++ed by:
Kevin Ryde
and 1 contributors

# NAME

Math::PlanePath::SquareArms -- four spiral arms

# SYNOPSIS

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

# DESCRIPTION

This path follows four spiral arms, each advancing successively,

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

^   ^   ^   ^   ^   ^   ^
-3  -2  -1  X=0  1   2   3 ...``````

Each arm is quadratic, with each loop 128 longer than the preceding. The perfect squares fall in eight straight lines 4, with the even squares on the X and Y axes and the odd squares on the diagonals X=Y and X=-Y.

Some novel straight lines arise from numbers which are a repdigit in one or more bases (Sloane's A167782). "111" in various bases falls on straight lines. Numbers "[16][16][16]" in bases 17,19,21,etc are a horizontal at Y=3 because they're perfect squares, and "[64][64][64]" in base 65,66,etc go a vertically downwards from X=12,Y=-266 similarly because they're squares.

Each arm is N=4*k+rem for a remainder rem=0,1,2,3, so sequences related to multiples of 4 or with a modulo 4 pattern may fall on particular arms.

# FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

`\$path = Math::PlanePath::SquareArms->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. For `\$n < 1` the return is an empty list, as the path starts at 1.

Fractional `\$n` gives a point on the line between `\$n` and `\$n+4`, that `\$n+4` being the next point on the same spiralling arm. This is probably of limited use, but arises fairly naturally from the calculation.

## Descriptive Methods

`\$arms = \$path->arms_count()`

Return 4.

# FORMULAS

## Rectangle N Range

Within a square X=-d...+d, and Y=-d...+d the biggest N is the end of the N=5 arm in that square, which is N=9, 25, 49, 81, etc, (2d+1)^2, in successive corners of the square. So for a rectangle find a surrounding d square,

``    d = max(abs(x1),abs(y1),abs(x2),abs(y2))``

from which

``````    Nmax = (2*d+1)^2
= (4*d + 4)*d + 1``````

This can be used for a minimum too by finding the smallest d covered by the rectangle.

``````    dlo = max (0,
min(abs(y1),abs(y2)) if x=0 not covered
min(abs(x1),abs(x2)) if y=0 not covered
)``````

from which the maximum of the preceding dlo-1 square,

``````    Nlo = /  1 if dlo=0
\  (2*(dlo-1)+1)^2 +1  if dlo!=0
= (2*dlo - 1)^2
= (4*dlo - 4)*dlo + 1``````

For a tighter maximum, horizontally N increases to the left or right of the diagonal X=Y line (or X=Y+/-1 line), which means one end or the other is the maximum. Similar vertically N increases above or below the off-diagonal X=-Y so the top or bottom is the maximum. This means for a rectangle the biggest N is at one of the four corners,

``````    Nhi = max (xy_to_n (x1,y1),
xy_to_n (x1,y2),
xy_to_n (x2,y1),
xy_to_n (x2,y2))``````

The current code uses a dlo for Nlo and the corners for Nhi, which means the high is exact but the low is not.

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