Kevin Ryde > Math-PlanePath-116 > Math::PlanePath::LTiling

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Module Version: 116   Source   Latest Release: Math-PlanePath-117

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.

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 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/>.

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