Math::PlanePath::Flowsnake(3) self-similar path through hexagons

SYNOPSIS

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

DESCRIPTION

This path is an integer version of the flowsnake curve by William Gosper.

A single arm of the curve fills 1/3 of the plane, spiralling around anti-clockwise ever fatter and with jagged self-similar edges.

```                         39----40----41                        8
\           \
32----33----34    38----37    42                     7
\           \        /     /
31----30    35----36    43    47----48            6
/                    \     \     \
28----29    17----16----15    44    46    49--..     5
/              \           \     \  /
27    23----22    18----19    14    45                  4
\     \     \        /     /
26    24    21----20    13    11----10               3
\  /                    \  /     /
25     4---- 5---- 6    12     9                  2
\           \         /
3---- 2     7---- 8                     1
/
0---- 1                                  Y=0
X=-4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11
```

The points are spread out on every second X coordinate to make little triangles with integer coordinates, per ``Triangular Lattice'' in Math::PlanePath.

The base pattern is the seven points 0 to 6,

```    4---- 5---- 6
\           \
3---- 2
/
0---- 1
```

This repeats at 7-fold increasing scale, with sub-sections rotated according to the edge direction and the 1, 2 and 6 sections in reverse. The next level can be seen at the multiple-of-7 points N=0,7,14,21,28,35,42,49.

```                                  42
-----------    ---
35                   ---
-----------                         ---
28                                        49 ---
---
----                  14
---   -----------  |
21              |
|
|
|
---- 7
-----
0 ---
```

Notice this is the same shape as N=0 to N=6, but rotated by atan(1/sqrt(7)) = 20.68 degrees anti-clockwise. Each level rotates further and for example after about 18 levels it goes all the way around and back to the first quadrant.

The rotation doesn't fill the plane though, only 1/3 of it. The shape fattens as it curls around, but leaves a spiral gap beneath (see ``Arms'' below).

Tiling

The base pattern corresponds to a tiling by hexagons as follows, with the ``***'' lines being the base figure.

```          .     .
/ \   / \
/   \ /   \
.     .     .
|     |     |
|     |     |
4*****5*****6
/*\   / \   /*\
/ * \ /   \ / * \
.   * .     .   * .
|   * |     |    *|
|    *|     |    *|
.     3*****2     7...
\   / \   /*\   /
\ /   \ / * \ /
.     . *   .
|     | *   |
|     |*    |
0*****1     .
\   / \   /
\ /   \ /
.     .
```

In the next level the parts corresponding to 1, 2 and 6 are reversed because they have their hexagon tile to the right of the line segment, rather than to the left.

Arms

The optional "arms" parameter can give up to three copies of the flowsnake, each advancing successively. For example "arms=>3" is as follows.

```    arms => 3                        51----48----45                 5
\           \
...   69----66    54----57    42              4
\     \     \        /     /
28----25----22    78    72    63----60    39    33----30     3
\           \     \  /                    \  /     /
31----34    19    75    12----15----18    36    27        2
/     /              \           \        /
40----37    16     4---- 1     9---- 6    21----24           1
/              \     \              /
43    55----58    13     7     0---- 3    74----77---...    <- Y=0
\     \     \     \  /                    \
46    52    61    10     2     8----11    71----68          -1
\  /     /              \  /     /           /
49    64    70----73     5    14    62----65             -2
\  /     /           /     /
67    76    20----17    59    53----50          -3
/     /              \  /     /
...   23    35----38    56    47             -4
\     \     \        /
26    32    41----44                -5
\  /
29                               -6
^
-9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8  9
```

The N=3*k points are the plain curve, N=3*k+1 is a copy rotated by 120 degrees (1/3 around), and N=3*k+2 is a copy rotated by 240 degrees (2/3 around). The initial N=1 of the second arm and N=2 of the third correspond to N=3 of the first arm, rotated around.

Essentially the flowsnake fills an ever expanding hexagon with one corner at the origin, and wiggly sides.

```            *---*
/     \
*---*   A   *         Plain curve in A.
/     \     /          Room for two more arms in B and C,
*   B   O---*           rotated 120 and 240 degrees.
\     /     \
*---*   C   *
\     /
*---*
```

The sides of these ``hexagons'' are not straight lines but instead wild wiggly spiralling S shapes, and the endpoints rotate around by the angle described above at each level. Opposing sides are symmetric, so they mesh perfectly and with three arms fill the plane.

