and 1 contributors

# 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

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 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.

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