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# NAME

Math::PlanePath::KnightSpiral -- integer points around a square, by chess knight moves

# SYNOPSIS

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

# DESCRIPTION

This path traverses the plane by an infinite "knight's tour" in the form of a square spiral.

``````                            ...
21   4   9  14  19                 2

10  15  20   3   8      28         1

5  22   1  18  13            <- Y=0

16  11  24   7   2  27             1

23   6  17  12  25                 2

26

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

Each step is a chess knight's move 1 across and 2 along, or vice versa. The pattern makes 4 cycles on a 2-wide path around a square before stepping outwards to do the same again to a now bigger square. The above sample shows the first 4-cycle around the central 1, then stepping out at 26 and beginning to go around the outside of the 5x5 square.

An attractive traced out picture of the path appeared in the past at `www.borderschess.org`,

(HTML colours might might make the text invisible. Try deleting, or browser option to ignore page colours, or a text browser.)

See math-image to draw the path lines too.

# FUNCTIONS

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

`\$path = Math::PlanePath::KnightSpiral->new ()`

Create and return a new knight spiral 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, 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.

# OEIS

This Knight's tour is in Sloane's OEIS following the Knight spiral and giving the resulting X,Y location by the `SquareSpiral` numbering. There's eight forms for 4 rotations and the two spirals same or opposite directions.

``````    permutations
A068608   same knight and square spiral directions
A068609   rotate 90 degrees
A068610   rotate 180 degrees
A068611   rotate 270 degrees
A068612   rotate 180 degrees, spiral opp dir (X negate)
A068613   rotate 270 degrees, spiral opp dir
A068614   spiral opposite direction (Y negate)
A068615   rotate 90 degrees, spiral opp dir (X,Y transpose)``````

See examples/knights-oeis.pl for a sample program printing the values of A068608.

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