Fractal

The flowsnake can also be thought of as successively subdividing line segments with suitably scaled copies of the 0 to 7 figure (or its reversal).

The code here could be used for that by taking points N=0 to N=7^level. The Y coordinates should be multiplied by sqrt(3) to make proper equilateral triangles, then a rotation and scaling to make the endpoint come out at some desired point, such as X=1,Y=0. With such a scaling the path is confined to a finite fractal boundary.

FUNCTIONS

See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes.
"\$path = Math::PlanePath::Flowsnake->new ()"
"\$path = Math::PlanePath::Flowsnake->new (arms => \$a)"
Create and return a new flowsnake path object.

The optional "arms" parameter gives between 1 and 3 copies of the curve successively advancing.

"(\$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.

Fractional positions give an X,Y position along a straight line between the integer positions.

"(\$n_lo, \$n_hi) = \$path->rect_to_n_range (\$x1,\$y1, \$x2,\$y2)"
In the current code the returned range is exact, meaning \$n_lo and \$n_hi are the smallest and biggest in the rectangle, but don't rely on that yet since finding the exact range is a touch on the slow side. (The advantage of which though is that it helps avoid very big ranges from a simple over-estimate.)

Level Methods

"(\$n_lo, \$n_hi) = \$path->level_to_n_range(\$level)"
Return "(0, 7**\$level)", or for multiple arms return "(0, \$arms * 7**\$level)".

There are 7^level + 1 points in a level, numbered starting from 0. On the second and third arms the origin is omitted (so as not to repeat that point) and so just 7^level for them, giving 7^level+1 + (arms-1)*7^level = arms*7^level + 1 many points starting from 0.

N to X,Y

The position of a given N can be calculated from the base-7 digits of N from high to low. At a given digit position the state maintained is

```    direction 0 to 5, multiple of 60-degrees
plain or reverse pattern
```

It's convenient to work in the ``i,j'' coordinates per ``Triangular Calculations'' in Math::PlanePath. This represents a point in the triangular grid as i+j*w where w=1/2+I*sqrt(3)/2 the a complex sixth root of unity at +60 degrees.

```    foreach base-7 digit high to low
(i,j) = (2i-j, i+3j)   # multiply by 2+w
(i,j) += position of digit in plain or reverse,
and rotated by "direction"
```

The multiply by 2+w scales up i,j by that vector, so for instance i=1,j=0 becomes i=2,j=1. This spreads the points as per the multiple-of-7 diagram shown above, so what was at N scales up to 7*N.

The digit is then added as either the plain or reversed base figure,

```      plain             reverse
4-->5-->6
^       ^
\       \
3-->2   *             6<---*
/                  ^
v                  /
0-->1             0   5<--4
\       \
v       v
1<--2<--3
```

The arrow shown in each part is whether the state becomes plain or reverse. For example in plain state at digit=1 the arrow in segment 1 to 2 points backwards so if digit=1 then the state changes to reverse for the next digit. The direction likewise follows the direction of each segment in the pattern.

Notice the endpoint ``*'' is at at 2+w in both patterns. When considering the rotation it's convenient to reckon the direction by this endpoint.

The combination of direction and plain/reverse is a total of 14 different states, and for each there's 7 digit values (0 to 6) for a total 84 i,j.

X,Y to N

The current approach uses the "FlowsnakeCentres" code. The tiling in "Flowsnake" and "FlowsnakeCentres" is the same so the X,Y of a given N are no more than 1 away in the grid of the two forms.

The way the two lowest shapes are arranged in fact means that if the Flowsnake N is at X,Y then the same N in "FlowsnakeCentres" is at one of three locations

```    X, Y         same
X-2, Y       left      (+180 degrees)
X-1, Y-1     left down (+240 degrees)
```

This is true even when the rotated ``arms'' multiple paths (the same number of arms in both paths).

Is there an easy way to know which of the three offsets is right? The current approach is to put each through "FlowsnakeCentres" to make an N, and put that N back through Flowsnake "n_to_xy()" to see if it's the target \$n.

