Row Ranges

The number of points in each row is always a power of 2. The power is the count of 1-bits in Y. (This count is sometimes called Gould's sequence.)

    rowpoints(Y) = 2^count_1_bits(Y)

For example Y=13 is binary 1101 which has three 1-bits so in row Y=13 there are 2^3=8 points.

Because the first point is N=0, the N at the left of each row is the cumulative count of preceding points,

    Ndepth(Y) = rowpoints(0) + ... + rowpoints(Y-1)

Since the powers of 2 are always even except for 2^0=1 in row Y=0, this Ndepth(Y) total is always odd. The self-similar nature of the triangle means the same is true of the sub-triangles, for example odd N=31,35,41,47,etc on the left of the triangle at X=8,Y=8. This means in particular the primes (being odd) fall predominately on the left side of the triangles and sub-triangles.

Replication Sizes

Counting the single point N=0 as level=0, then N=0,1,2 as level 1, each replication level goes from

    Nstart = 0
    Nlevel = 3^level - 1     inclusive

For example level 2 is from N=0 to N=3^2-1=8. Each level doubles in size,

               0  <= Y <= 2^level - 1
    - 2^level + 1 <= X <= 2^level - 1

Align Right

Optional "align=>"right"" puts points to the right of the Y axis, packed next to each other and so using an eighth of the plane.

    align => "right"
      7  | 19 20 21 22 23 24 25 26 
      6  | 15    16    17    18    
      5  | 11 12       13 14       
      4  |  9          10          
      3  |  5  6  7  8             
      2  |  3     4                
      1  |  1  2                   
    Y=0  |  0                      
         +-------------------------
          X=0  1  2  3  4  5  6  7

Align Left

Optional "align=>"left"" puts points to the left of the Y axis, ie. into negative X. The rows are still numbered starting from the left, so it's a shift across, not a negate of X.

    align => "left"
    19 20 21 22 23 24 25 26  |     7
       15    16    17    18  |     6
          11 12       13 14  |     5
              9          10  |     4
                 5  6  7  8  |     3
                    3     4  |     2
                       1  2  |     1
                          0  | <- Y=0
    -------------------------+
    -7 -6 -5 -4 -3 -2 -1 X=0

Align Diagonal

Optional "align=>"diagonal"" puts rows on diagonals down from the Y axis to the X axis. This uses the whole of the first quadrant, with gaps according to the pattern.

    align => "diagonal"
     15 | 65       ...
     14 | 57 66
     13 | 49    67
     12 | 45 50 58 68
     11 | 37          69
     10 | 33 38       59 70
      9 | 29    39    51    71
      8 | 27 30 34 40 46 52 60 72
      7 | 19                      73
      6 | 15 20                   61 74
      5 | 11    21                53    75
      4 |  9 12 16 22             47 54 62 76
      3 |  5          23          41          77       ...
      2 |  3  6       17 24       35 42       63 78
      1 |  1     7    13    25    31    43    55    79
    Y=0 |  0  2  4  8 10 14 18 26 28 32 36 44 48 56 64 80
        +-------------------------------------------------
         X=0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15

This form visits all points X,Y where X and Y written in binary have no 1-bits in the same bit positions, ie. where X bitand Y == 0. For example X=13,Y=3 is not visited because 13="1011" and 6="0110" both have bit "0010" set.

This bit-and rule is an easy way to test for visited or not visited cells of the pattern. The visited cells can be calculated by this diagonal X,Y bitand, but then plotted X,X+Y for the "right" align or X-Y,X+Y for "triangular".

Cellular Automaton

The triangle arises in Stephen Wolfram's 1-D cellular automatons (per Math::PlanePath::CellularRule and Cellular::Automata::Wolfram).

    align           rule
    -----           ----
    "triangular"    18,26,82,90,146,154,210,218
    "right"         60
    "left"          102

In each row the rule 18 etc pattern turns a cell "on" in the next row if one but not both its diagonal predecessors are "on". This is a mod 2 sum giving Pascal's triangle mod 2.

