++ed by:
Kevin Ryde
and 1 contributors

# NAME

Math::PlanePath::SquareSpiral -- integer points drawn around a square (or rectangle)

# SYNOPSIS

`````` use Math::PlanePath::SquareSpiral;
my \$path = Math::PlanePath::SquareSpiral->new;
my (\$x, \$y) = \$path->n_to_xy (123);``````

# DESCRIPTION

This path makes a square spiral,

``````    37--36--35--34--33--32--31         3
|                       |
38  17--16--15--14--13  30         2
|   |               |   |
39  18   5---4---3  12  29         1
|   |   |       |   |   |
40  19   6   1---2  11  28    <- y=0
|   |   |           |   |
41  20   7---8---9--10  27        -1
|   |                   |
42  21--22--23--24--25--26        -2
|
43--44--45--46--47 ...

^
-3  -2  -1  x=0  1   2   3``````

The perfect squares 1,4,9,16,25 fall on diagonals with the even perfect squares going to the upper left and the odd ones to the lower right. The pronic numbers 2,6,12,20,30,42 etc (k^2+k) half way between the squares fall on similar diagonals to the upper right and lower left.

This path is well known from Stanislaw Ulam finding interesting straight lines plotting the prime numbers on it. See examples/ulam-spiral-xpm.pl in the sources for a program generating that, or see math-image using this SquareSpiral to draw Ulam's pattern and more.

In general straight lines in this spiral and other stepped paths (meaning everything except the VogelFloret currently) are quadratics a*k^2+b*k+c, with a=step/2 where step is how much longer each loop takes than the preceding (8 in the case of the SquareSpiral). There are various interesting properties of primes in quadratic progressions like this. Some seem to have more primes than others, for instance see PyramidSides for Euler's k^2+k+41. Many quadratics have no primes at all, or above a certain point, either trivially if always a multiple of 2 or similar, or by a more sophisticated reasoning. See PyramidRows with step 3 for an example of a factorization by the roots making a no-primes gap.

## Wider

An optional `wider` parameter makes the path wider, becoming a rectangle instead of a square. For example

``    \$path = Math::PlanePath::SquareSpiral->new (wider => 3);``

gives

``````    29--28--27--26--25--24--23--22        2
|                           |
30  11--10-- 9-- 8-- 7-- 6  21        1
|   |                   |   |
31  12   1-- 2-- 3-- 4-- 5  20   <- y=0
|   |                       |
32  13--14--15--16--17--18--19       -1
|
33--34--35--36-...                   -2

^
-4  -3  -2  -1  x=0  1   2   3``````

The centre horizontal 1 to 2 is extended by `wider` many further places, then the path loops around that shape. The starting point 1 is shifted to the left by wider/2 places (rounded up to an integer) to keep the spiral centred on the origin x=0,y=0.

Widening doesn't change the nature of the straight lines which arise, it just rotates them around. For example in this wider=3 example the perfect squares are still on diagonals, but the even squares go towards the bottom left (instead of top left when wider=0) and the odd squares to the top right (instead of the bottom right).

Each loop is still 8 longer than the previous, since the widening is basically a constant amount added into each loop.

## Corners

Other spirals can be formed by cutting the corners of the square so as to go around faster. See the following module,

``````    Corners Cut    Class
-----------    -----
1        HeptSpiralSkewed
2        HexSpiralSkewed
3        PentSpiralSkewed
4        DiamondSpiral``````

The PyramidSpiral is a re-shaped SquareSpiral looping at the same rate.

# FUNCTIONS

`\$path = Math::PlanePath::SquareSpiral->new ()`
`\$path = Math::PlanePath::SquareSpiral->new (wider => \$w)`

Create and return a new square spiral object. An optional `wider` parameter widens the spiral path, it defaults to 0 which is no widening.

`(\$x,\$y) = \$path->n_to_xy (\$n)`

Return the x,y coordinates of point number `\$n` on the path.

For `\$n < 1` the return is an empty list, it being considered the path starts at 1.

`\$n = \$path->xy_to_n (\$x,\$y)`

Return the point number for coordinates `\$x,\$y`. `\$x` and `\$y` are each rounded to the nearest integer, which has the effect of treating each N in the path as centred in a square of side 1, so the entire plane is covered.