Rectangle to N Range

The current code calculates an exact "rect_to_n_range()" by searching for the highest and lowest Ns which are in the rectangle.

The curve at a given level is bounded by the Gosper island shape but the wiggly sides make it difficult to calculate, so a bounding radius sqrt(7)^level, plus a bit, is used. The portion of the curve comprising a given set of high digits of N can be excluded if the N point is more than that radius away from the rectangle.

When a part of the curve is excluded it prunes a whole branch of the digits tree. When the lowest digit is reached then a check for that point being actually within the rectangle is made. The radius calculation is a bit rough, and it doesn't take into account the direction of the curve, so it's a rather large over-estimate, but it works.

The same sort of search can be applied for highest and lowest N in a non-rectangular shapes, calculating a radial distance away from the shape. The distance calculation doesn't have to be exact either, it can go from something bounding the shape until the lowest digit is reached and an individual X,Y is considered as a candidate for high or low N.

N to Turn

The turn made by the curve at a point N>=1 can be calculated from the lowest non-0 digit and the plain/reverse state per the lowest non-3 above there.

```   N digits in base 7
strip low 0-digits, zcount many of them
zdigit = take next digit (lowest non-0)
strip 3-digits
tdigit = take next digit (0 if no digits left)
plain if tdigit=0,4,5, reverse if tdigit=1,2,6
zcount
---+--------+--------+--------+--------+
high    | tdigit |   3s   | zdigit |   0s   |   low
---+--------+--------+--------+--------+
if zcount=0       if zcount>=1
ie. no low 0s     ie. some low 0s
zdigit   plain reverse     plain   reverse
------   ----- -------     -----   -------
1        1      1          1        1
2        2      0          1       -1       turn left
3       -1      2         -1        1       multiple of
4       -2      1         -1        1       60 degrees
5        0     -2          1       -1
6        1     -1         -1       -1
```

For example N=9079 is base-7 ``35320'' so a single low 0 for zcount=1 and strip it to ``3532''. Take zdigit=2 leaving ``353''. Skip low 3s leaving ``35''. Take tdigit=5 which is ``plain''. So table ``plain'' with zcount>=1 is the third column and there zdigit=2 is turn=+1.

The turns in the zcount=0 ``no low 0s'' columns follow the turns of the base pattern shown above. For example zdigit=1 is as per N=1 turning 120 degrees left, so +2. For the reverse pattern the turns are negated and the zdigit value reversed, so the ``reverse'' column read 6 to 1 is the same as the plain column negated and read 1 to 6.

Low 0s are stripped because the turn at a point such as N=7 (``10'' in base-7) is determined by the pattern above it, the self-similar multiple-of-7s shape. But when there's low 0s in the way there's an adjustment to apply because the last segment of the base pattern is not in the same direction as the first, but instead at -60 degrees. Likewise the first segment of the reverse pattern. At some zdigit positions those two cancel out, such as at zdigit=1 where a plain and reverse meet, but others it's not so and hence separate table columns for with or without low 0s.

The plain or reverse pattern is determined by the lowest non-3 digit. This works because the digit=0, digit=4, and digit=5 segments of the base pattern have their sub-parts ``plain'' in all cases, both the plain and reverse forms. Conversely digit=1, digit=2 and digit=6 segments are ``reverse'' in all cases, both plain and reverse forms. But the digit=3 part is plain in plain and reverse in reverse, so it inherits the orientation of the digit above and it's therefore necessary to skip across any 3s.

When taking digits, N is treated as having infinite 0-digits at the high end. This only affects the tdigit plain/reverse step. If N has a single non-zero such as ``5000'' then it's taken as zdigit=5 and above that for the plain/reverse a tdigit=0 is then assumed. The first turn is at N=1 so there's always at least one non-0 for the zdigit.

OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

<http://oeis.org/A261180> (etc)

```    A261180   direction 0 to 5
A261185   direction mod 2
A229214   direction 1,2,3,-1,-2,-3 clockwise
A261120   count of triple-visited points in the fractal limit
A262147   \ fractions making a spiral in the fractal
A262148   /
```