Radix

The "radix" parameter controls the radix used to break N into X,Y. For example "radix => 4" gives 4x4 blocks, with radix-1 replications of the preceding level at each stage.

     radix => 4  
     60 61 62 63 44 45 46 47 28 29 30 31 12 13 14 15      3
     56 57 58 59 40 41 42 43 24 25 26 27  8  9 10 11      2
     52 53 54 55 36 37 38 39 20 21 22 23  4  5  6  7      1
     48 49 50 51 32 33 34 35 16 17 18 19  0  1  2  3  <- Y=0
    --------------------------------------^-----------
    -12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3

Notice for X negative the parts replicate successively towards -infinity, so the block N=16 to N=31 is first at X=-4, then N=32 at X=-8, N=48 at X=-12, and N=64 at X=-16 (not shown).

Digit Order

The "digit_order" parameter controls the order digits from N are applied to X and Y. The default above is "XYX" so the replications go X then Y then negative X.

"XXY" goes to negative X before Y, so N=2,N=3 goes to negative X before repeating N=4 to N=7 in the Y direction.

    digit_order => "XXY"
    38  39  36  37  46  47  44  45
    34  35  32  33  42  43  40  41
     6   7   4   5  14  15  12  13
     2   3   0   1  10  11   8   9
    ---------^--------------------
    -2  -1  X=0  1   2   3   4   5

The further options are as follows, for six permutations of each 3 digits from N.

    digit_order => "YXX"               digit_order => "XnYX"   
    38 39 36 37 46 47 44 45            19 23 18 22 51 55 50 54
    34 35 32 33 42 43 40 41            17 21 16 20 49 53 48 52
     6  7  4  5 14 15 12 13             3  7  2  6 35 39 34 38
     2  3  0  1 10 11  8  9             1  5  0  4 33 37 32 36
    digit_order => "XnXY"              digit_order => "YXnX"   
    37 39 36 38 53 55 52 54            11 15  9 13 43 47 41 45
    33 35 32 34 49 51 48 50            10 14  8 12 42 46 40 44
     5  7  4  6 21 23 20 22             3  7  1  5 35 39 33 37
     1  3  0  2 17 19 16 18             2  6  0  4 34 38 32 36

"Xn" means the X negative direction. It's still spaced 2 apart (or whatever radix), so the result is not simply a -X,Y.

Axis Values

N=0,1,4,5,8,9,etc on the X axis (positive and negative) are those integers with a 0 at every third bit starting from the second least significant bit. This is simply demanding that the bits going to the Y coordinate must be 0.

    X axis Ns = binary ...__0__0__0_     with _ either 0 or 1
    in octal, digits 0,1,4,5 only

N=0,1,8,9,etc on the X positive axis have the highest 1-bit in the first slot of a 3-bit group. Or N=0,4,5,etc on the X negative axis have the high 1 bit in the third slot,

    X pos Ns = binary    1_0__0__0...0__0__0_
    X neg Ns = binary  10__0__0__0...0__0__0_
                       ^^^
                       three bit group
    X pos Ns in octal have high octal digit 1
    X neg Ns in octal high octal digit 4 or 5

N=0,2,16,18,etc on the Y axis are conversely those integers with a 0 in two of each three bits, demanding the bits going to the X coordinate must be 0.

    Y axis Ns = binary ..._00_00_00_0    with _ either 0 or 1
    in octal has digits 0,2 only

For a radix other than binary the pattern is the same. Each "_" is any digit of the given radix, and each 0 must be 0. The high 1 bit for X positive and negative become a high non-zero digit.

Level Ranges

Because the X direction replicates twice for each once in the Y direction the width grows at twice the rate, so after each 3 replications

    width = height*height

For this reason N values for a given Y grow quite rapidly.

Proth Numbers

The Proth numbers, k*2^n+1 for k<2^n, fall in columns on the path.

    *                               *                               *
    *                               *                               *
    *                               *                               *
    *               *               *               *               *
    *               *               *               *               *
                            *       *       *       *
    *       *       *       *       *       *       *       *       *
                            *   *   *   *   *       *
                                    *
    *       *       *       *   * *   * *   *       *       *       *
    -----------------------------------------------------------------
    -31    -23     -15     -7  -3-1 0 3 5   9      17       25     33

The height of the column is from the zeros in X ending binary ...1000..0001 since this limits the "k" part of the Proth numbers which can have N ending suitably. Or for X negative ending ...10111...11.

Level Methods

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

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