Level Ranges¶

Each replication adds 3 copies of the existing points and is thus 4 times bigger, so if N=0 to N=4 is reckoned as level 1 then a given replication level goes from

    Nstart = 0
    Nlevel = 4^level   (inclusive)

Each replication is 3 times the width. The initial N=0 to N=4 figure is 6 wide and in general a level runs from

    Xstart = 0
    Xlevel = 2*3^level   at N=Nlevel

The highest Y is 3 times greater at each level similarly. The peak is at the midpoint of each level,

    Npeak = (4^level)/2
    Ypeak = 3^level
    Xpeak = 3^level

It can be seen that the N=6 point backtracks horizontally to the same X as the start of its section N=4 to N=8. This happens in the further replications too and is the maximum extent of the backtracking.

The Nlevel is multiplied by 4 to get the end of the next higher level. The same 4*N can be applied to all points N=0 to N=Nlevel to get the same shape but a factor of 3 bigger X,Y coordinates. The in-between points 4*N+1, 4*N+2 and 4*N+3 are then new finer structure in the higher level.

Fractal¶

Koch conceived the curve as having a fixed length and infinitely fine structure, making it continuous everywhere but differentiable nowhere. The code here can be pressed into use for that sort of construction for a given level of granularity by scaling

    X/3^level
    Y/3^level

which makes it a fixed 2 wide by 1 high. Or for unit-side equilateral triangles then apply further factors 1/2 and sqrt(3)/2, as noted in "Triangular Lattice" in Math::PlanePath.

    (X/2) / 3^level
    (Y*sqrt(3)/2) / 3^level

Area¶

The area under the curve to a given level can be calculated from its self-similar nature. The curve at level+1 is 3 times wider and higher and adds a triangle of unit area onto each line segment. So reckoning the line segment N=0 to N=1 as level=0 (which is area[0]=0),

    area[level] = 9*area[level-1] + 4^(level-1)
                = 4^(level-1) + 9*4^(level-2) + ... + 9^(level-1)*4^0
                  9^level - 4^level
                = -----------------
                          5
                = 0, 1, 13, 133, 1261, 11605, 105469, ...  (A016153)

The sides are 6 different angles. The triangles added on the sides are always the same shape either pointing up or down. Base width=2 and height=1 gives area=1.

       *            *-----*   ^
      / \            \   /    | height=1
     /   \            \ /     |
    *-----*            *      v     triangle area = 2*1/2 = 1
    <-----> width=2

If the Y coordinates are stretched to make equilateral triangles then the number of triangles is not changed and so the area increases by a factor of the area of the equilateral triangle, sqrt(3)/4.

Level Methods¶

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

Return "(0, 4**$level)".

N to Turn¶

The curve always turns either +60 degrees or -120 degrees, it never goes straight ahead. In the base 4 representation of N, the lowest non-zero digit gives the turn. The first turn is at N=1 so there's always a non-zero digit in N.

   low digit
    base 4         turn
   ---------   ------------
      1         +60 degrees (left)
      2        -120 degrees (right)
      3         +60 degrees (left)

For example N=8 is 20 base 4, so lowest nonzero "2" means turn -120 degrees for the next segment.

If the least significant digit is non-zero then it determines the turn, making the base N=0 to N=4 shape. If the least significant is zero then the next level up is in control, eg. N=0,4,8,12,16, making a turn according to the base shape again at that higher level. The first and last segments of the base shape are "straight" so there's no extra adjustment to apply in those higher digits.

This base 4 digit rule is equivalent to counting low 0-bits. A low base-4 digit 1 or 3 is an even number of low 0-bits and a low digit 2 is an odd number of low 0-bits.

    count low 0-bits         turn
    ----------------     ------------
         even             +60 degrees (left)
         odd             -120 degrees (right)

For example N=8 in binary "1000" has 3 low 0-bits and 3 is odd so turn -120 degrees (right).

See "Turn" in Math::PlanePath::GrayCode for a similar turn sequence arising from binary Gray code.

N to Next Turn¶

The turn at N+1, ie the next turn, can be found from the base-4 digits by considering how the digits of N change when 1 is added, and the low-digit turn calculation is applied on those changed digits.