Some other cellular rules are variations on the triangle,

• Rule 22 is "triangular" but filling the gap between leaf points such as N=5 and N=6.
• Rule 126 adds an extra point on the inward side of each visited.
• Rule 182 fills in the big gaps leaving just a single-cell empty border delimiting them.

N Start

The default is to number points starting N=0 as shown above. An optional "n_start" parameter can give a different start, with the same shape. For example starting at 1, which is the numbering of "CellularRule" rule=60,

    n_start => 1
    20    21    22    23    24    25    26    27
       16          17          18          19
          12    13                14    15
             10                      11
                 6     7     8     9
                    4           5
                       2     3
                          1

Descriptive Methods

"$n = $path->n_start()"

Return the first N in the path. This is 0 by default, or the given "n_start" parameter.

Tree Methods

"@n_children = $path->tree_n_children($n)"

Return the children of $n, or an empty list if "$n < n_start" (ie. before the start of the path).

The children are the points diagonally up left and right on the next row (Y+1). There can be 0, 1 or 2 such points. At even depth there's 2, on depth=1mod4 there's 1. On depth=3mod4 there's some 0s and some 1s. See "N to Number of Children" below.

For example N=3 has two children N=5,N=6. Then in turn N=5 has just one child N=9 and N=6 has no children. The way points are numbered across a row means that when there's two children they're consecutive N values.

"$n_parent = $path->tree_n_parent($n)"

Return the parent node of $n, or "undef" if "$n <= n_start" (the top of the triangle).

"$depth = $path->tree_n_to_depth($n)"

Return the depth of node $n, or "undef" if there's no point $n. In the "triangular", "right" and "left" alignments this is the same as the Y coordinate from "n_to_xy()". In the "diagonal" alignment it's X+Y.

"$n = $path->tree_depth_to_n($depth)"

"$n = $path->tree_depth_to_n_end($depth)"

Return the first or last N at tree level $depth. The start of the tree is depth=0 at the origin X=0,Y=0.

This is the N at the left end of each row. So in the default triangular alignment it's the same as "xy_to_n(-$depth,$depth)".

Level Methods

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

Return "(0, 3**$level - 1)".

X,Y to N

For calculation it's convenient to turn the X,Y coordinates into the "right" alignment style, so that Y is the depth and X is in the range 0<=X<=Y.

The starting position of each row of the triangle is given by turning 1-bits of the depth into powers-of-3.

    Y = depth = 2^a + 2^b + 2^c + 2^d ...       a>b>c>d...
    Ndepth = first N at this depth
           =         3^a
             +   2 * 3^b
             + 2^2 * 3^c
             + 2^3 * 3^d
             + ...

For example depth=6=2^2+2^1 starts at Ndepth=3^2+2*3^1=15. The powers-of-3 are the three parts of the triangle replication. The power-of-2 doubling is the doubling of the row Y when replicated.

Then the bits of X at the positions of the 1-bits of the depth become an N offset into the row.

               a  b  c  d
    depth    = 10010010010     binary
    X        = m00n00p00q0
    Noffset  =        mnpq     binary
    N = Ndepth + Noffset

For example in depth=6 binary "110" then at X=4="100" take the bits of X where depth has 1-bits, which is X="10_" so Noffset="10" binary and N=15+2=17, as per the "right" table above at X=4,Y=6.

If X has any 1-bits which are a 0-bits in the depth depth then that X,Y is not visited. For example if depth=6="110" then X=3="11" is not visited because the low bit X="__1" has depth="__0" at that position.

N to Depth

The row containing N can be found by working down the Ndepth formula shown above. The "a" term is the highest 3^a <= N, thus giving a bit 2^a for the depth. Then for the remaining Nrem = N - 3^a find the highest "b" where 2*3^b <= Nrem. And so on until reaching an Nrem which is too small to subtract any more terms.

