Math::PlanePath::SacksSpiral -- circular spiral squaring each revolution
use Math::PlanePath::SacksSpiral; my $path = Math::PlanePath::SacksSpiral->new; my ($x, $y) = $path->n_to_xy (123);
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.
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.
$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
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
$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 ()
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.
Copyright 2010, 2011, 2012, 2013, 2014, 2015 Kevin Ryde
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