NAME¶
Math::PlanePath::SacksSpiral -- circular spiral squaring each revolution
SYNOPSIS¶
use Math::PlanePath::SacksSpiral;
my $path = Math::PlanePath::SacksSpiral->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION¶
The Sacks spiral by Robert Sacks is an Archimedean spiral with points N placed
on the spiral so the perfect squares fall on a line going to the right. Read
more at
An Archimedean spiral means each loop is a constant distance from the preceding,
in this case 1 unit. The polar coordinates are
R = sqrt(N)
theta = sqrt(N) * 2pi
which comes out roughly as
18
19 11 10 17
5
20 12 6 2
0 1 4 9 16 25
3
21 13 7 8
15 24
14
22 23
The X,Y positions returned are fractional, except for the perfect squares on the
positive X axis at X=0,1,2,3,etc. The perfect squares are the closest points,
at 1 unit apart. Other points are a little further apart.
The arms going to the right like N=5,10,17,etc or N=8,15,24,etc are constant
offsets from the perfect squares, ie. d^2 + c for positive or
negative integer c. To the left the central arm N=2,6,12,20,etc is the pronic
numbers d^2 + d = d*(d+1), half way between the successive
perfect squares. Other arms going to the left are offsets from that, ie.
d*(d+1) + c for integer c.
Euler's quadratic d^2+d+41 is one such arm going left. Low values loop around a
few times before straightening out at about y=-127. This quadratic has
relatively many primes and in a plot of primes on the spiral it can be seen
standing out from its surrounds.
Plotting various quadratic sequences of points can form attractive patterns. For
example the triangular numbers k*(k+1)/2 come out as spiral arcs going
clockwise and anti-clockwise.
See
examples/sacks-xpm.pl in the Math-PlanePath sources for a complete
program plotting the spiral points to an XPM image.
FUNCTIONS¶
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
classes.
- "$path = Math::PlanePath::SacksSpiral->new ()"
- Create and return a new path object.
- "($x,$y) = $path->n_to_xy ($n)"
- Return the X,Y coordinates of point number $n on the path.
$n can be any value "$n >= 0" and fractions give positions on
the spiral in between the integer points.
For "$n < 0" the return is an empty list, it being considered
there are no negative points in the spiral.
- "$rsquared = $path->n_to_rsquared ($n)"
- Return the radial distance R^2 of point $n, or "undef" if
there's no point $n. This is simply $n itself, since R=sqrt(N).
- "$n = $path->xy_to_n ($x,$y)"
- Return an integer point number for coordinates "$x,$y". Each
integer N is considered the centre of a circle of diameter 1 and an
"$x,$y" within that circle returns N.
The unit spacing of the spiral means those circles don't overlap, but they
also don't cover the plane and if "$x,$y" is not within one then
the return is "undef".
Descriptive Methods¶
- "$dx = $path->dx_minimum()"
- "$dx = $path->dx_maximum()"
- "$dy = $path->dy_minimum()"
- "$dy = $path->dy_maximum()"
- dX and dY have minimum -pi=-3.14159 and maximum pi=3.14159. The loop
beginning at N=2^k is approximately a polygon of 2k+1 many sides and
radius R=k. Each side is therefore
side = sin(2pi/(2k+1)) * k
-> 2pi/(2k+1) * k
-> pi
- "$str = $path->figure ()"
- Return "circle".
Rectangle to N Range¶
R=sqrt(N) here is the same as in the "TheodorusSpiral" and the code is
shared here. See "Rectangle to N Range" in
Math::PlanePath::TheodorusSpiral.
The accuracy could be improved here by taking into account the polar angle of
the corners which are candidates for the maximum radius. On the X axis the
stripes of N are from X-0.5 to X+0.5, but up on the Y axis it's 0.25 further
out at Y-0.25 to Y+0.75. The stripe the corner falls in can thus be biased by
theta expressed as a fraction 0 to 1 around the plane.
An exact theta 0 to 1 would require an arctan, but approximations 0, 0.25, 0.5,
0.75 from the quadrants, or eighths of the plane by X>Y etc diagonals. As
noted for the Theodorus spiral the over-estimate from ignoring the angle is at
worst R many points, which corresponds to a full loop here. Using the angle
would reduce that to 1/4 or 1/8 etc of a loop.
SEE ALSO¶
Math::PlanePath, Math::PlanePath::PyramidRows,
Math::PlanePath::ArchimedeanChords, Math::PlanePath::TheodorusSpiral,
Math::PlanePath::VogelFloret
HOME PAGE¶
<
http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE¶
Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde
This file is part of Math-PlanePath.
Math-PlanePath is free software; you can redistribute it and/or modify it under
the terms of the GNU General Public License as published by the Free Software
Foundation; either version 3, or (at your option) any later version.
Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with
Math-PlanePath. If not, see <
http://www.gnu.org/licenses/>.