It's convenient to go by bits high to low of the prospective depth, deciding at each bit whether Nrem is big enough to give the depth a 1-bit there, or whether it must be a 0-bit.

    a = floor(log3(N))     round down to power-of-3
    pow = 3^a
    Nrem = N - pow
    depth = high 1-bit at bit position "a" (counting from 0)
    factor = 2
    loop bitpos a-1 down to 0
      pow /= 3
      if pow*factor <= Nrem
      then depth 0-bit, factor *= 2
      else depth 1-bit
    factor is 2^count1bits(depth)
    Noffset = Nrem     offset into row
    0 <= Noffset < factor

N to X,Y

N is turned into depth and Noffset as per above. X in "right" alignment style is formed by spreading the bits of Noffset out according to the 1-bits of the depth.

    depth   = 100110  binary
    Noffset =    abc  binary
    Xright  = a00bc0

For example in depth=5 this spreads an Noffset=0to3 to make X=000, 001, 100, 101 in binary and in "right" alignment style.

From an X,Y in "right" alignment the other alignments are formed

    alignment   from "right" X,Y
    ---------   ----------------
    triangular     2*X-Y, Y       so -Y <= X < Y
    right          X,     Y       unchanged
    left           X-Y,   Y       so -Y <= X <= 0
    diagonal       X,   Y-X       downwards sloping

N to Number of Children

The number of children follows a pattern based on the depth.

    depth      number of children
    -----      ------------------
     12    2       2       2       2   
     11     1 0 0 1         1 0 0 1
     10      2   2           2   2
      9       1 1             1 1
      8        2               2
      7         1 0 0 0 0 0 0 1   
      6          2   2   2   2 
      5           1 1     1 1  
      4            2       2   
      3             1 0 0 1   
      2              2   2
      1               1 1
      0                2

If depth is even then all points have 2 children. For example row depth=6 has 4 points and all have 2 children each.

At odd depth the number of children is either 1 or 0 according to the Noffset position in the row masked down by the trailing 1-bits of the depth.

    depth  = ...011111 in binary, its trailing 1s
    Noffset = ...00000   \ num children = 1
            = ...11111   /
            = ...other   num children = 0

For example depth=11 is binary "1011" which has low 1-bits "11". If those two low bits of Noffset are "00" or "11" then 1 child. Any other bit pattern in Noffset ("01" or "10" in this case) is 0 children. Hence the pattern 1,0,0,1,1,0,0,1 reading across the depth=11 row.

In general when the depth doubles the triangle is replicated twice and the number of children is carried with the replications, except the middle two points are 0 children. For example the triangle of depth=0to3 is repeated twice to make depth=4to7, but the depth=7 row is not children 10011001 of a plain doubling from the depth=3 row, but instead 10000001 which is the middle two points becoming 0.

N to Number of Siblings

Each node N has either 0 or 1 siblings. This is determined by depth,

    depth      number of siblings
    -----      ------------------
      4            0       0   
      3             1 1 1 1   
      2              0   0
      1               1 1
      0                0   
    depth     number of siblings
    -----     ------------------
     odd             1
     even            0

In an even row the points are all spread apart so there are no siblings. The points in such a row are cousins or second cousins, etc, but none share a parent.

In an odd row each parent node (an even row) has 2 children and so each of those points has 1 sibling.

The effect is to conflate the NumChildren=1 and NumChildren=0 cases in the children picture above, those two becoming a single sibling.

    num children of N      num siblings of N
    -----------------      -----------------
          0 or 1                   1
            2                      0

Rectangle to N Range

An easy range can be had just from the Y range by noting the diagonals X=Y and X=-Y are always visited, so just take the Ndepth of Ymin and Nend of Ymax,

    # align="triangular"
    Nmin = N at X=-Ymin,Y=Ymin
    Nmax = N at X=Ymax,Y=Ymax

Or in "right" style the left end is at X=0 instead,

    # align="right"
    Nmin = N at X=0,Ymin
    Nmax = N at Ymax,Ymax

For less work but a bigger over-estimate, invert the Nlevel formulas given in "Row Ranges" above to round up to the end of a depth=2^k replication,

    level = floor(log2(Ymax)) + 1
    Nmax = 3^level - 1

For example Y=11, level=floor(log2(11))+1=4, so Nmax=3^4-1=80, which is the end of the Y=15 row, ie. rounded up to the top of the replication block Y=8 to Y=15.