Provided by: libmath-planepath-perl_125-1_all bug

NAME

       Math::PlanePath::LTiling -- 2x2 self-similar of four pattern parts

SYNOPSIS

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

DESCRIPTION

       This is a self-similar tiling by "L" shapes.  A base "L" is replicated four times with end
       parts turned +90 and -90 degrees to make a larger L,

                                               +-----+-----+
                                               |12   |   15|
                                               |  +--+--+  |
                                               |  |14   |  |
                                               +--+  +--+--+
                                               |  |  |11   |
                                               |  +--+  +--+
                                               |13   |  |  |
                          +-----+              +-----+--+  +--+--+-----+
                          | 3   |              | 3   |  |10   |  |    5|
                          |  +--+        -->   |  +--+  +--+--+  +--+  |
                          |  |  |              |  |  | 8   |   9 |  |  |
           +--+           +--+  +--+--+        +--+  +--+--+--+--+  +--+
           |  |     -->   |  | 2   |  |        |  | 2   |  |  |   6 |  |
           |  +--+        |  +--+--+  |        |  +--+--+  |  +--+--+  |
           | 0   |        | 0   |   1 |        | 0   |   1 | 7   |   4 |
           +-----+        +-----+-----+        +-----+-----+-----+-----+

       The parts are numbered to the left then middle then upper.  This relative numbering is
       maintained when rotated at the next replication level, as for example N=4 to N=7.

       The result is to visit 1 of every 3 points in the first quadrant with a subtle layout of
       points and spaces making diagonal lines and little 2x2 blocks.

           15  |  48          51  61          60 140         143 163
           14  |      50                  62         142                 168
           13  |          56          59                 139         162
           12  |  49          58              63 141             160
           11  |  55              44          47 131         138
           10  |          57          46                 136         137
            9  |      54                  43         130                 134
            8  |  52          53  45             128         129 135
            7  |  12          15  35          42              37  21
            6  |      14                  40          41                  22
            5  |          11          34                  38          25
            4  |  13              32          33  39          36
            3  |   3          10               5  31              26
            2  |           8           9                  27          24
            1  |       2                   6          30                  18
           Y=0 |   0           1   7           4  28          29  19
               +------------------------------------------------------------
                 X=0   1   2   3   4   5   6   7   8   9  10  11  12  13  14

       On the X=Y leading diagonal N=0,2,8,10,32,etc is the integers made from only digits 0 and
       2 in base 4.  Or equivalently integers which have zero bits at all even numbered
       positions, binary c0d0e0f0.

   Left or Upper
       Option "L_fill => "left"" or "L_fill => "upper"" numbers the tiles instead at their left
       end or upper end respectively.

           L_fill => 'left'           8  |      52              45  43
                                      7  |          15                      42
           +-----+                    6  |  12              35          40
           |     |                    5  |      14                  34  33
           |  +--+                    4  |      13  11          32
           | 3|  |                    3  |                  10   9   5
           +--+  +--+--+              2  |   3           8           6      31
           |  |    2| 1|              1  |           2   1               4
           |  +--+--+  |             Y=0 |       0               7
           |    0|     |                 +------------------------------------
           +-----+-----+                   X=0   1   2   3   4   5   6   7   8

           L_fill => 'upper'          8  |          53                  42
                                      7  |      12              35  40
           +-----+                    6  |          14  15      34          41
           |    3|                    5  |  13          11  32              39
           |  +--+                    4  |              10          33
           |  | 2|                    3  |       3   8
           +--+  +--+--+              2  |       2           9           5
           | 0|     |  |              1  |   0               7   6          28
           |  +--+--+  |             Y=0 |           1               4
           |     | 1   |                 +------------------------------------
           +-----+-----+                   X=0   1   2   3   4   5   6   7   8

       The effect is to disrupt the pattern a bit though the overall structure of the
       replications is unchanged.

       "left" is as viewed looking towards the L from above.  It may have been better to call it
       "right", but won't change that now.

   Ends
       Option "L_fill => "ends"" numbers the two endpoints within each "L", first the left then
       upper.  This is the inverse of the default middle shown above, ie. it visits all the
       points which the middle option doesn't, and so 2 of every 3 points in the first quadrant.

