Complex Base

This tiling corresponds to expressing a complex integer X+i*Y as

    base b=2+i
    X+Yi = a[n]*b^n + ... + a[2]*b^2 + a[1]*b + a[0]

where each digit position factor a[i] corresponds to N digits

    N digit     a[i]
    -------    ------
       0          0
       1          1
       2          i
       3         -1
       4         -i

The base b is at an angle arg(b) = atan(1/2) = 26.56 degrees as seen at N=5 above. Successive powers b^2, b^3, b^4 etc at N=5^level rotate around by that much each time.

    Npow = 5^level  at b^level
    angle(Npow) = level*26.56 degrees
    radius(Npow) = sqrt(5) ^ level

The path can be reckoned bottom-up as a new low digit of N expanding each unit square to the base "+" shape.

                             +---C      
    D-------C                | 2 |      
    |       |            D---+---+---+  
    |       |     =>     | 3 | 0 | 1 |  
    |       |            +---+---+---B  
    A-------B                | 4 |      
                             A---+

Side A-B becomes a 3-segment S. Such an expansion is the same as the TerdragonCurve or GosperSide, but here turns of 90 degrees. Like GosperSide there is no touching or overlap of the sides expansions, so boundary length 4*3^level.

Rotate Numbering

Parameter "numbering_type => 'rotate'" applies a rotation to the numbering in each sub-part according to its location around the preceding level.

The effect can be illustrated by writing N in base-5. Part 10-14 is the same as the middle 0-4. Part 20-24 has a rotation by +90 degrees. Part 30-34 has rotation by +180 degrees, and part 40-44 by +270 degrees.

            21
          /  |                   
        22  20  24      12           numbering_type => 'rotate' 
          \    /      /    \             N shown in base-5
            23   2  13  10--11
               /   \   \
        34   3   0-- 1  14
           \   \  
    31--30  33   4  41
      \    /       /   \
        32      43  40  42
                     | /
                    41

Notice this means in each part the 11, 21, 31, etc, points are directed away from the middle in the same way, relative to the sub-part locations.

Working through the expansions gives the following rule for when an N is on the boundary of level k,

    write N in base-5 digits  (empty string if k=0)
    if length < k then non-boundary
    ignore high digit and all 1 digits
    if any pair 32, 33, 44 then non-boundary

A 0 digit is the middle of a block, so always non-boundary. After that the 4,1,2,3 parts variously expand with rotations so that a 44 is enclosed on the clockwise side and 32 and 33 on the anti-clockwise side.

Level Methods

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

Return "(0, 5**$level - 1)".

Axis Rotations

The digits positions 1,2,3,4 go around +90deg each, so the N for rotation by +90 is each digit +1, cycling around.

    rot+90(N) = 0, 2, 3, 4, 1, 10, 12, 13, 14, 11, 15, ... decimal
              = 0, 2, 3, 4, 1, 20, 22, 23, 24, 21, 30, ... base5
    rot-90(N) = 0, 4, 1, 2, 3, 20, 24, 21, 22, 23,  5, ... decimal
              = 0, 4, 1, 2, 3, 40, 44, 41, 42, 43, 10, ... base5
    rot180(N) = 0, 3, 4, 1, 2, 15, 18, 19, 16, 17, 20, ... decimal
              = 0, 3, 4, 1, 2, 30, 33, 34, 31, 32, 40, ... base5

X,Y Extents

The maximum X in a given level N=0 to 5^k-1 can be calculated from the replications. A given high digit 1 to 4 has sub-parts located at b^k*i^(d-1). Those sub-parts are all the same, so the one with maximum real(b^k*i^(d-1)) contains the maximum X.

    N_xmax_digit(j) = d=1,2,3,4 where real(i^(d-1) * b^j) is maximum
                    = 1,1,4,4,4,4,3,3,3,2,2,2,1,1, ...
                 k-1
    N_xmax(k) = digits N_xmax_digit(j)    low digit j=0
                 j=0
              = 0, 1, 6, 106, 606, 3106, 15606, ...    decimal
              = 0, 1, 11, 411, 4411, 44411, 444411, ...  base5
                k-1
    z_xmax(k) = sum  i^d[j] * b^j
                j=0      each d[j] with real(i^d[j] * b^j) maximum
              = 0, 1, 3+i, 7-2*i, 18-4*i, 42+3*i, 83+41*i, ...
    xmax(k) = real(z_xmax(k))
            = 0, 1, 3, 7, 18, 42, 83, 200, 478, 1005, ...

For computer calculation these maximums can be calculated by the powers. The digit parts can also be written in terms of the angle arg(b) = atan(1/2). For successive k, if adding atan(1/2) pushes the b^k angle past +45deg then the preceding digit goes past -45deg and becomes the new maximum X. Write the angle as a fraction of 90deg (pi/2),

    F = atan(1/2) / (pi/2)  = 0.295167 ...

This is irrational since b^k is never on the X or Y axes. That can be seen since imag(b^k) mod 5 == 1 if k odd and == 4 if k even >= 2. Similarly real(b^k) mod 5 == 2,3 so not on the Y axis, or also anything on the Y axis would have 3*k fall on the X axis.

Digits low to high successively step back in a cycle 4,3,2,1 so that (with mod giving 0 to 3),

    N_xmax_digit(j) = (-floor(F*j+1/2) mod 4) + 1

The +1/2 is since initial direction b^0=1 is angle 0 which is half way between -45 and +45 deg.

Similarly the X,Y location, using -i for rotation back

    z_xmax_exp(j) = floor(F*j+1/2)
                  = 0,0,1,1,1,1,2,2,2,3,3,3,4,4,4,4,5,5, ...
    z_xmax(k) = sum(j=0,k-1, (-i)^z_xmax_exp(j) * b^j)

By symmetry the maximum extent is the same for Y vertically and for X or Y negative, suitably rotated. The N in those cases has the digits 1,2,3,4 cycled around as per "Rotations" above.

If the +1/2 in the floor is omitted then the effect is to find the maximum point in direction +45deg, so the point(s) with maximum sum S = X+Y.

    N_smax_digit(j) = (-floor(F*j) mod 4) + 1
                    = 1,1,1,1,4,4,4,3,3,3,3,2,2,2,1, ...
                 k-1
    N_smax(k) = digits N_smax_digit(j)    low digit j=0
                 j=0
              = 0, 1, 6, 31, 156, 2656, 15156, ...     decimal
              = 0, 1, 11, 111, 1111, 41111, 441111, ...  base5
    and also N_smax() + 1
    z_smax_exp(j) = floor(F*j)
                  = 0,0,0,0,1,1,1,2,2,2,2,3,3,3,4,4,4,5,5,5, ...
    z_smax(k) = sum(j=0,k-1, (-i)^z_smax_exp(j) * b^j)
              = 0, 1, 3+i, 6+5*i, 8+16*i, 32+23*i, 73+61*i, ...
    and also z_smax() + 1+i
    smax(k) = real(z_smax(k)) + imag(z_smax(k))
            = 0, 1, 4, 11, 24, 55, 134, 295, 602, 1465, ...

In the base figure points 1 and 2 are both on the same 45deg line and this remains so in subsequent levels, so that for k>=1 N_smax(k) and N_smax(k)+1 are equal maximums.