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

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 right axis at X=0,1,2,3,etc. Those perfect squares are spaced 1 apart, other pointer are a little further apart.

The arms going to the right like 5,10,17,etc or 8,15,24,etc are constant offsets from the perfect squares, ie. s**2 + c for a positive or negative integer c. The central arm 2,6,12,20,etc going left is the pronic numbers s**2 + s, half way between the successive perfect squares. Other arms going to the left are offsets from that, ie. s**2 + s + c for integer c.

Plotting quadratic sequences in the points can form attractive patterns. For example the triangular numbers (s**2 + s)/2 come out as spiral arms going clockwise and counter-clockwise.

FUNCTIONS

$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.

$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.

SEE ALSO

Math::PlanePath, Math::PlanePath::PyramidRows Math::PlanePath::VogelFloret

HOME PAGE

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

LICENSE

Math-PlanePath is Copyright 2010 Kevin Ryde

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/>.