           +-----+
           |    7|
           |  +--+
           | 6| 5|
           +--+  +--+--+
           | 1|    4| 2|
           |  +--+--+  |
           |    0| 3   |
           +-----+-----+

            15  |      97 102         123 120         281 286         327 337
            14  |  96     101 103 122 124     121 280     285 287 326 325
            13  |  99 100     113 118     125 126 283 284     279 321     324
            12  |      98 112     117 119 127         282 278 277     320 323
            11  |     111 115 116      89  94         263 273     276 274 266
            10  | 110 109     114  88      93  95 262 261     272 275     268
             9  | 105     108 106  91  92      87 257     260 258 271 269
             8  |     104 107          90  86  85     256 259         270 265
             7  |      25  30          71  81      84  82  74          43  40
             6  |  24      29  31  70  69      80  83      76  75  42  44
             5  |  27  28      23  65      68  66  79  77      72  50      45
             4  |      26  22  21      64  67          78  73      52  51  47
             3  |       7  17      20  18  10          63  55  53      48  34
             2  |   6   5      16  19      12  11  62  61      54  49      36
             1  |   1       4   2  15  13       8  57      60  58  39  37
            Y=0 |       0   3          14   9          56  59          38  33
                +------------------------------------------------------------
                  X=0   1   2   3   4   5   6   7   8   9  10  11  12  13  14

   All
       Option "L_fill => "all"" numbers all three points of each "L", as middle, left then right.
       With this the path visits all points of the first quadrant.

                                  7  |  36  38  46  45 105 107 122 126
           +-----+                6  |  37  42  44  47 106 104 120 121
           | 9 11|                5  |  41  43  33  35  98 102 103 100
           |  +--+                4  |  39  40  34  32  96  97 101  99
           |10| 8|                3  |   9  11  26  30  31  28  16  15
           +--+  +--+--+          2  |  10   8  24  25  29  27  19  17
           | 2| 6  7| 4|          1  |   2   6   7   4  23  20  18  13
           |  +--+--+  |         Y=0 |   0   1   5   3  21  22  14  12
           | 0  1| 5  3|             +--------------------------------
           +-----+-----+               X=0   1   2   3   4   5   6   7

       Along the X=Y leading diagonal N=0,6,24,30,96,etc are triples of the values from the
       single-point case, so 3* numbers using digits 0 and 2 in base 4, which is the same as 2*
       numbers using 0 and 3 in base 4.

   Level Ranges
       For the "middles", "left" or "upper" cases with one N per tile, and taking the initial N=0
       tile as level 0, a replication level is

           Nstart = 0
            to
           Nlevel = 4^level - 1      inclusive

           Xmax = Ymax = 2 * 2^level - 1

       For example level 2 which is the large tiling shown in the introduction is N=0 to
       N=4^2-1=15 and extends to Xmax=Ymax=2*2^2-1=7.

       For the "ends" variation there's two points per tile, or for "all" there's three, in which
       case the Nlevel increases to

           Nlevel_ends = 2 * 4^level - 1
           Nlevel_all  = 3 * 4^level - 1

FUNCTIONS

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

       "$path = Math::PlanePath::LTiling->new ()"
       "$path = Math::PlanePath::LTiling->new (L_fill => $str)"
           Create and return a new path object.  The "L_fill" choices are

               "middle"    the default
               "left"
               "upper"
               "ends"
               "all"

       "($x,$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,   4**$level - 1      middle, left, upper
               0, 2*4**$level - 1      ends
               0, 3*4**$level - 1      all

           There are 4^level L shapes in a level, each containing 1, 2 or 3 points, numbered
           starting from 0.

OEIS

       Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

           <http://oeis.org/A062880> (etc)

           L_fill=middle
             A062880    N on X=Y diagonal, base 4 digits 0,2 only
             A048647    permutation N at transpose Y,X
                          base4 digits 1<->3 and 0,2 unchanged
             A112539    X+Y+1 mod 2, parity inverted

           L_fill=left or upper
             A112539    X+Y mod 2, parity

       A112539 is a parity of bits at even positions in N, ie. count 1-bits at even bit positions
       (least significant is bit position 0), then add 1 and take mod 2.  This works because in
       the pattern sub-blocks 0 and 2 are unchanged and 1 and 3 are turned so as to be on
       opposite X,Y odd/even parity, so a flip for every even position 1-bit.  L_fill=middle
       starts on a 0 even parity, and L_fill=left and upper start on 1 odd parity.  The latter is
       the form in A112539 and L_fill=middle is the bitwise 0<->1 inverse.

SEE ALSO

       Math::PlanePath, Math::PlanePath::CornerReplicate, Math::PlanePath::SquareReplicate,
       Math::PlanePath::QuintetReplicate, Math::PlanePath::GosperReplicate

HOME PAGE

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

LICENSE

       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017 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/>.