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# NAME

Math::PlanePath::CincoCurve -- 5x5 self-similar curve

# SYNOPSIS

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

# DESCRIPTION

This is the 5x5 self-similar Cinco curve

It makes a 5x5 self-similar traversal of the first quadrant X>0,Y>0.

``````                                                    |
4  |  10--11  14--15--16  35--36  39--40--41  74  71--70  67--66
|   |   |   |       |   |   |   |       |   |   |   |   |   |
3  |   9  12--13  18--17  34  37--38  43--42  73--72  69--68  65
|   |           |       |           |                       |
2  |   8   5-- 4  19--20  33  30--29  44--45  52--53--54  63--64
|   |   |   |       |   |   |   |       |   |       |   |
1  |   7-- 6   3  22--21  32--31  28  47--46  51  56--55  62--61
|           |   |               |   |       |   |           |
Y=0  |   0-- 1-- 2  23--24--25--26--27  48--49--50  57--58--59--60
|
+--------------------------------------------------------------
X=0  1   2   3   4   5   6   7   8   9  10  11  12  13  14``````

The base pattern is the N=0 to N=24 part. It repeats transposed and rotated to make the ends join. N=25 to N=49 is a repeat of the base, then N=50 to N=74 is a transpose to go upwards. The sub-part arrangements are as follows.

``````    +------+------+------+------+------+
|  10  |  11  |  14  |  15  |  16  |
|      |      |      |      |      |
|----->|----->|----->|----->|----->|
+------+------+------+------+------+
|^  9  |  12 ||^ 13  |  18 ||<-----|
||  T  |  T  |||  T  |  T  ||  17  |
||     |     v||     |     v|      |
+------+------+------+------+------+
|^  8  |  5  ||^  4  |  19 ||  20  |
||  T  |  T  |||  T  |  T  ||      |
||     |     v||     |     v|----->|
+------+------+------+------+------+
|<-----|<---- |^  3  |  22 ||<-----|
|  7   |  6   ||  T  |  T  ||  21  |
|      |      ||     |     v|      |
+------+------+------+------+------+
|  0   |  1   |^  2  |  23 ||  24  |
|      |      ||  T  |  T  ||      |
|----->|----->||     |     v|----->|
+------+------+------+------+------+``````

Parts such as 6 going left are the base rotated 180 degrees. The verticals like 2 are a transpose of the base, ie. swap X,Y, and downward vertical like 23 is transpose plus rotate 180 (which is equivalent to a mirror across the anti-diagonal). Notice the base shape fills its sub-part to the left side and the transpose instead fills on the right.

The N values along the X axis are increasing, as are the values along the Y axis. This occurs because the values along the sub-parts of the base are increasing along the X and Y axes, and the other two sides are increasing too when rotated or transposed for sub-parts such as 2 and 23, or 7, 8 and 9.

Dennis conceives this for use in combination with 2x2 Hilbert and 3x3 meander shapes so that sizes which are products of 2, 3 and 5 can be used for partitioning. Such mixed patterns can't be done with the code here, mainly since a mixture depends on having a top-level target size rather than the unlimited first quadrant here.

# FUNCTIONS

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

`\$path = Math::PlanePath::CincoCurve->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. Points begin at 0 and if `\$n < 0` then the return is an empty list.

## Level Methods

`(\$n_lo, \$n_hi) = \$path->level_to_n_range(\$level)`

Return `(0, 25**\$level - 1)`.