Kevin Ryde > Math-PlanePath-Toothpick-12 > Math::PlanePath::LCornerReplicate
Module Version: 12   Source   Latest Release: Math-PlanePath-Toothpick-13

# NAME

Math::PlanePath::LCornerReplicate -- self-similar growth at exposed corners

# SYNOPSIS

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

# DESCRIPTION

This is a self-similar "L" shaped corners,

```     7  |   58  57  55  54  46  45  43  42  64
6  |   59  56  52  53  47  44  40  41  ...
5  |   61  60  50  49  35  34  36  39
4  |   62  63  51  48  32  33  37  38
3  |   14  13  11  10  16  19  31  30
2  |   15  12   8   9  17  18  28  29
1  |    3   2   4   7  21  20  24  27
Y=0 |    0   1   5   6  22  23  25  26
+-------------------------------------
X=0   1   2   3   4   5   6   7   8```

The base pattern is the initial N=0,1,2,3 and then when replicating the 1 and 3 sub-blocks are rotated -90 and +90 degrees,

```    +----------------+
|     3 |  2     |
|  ^    |    ^   |
|   \   |   /    |
|    \  |  /     |
| +90   |        |
|-------+--------|
|       |    -90 |
|    ^  |  \     |
|   /   |   \    |
|  /    |    v   |
| /  0  |  1     |
+----------------+```

Groups of 3 points such as N=13,14,15 make little L-shaped blocks, except at the middle single points where a replication begins such as N=4,8,12.

The sub-block layout is like `CornerReplicate` (Math::PlanePath::CornerReplicate) but its blocks are not rotated.

# FUNCTIONS

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

`\$path = Math::PlanePath::LCornerReplicate->new ()`

Create and return a new path object.

# FORMULAS

## Direction Maximum

The `dir_maximum_dxdy()` seems to occur at

```          base-4
N  = 323333...333
dX = 2121...2121212   (or 2121...12121)
dY = -3030...303031   (or  303...30310)```

which are

```    dX =    2*16^k + 6*(16^k -1)/15
dY = - (        12*(16^k -1)/15 + 1)

dY/dX -> -1/3   as k->infinity```

The pattern N=3233..333 [base4] probably has a geometric interpretation in the sub-blocks described above, but in any case the angles made by steps dX,dY approach a supremum dX=3,dY=-1.

# OEIS

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

```    http://oeis.org/A062880    (etc)

A062880    N values on diagonal X=Y (digits 0,2 in base-4)

A048647    permutation N at transpose Y,X
base-4 digit change 1<->3```

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

This file is part of Math-PlanePath-Toothpick.

Math-PlanePath-Toothpick 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-Toothpick 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-Toothpick. If not, see <http://www.gnu.org/licenses/>.

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