Ubuntu Manpage: Math::PlanePath::SacksSpiral -- circular spiral squaring each revolution

Provided by: libmath-planepath-perl_113-1_all

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

<http://www.numberspiral.com>

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".

FORMULAS

   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.

HOME PAGE

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

LICENSE

       Copyright 2010, 2011, 2012, 2013 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/>.