and 1 contributors

# 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

``````    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.

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