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 isundef
.
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/>.