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 radial distance a constant factor of the angle, and so each loop a constant distance out from the preceding loop, in this case 1 unit out. The polar coordinates are

    R = sqrt(N)
    theta = sqrt(N) * 2pi

which comes out roughly as

          19   11        10  17
    20  12  6   2
                   0  1   4   9  16  25

      21   13   7        8
                             15   24
               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/ 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.

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.


Math::PlanePath, Math::PlanePath::PyramidRows, Math::PlanePath::ArchimedeanChords, Math::PlanePath::TheodorusSpiral, Math::PlanePath::VogelFloret



Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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 <>.