Provided by: libmath-planepath-perl_129-1_all bug

NAME

       Math::PlanePath::GosperReplicate -- self-similar hexagon replications

SYNOPSIS

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

DESCRIPTION

       This is a self-similar hexagonal tiling of the plane.  At each level the shape is the
       Gosper island.

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

                                 ^
           -7 -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7

       Points are spread out on every second X coordinate to make a triangular lattice in integer
       coordinates (see "Triangular Lattice" in Math::PlanePath).

       The base pattern is the inner N=0 to N=6, then six copies of that shape are arranged
       around as the blocks N=7,14,21,28,35,42.  Then six copies of the resulting N=0 to N=48
       shape are replicated around, etc.

       Each point can be taken as a little hexagon, so that all points tile the plane with
       hexagons.  The innermost N=0 to N=6 are for instance,

                 *     *
                / \   / \
               /   \ /   \
              *     *     *
              |  3  |  2  |
              *     *     *
             / \   / \   / \
            /   \ /   \ /   \
           *     *     *     *
           |  4  |  0  |  1  |
           *     *     *     *
            \   / \   / \   /
             \ /   \ /   \ /
              *     *     *
              |  5  |  6  |
              *     *     *
               \   / \   /
                \ /   \ /
                 *     *

       The further replications are the same arrangement, but the sides become ever wigglier and
       the centres rotate around.  The rotation can be seen N=7 at X=5,Y=1 which is up from the X
       axis.

       The "FlowsnakeCentres" path is this same replicating shape, but starting from a side
       instead of the middle and traversing in such as way as to make each N adjacent.  The
       "Flowsnake" curve itself is this replication too, but segments across hexagons.

   Complex Base
       The path corresponds to expressing complex integers X+i*Y in a base

           b = 5/2 + i*sqrt(3)/2

       with coordinates scaled to put equilateral triangles on a square grid.  So for integer X,Y
       on the triangular grid (X,Y either both odd or both even),

           X/2 + i*Y*sqrt(3)/2 = a[n]*b^n + ... + a[2]*b^2 + a[1]*b + a[0]

       where each digit a[i] is either 0 or a sixth root of unity encoded into base-7 digits of
       N,

            w6 = e^(i*pi/3)            sixth root of unity, b = 2 + w6
               = 1/2 + i*sqrt(3)/2

            N digit     a[i] complex number
            -------     -------------------
              0          0
              1         w6^0 =  1
              2         w6^1 =  1/2 + i*sqrt(3)/2
              3         w6^2 = -1/2 + i*sqrt(3)/2
              4         w6^3 = -1
              5         w6^4 = -1/2 - i*sqrt(3)/2
              6         w6^5 =  1/2 - i*sqrt(3)/2

       7 digits suffice because

            norm(b) = (5/2)^2 + (sqrt(3)/2)^2 = 7

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

       The effect can be illustrated by writing N in base-7.  Part 10-16 is the same as the
       middle 0-6.  Part 20-26 has a rotation by +60 degrees.  Part 30-36 has rotation by +120
       degrees, and so on.

                                22----21
                               /     /           numbering_type => 'rotate'
                 31    36    23    20    26          N shown in base-7
                /  \     \     \        /
              32    30    35    24----25    13----12
                \        /                 /        \
                 33----34     3---- 2    14    10----11
                            /        \     \
              46----45     4     0---- 1    15----16
                      \     \
           41----40    44     5---- 6    64----63
             \        /                 /        \
              42----43    55----54    65    60    62
                         /        \     \     \  /
                       56    50    53    66    61
                            /     /
                          51----52

       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 k many base-7 digits  (empty string if k=0)
           if any 0 digit then non-boundary
           ignore high digit and all 1 digits
           if any 4 or 5 digit then non-boundary
           if any 32, 33, 66 pair then non-boundary

       A 0 digit is the middle of a block, or 4 or 5 digit the inner side of a block, for k>=1,
       hence non-boundary.  After that the 6,1,2,3 parts variously expand with rotations so that
       a 66 is enclosed on the clockwise side and 32 and 33 on the anti-clockwise side.

FUNCTIONS

       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

       "$path = Math::PlanePath::GosperReplicate->new ()"
       "$path = Math::PlanePath::GosperReplicate->new (numbering_type => $str)"
           Create and return a new path object.  The "numbering_type" parameter can be

               "fixed"        (default)
               "rotate"

       "($x,$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.

   Level Methods
       "($n_lo, $n_hi) = $path->level_to_n_range($level)"
           Return "(0, 7**$level - 1)".

FORMULAS

   Axis Rotations
       In the fixed numbering, digit positions 1,2,3,4,5,6 go around +60deg each, so the N for
       rotation of X,Y by +60 degrees is each digit +1.

           N          = 0, 1, 2, 3, 4, 5, 6, 10, 11, 12

           rot+60(N)  = 0, 2, 3, 4, 5, 6, 1, 14, 16, 17, ... decimal
                      = 0, 2, 3, 4, 5, 6, 1, 20, 22, 23, ... base7

           rot+120(N) = 0, 3, 4, 5, 6, 1, 2, 21, 24, 25, ... decimal
                      = 0, 3, 4, 5, 6, 1, 2, 30, 33, 34, ... base7

           etc

       In the rotate numbering, just adding +1 (etc) at the high digit alone is rotation.

