Kevin Ryde > Math-PlanePath-Toothpick > Math::PlanePath::LCornerReplicate

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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.

Level Methods

($n_lo, $n_hi) = $path->level_to_n_range($level)

Return (0, 4**$level - 1).

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.

Level End

The last point in each level is N=4^k-1 and is located at

    X(k) = binary 00110011001100... take first k many bits
         = 0, 0, 0, 1, 3, 6, 12, 25, 51, 102, 204, 409, ...   (A077854)
         = binary: empty, 0, 00, 001, 0011, 00110, 001100, 0011001, ...

    Y(k) = binary 10011001100110... take first k many bits
         = 0, 1, 2, 4, 9, 19, 38, 76, 153, 307, 614, 1228, ...
         = binary: empty, 1, 10, 100, 1001, 10011, 100110, 1001100, ...

N=4^k-1 is 333...3 in base-4 and so is part 3 each time. Each part 3 is a transformation (H,L)->(~L,H), where ~ is a ones-complement reversal 0<->1. Applying that transform down each digit of N gives a repeating pattern 1,1,0,0.

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,
                 being base-4 digits 0,2 only

    A077854    level last X(k), per bit pattern above
    A048647    permutation N at transpose Y,X,
                 being base-4 digit change 1<->3

SEE ALSO ^

Math::PlanePath, Math::PlanePath::LCornerTree, Math::PlanePath::CornerReplicate, Math::PlanePath::ToothpickReplicate

HOME PAGE ^

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

LICENSE ^

Copyright 2012, 2013, 2014 Kevin Ryde

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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