should have received a copy of the GNU General Public License along
155 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
156perl v5.28.0 2017-12-03 Math::PlanePath::SacksSpiral(3)