   X,Y Extents
       The maximum X in a given level N=0 to 7^k-1 can be calculated from the replications.  A
       given high digit 1 to 6 has sub-parts located at b^k*w6^(d-1).  Those sub-parts are all
       the same, so the one with maximum real(b^k*w6^(d-1)) contains the maximum X.

           N_xmax_digit(j) = d=1to6 where real(w6^(d-1) * b^j) is maximum
                           = 1,1,6,6,6,5,5,5,4,4,4,3,3,3,3,2,2, ...

                        k-1
           N_xmax(k) = digits N_xmax_digit(j)    low digit j=0
                        j=0
                     = 0, 1, 8, 302, 2360, 16766, 100801, ...  decimal
                     = 0, 1, 11, 611, 6611, 66611, 566611, ...  base7

                       k-1
           z_xmax(k) = sum  w6^d[j] * b^j
                       j=0      each d[j] with real(w6^d[j] * b^j) maximum
                 = 0, 1, 7/2+1/2*sqrt3*i, 10-sqrt3*i, 57/2-3/2*sqrt3*i,...

           xmax(k) = 2*real(z_xmax(k))
                   = 0, 2, 7, 20, 57, 151, 387, 1070, 2833, 7106, ...

       For computer calculation these maximums can be calculated from the powers.  The parts
       resulting can also be written in terms of the angle

           arg(b) = atan(sqrt(3)/5) = 19.106... degrees

       For successive k, if adding this pushes the b^k angle past +30deg then the preceding digit
       goes past -30deg and becomes the new maximum X.  Write the angle as a fraction of 60deg
       (pi/3),

           F = atan(sqrt(3)/5) / (pi/3)  = 0.318443 ...

       This is irrational since b^k is never on the X or Y axes.  That can be seen since
       2/sqrt3*imag(b^k) mod 7 goes in a repeating pattern 1,5,4,6,2,3.  Similarly 2*real(b^k)
       mod 7 so not on the Y axis, and also anything on the Y axis would have 3*k fall on the X
       axis.

       Digits low to high are successive steps back cyclically 6,5,4,3,2,1 so that (with mod
       giving 0 to 5),

           N_xmax_digit(j) = (-floor(F*j+1/2) mod 6) + 1

       The +1/2 is since initial direction b^0=1 is angle 0 which is half way between -30 and +30
       deg.

       Similarly for the location, using conj(w6) for rotation back

           z_xmax_exp(j) = floor(F*j+1/2)
                         = 0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5, ...
           z_xmax(k) = sum(j=0,k-1, conj(w6)^z_xmax_exp(j) * b^j)

       By symmetry the maximum extent is the same in 60deg, 120deg, etc directions, suitably
       rotated.  The N in those cases has the digits 1,2,3,4,5,6 cycled around for the rotation.
       In PlanePath triangular X,Y coordinates direction 60deg means when sum X+3*Y is a maximum,
       etc.

       If the +1/2 in the floor is omitted then the effect is to find the maximum point in
       direction +30deg.  In the PlanePath coordinates this means maximum sum S = X+Y.

           N_smax_digit(j) = (-floor(F*j) mod 6) + 1
                           = 1,1,1,1,6,6,6,5,5,5,4,4,4,3,3, ...

                        k-1
           N_smax(k) = digits N_smax_digit(j)    low digit j=0
                        j=0
                     = 0, 1, 8, 57, 400, 14806, 115648, ...     decimal
                     = 0, 1, 11, 111, 1111, 61111, 661111, ...  base7
           and also N_smax() + 1

           z_smax_exp(j) = floor(F*j)
                         = 0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6, ...
           z_smax(k) = sum(j=0,k-1, conj(w6)^z_smax_exp(j) * b^j)
                     = 0, 1, 7/2+1/2*sqrt3*i, 9+3*sqrt3*i, 19+12*sqrt3*i, ...
           and also z_smax() + w6^2

           smax(k) = 2*real(z_smax(k)) + imag(z_smax(k))*2/sqrt3
                   = 0, 2, 8, 24, 62, 172, 470, 1190, 3202, 8740, ...
                     coordinate sum X+Y max

       In the base figure, points 1 and 2 have the same X+Y=2 and this remains so in subsequent
       levels, so that for k>=1 N_smax(k) and N_smax(k)+1 are equal maximums.

SEE ALSO

       Math::PlanePath, Math::PlanePath::GosperIslands, Math::PlanePath::Flowsnake,
       Math::PlanePath::FlowsnakeCentres, Math::PlanePath::QuintetReplicate,
       Math::PlanePath::ComplexPlus

HOME PAGE

       <http://user42.tuxfamily.org/math-planepath/index.html>

LICENSE

       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde

       This file is part of Math-PlanePath.

       Math-PlanePath is free software; you can redistribute it and/or modify it under the terms
       of the GNU General Public License as published by the Free Software Foundation; either
       version 3, or (at your option) any later version.

       Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY
       WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
       PURPOSE.  See the GNU General Public License for more details.

       You should have received a copy of the GNU General Public License along with Math-
       PlanePath.  If not, see <http://www.gnu.org/licenses/>.