Adding 1 means low digit 0, 1 or 2 will become non-zero. Any low 3s wrap around to become low 0s. So the turn at N+1 can be found from the digits of N by seeking the lowest non-3

   lowest non-3       turn
    digit of N       at N+1
   ------------   ------------
        0          +60 degrees (left)
        1         -120 degrees (right)
        2          +60 degrees (left)

N to Direction¶

The total turn at a given N can be found by counting digits 1 and 2 in base 4.

    direction = ((count of 1-digits in base 4)
                 - (count of 2-digits in base 4)) * 60 degrees

For example N=11 is "23" in base 4, so 60*(0-1) = -60 degrees.

In this formula the count of 1s and 2s can go past 360 degrees, representing a spiralling around which occurs at progressively higher replication levels. The direction can be taken mod 360 degrees, or the count mod 6, for a direction 0 to 5 if desired.

N to abs(dX),abs(dY)¶

The direction expressed as abs(dX) and abs(dY) can be calculated simply from N modulo 3. abs(dX) is a repeating pattern 2,1,1 and abs(dY) repeating 0,1,1.

    N mod 3     abs(dX),abs(dY)
    -------     ---------------
       0             2,0            horizontal, East or West
       1             1,1            slope North-East or South-West
       2             1,1            slope North-West or South-East

This works because the direction calculation above corresponds to N mod 3. Each N digit in base 4 becomes

    N digit
    base 4    direction add
    -------   -------------
       0            0
       1            1
       2           -1
       3            0

Notice that direction == Ndigit mod 3. Then because 4==1 mod 3 the power-of-4 for each digit reduces down to 1,

    N = 4^k * digit_k + ... 4^0 * digit_0
    N mod 3 = 1 * digit_k + ... 1 * digit_0
            = digit_k + ... digit_0
    same as
    direction = digit_k + ... + digit_0    taken mod 3

Rectangle to N Range -- Level¶

An easy over-estimate of the N values in a rectangle can be had from the Xlevel formula above. If Xlevel>rectangleX then Nlevel is past the rectangle extent.

    X = 2*3^level

so

    floorlevel = floor log_base_3(X/2)
    Nhi = 4^(floorlevel+1) - 1

For example a rectangle extending to X=13 has floorlevel = floor(log3(13/2))=1 and so Nhi=4^(1+1)-1=15.

The rounding-down of the log3 ensures a point such as X=18 which is the first in the next Nlevel will give that next level. So floorlevel=log3(18/2)=2 (exactly) and Nhi=4^(2+1)-1=63.

The worst case for this over-estimate is when rectangleX==Xlevel, ie. the rectangle is just into the next level. In that case Nhi is nearly a factor 4 bigger than it needs to be.

Rectangle to N Range -- Exact¶

The exact Nlo and Nhi in a rectangle can be found by searching along the curve. For Nlo search forward from the origin N=0. For Nhi search backward from the Nlevel over-estimate described above.

At a given digit position in the prospective N the sub-part of the curve comprising the lower digits has a certain triangular extent. If it's outside the target rectangle then step to the next digit value, and to the next of the digit above when past digit=3 (or below digit=0 when searching backwards).

There's six possible orientations for the curve sub-part. In the following pictures "o" is the start and the surrounding lines show the triangular extent. There's just four curve parts shown in each, but these triangles bound a sub-curve of any level.

   rot=0   -+-               +-----------------+
         --   --              - .-+-*   *-+-o -
       --   *   --             --    \ /    --
     --    / \    --             --   *   --
    - o-+-*   *-+-. -              --   --
   +-----------------+       rot=3   -+-
   rot=1
   +---------+               rot=4    /+
   |      . /                        / |
   |     / /                        / o|
   |*-+-* /                        / / |
   | \   /                        / *  |
   |  * /                        /   \ |
   | / /                        / *-+-*|
   |o /                        / /     |
   | /                        / .      |
   +/                        +---------+
   +\  rot=2                 +---------+
   | \                        \ o      |
   |. \                        \ \     |
   | \ \                        \ *-+-*|
   |  * \                        \   / |
   | /   \                        \ *  |
   |*-+-* \                        \ \ |
   |     \ \                        \ .|
   |      o \                rot=5   \ |
   +---------+                        \+

The "." is the start of the next sub-curve. It belongs to the next digit value and so can be excluded. For rot=0 and rot=3 this means simply shortening the X range permitted. For rot=1 and rot=4 similarly the Y range. For rot=2 and rot=5 it would require a separate test.

Tight sub-part extent checking reduces the sub-parts which are examined, but it works perfectly well with a looser check, such as a square box for the sub-curve extents. Doing that might be easier if the target region is not a rectangle but instead some trickier shape.