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Math-PlanePath-125
River stage one • 3 direct dependents • 4 total dependents
2 ++
/ Math::PlanePath::SquareReplicate

NAME

Math::PlanePath::SquareReplicate -- replicating squares

SYNOPSIS

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

DESCRIPTION

This path is a self-similar replicating square,

    40--39--38  31--30--29  22--21--20         4
     |       |   |       |   |       |
    41  36--37  32  27--28  23  18--19         3
     |           |           |
    42--43--44  33--34--35  24--25--26         2

    49--48--47   4-- 3-- 2  13--12--11         1
     |       |   |       |   |       |
    50  45--46   5   0-- 1  14   9--10     <- Y=0
     |           |           |
    51--52--53   6-- 7-- 8  15--16--17        -1

    58--57--56  67--66--65  76--75--74        -2
     |       |   |       |   |       |
    59  54--55  68  63--64  77  72--73        -3
     |           |           |
    60--61--62  69--70--71  78--79--80        -4

                     ^
    -4  -3  -2  -1  X=0  1   2   3   4

The base shape is the initial N=0 to N=8 section,

   4  3  2
   5  0  1
   6  7  8

It then repeats with 3x3 blocks arranged in the same pattern, then 9x9 blocks, etc.

    36 --- 27 --- 18
     |             |
     |             |
    45      0 ---  9
     |
     |
    54 --- 63 --- 72

The replication means that the values on the X axis are those using only digits 0,1,5 in base 9. Those to the right have a high 1 digit and those to the left a high 5 digit. These digits are the values in the initial N=0 to N=8 figure which fall on the X axis.

Similarly on the Y axis digits 0,3,7 in base 9, or the leading diagonal X=Y 0,2,6 and opposite diagonal 0,4,8. The opposite diagonal digits 0,4,8 are 00,11,22 in base 3, so is all the values in base 3 with doubled digits aabbccdd, etc.

Level Ranges

A given replication extends to

    Nlevel = 9^level - 1
    - (3^level - 1) <= X <= (3^level - 1)
    - (3^level - 1) <= Y <= (3^level - 1)

Complex Base

This pattern corresponds to expressing a complex integer X+i*Y with axis powers of base b=3,

    X+Yi = a[n]*b^n + ... + a[2]*b^2 + a[1]*b + a[0]

using complex digits a[i] encoded in N in integer base 9,

    a[i] digit     N digit
    ----------     -------
          0           0
          1           1
        i+1           2
        i             3
        i-1           4
         -1           5
       -i-1           6
       -i             7
       -i+1           8

Numbering Rotate-4

Parameter numbering_type => 'rotate-4' applies a rotation to 4 directions E,N,W,S for each sub-part according to its position around the preceding level.

         ^   ^
         |   |
       +---+---+---+
       | 4   3 | 2 |-->
       +---+---+   +
    <--| 5 | 0>| 1 |-->
       +   +---+---+
    <--| 6 | 7   8 |
       +---+---+---+
             |   |
             v   v

The effect can be illustrated by writing N in base-9.

    42--41  48  32--31  38  24--23--22
     |   |   |   |   |   |   |       |
    43  40  47  33  30  37  25  20--21      numbering_type => 'rotate-4'
     |       |   |       |   |                  N shown in base-9
    44--45--46  34--35--36  26--27--28
                                   
    58--57--56   4---3---2  14--13--12
             |   |       |   |       |
    51--50  55   5   0---1  15  10--11
     |       |   |           |     
    52--53--54   6---7---8  16--17--18
                                   
    68--67--66  76--75--74  86--85--84
             |   |       |   |       |
    61--60  65  77  70  73  87  80  83
     |       |   |   |   |   |   |   |
    62--63--64  78  71--72  88  81--82

Parts 10-18 and 20-28 are the same as the middle 0-8. Parts 30-38 and 40-48 have a rotation by +90 degrees. Parts 50-58 and 60-68 rotation by +180 degrees, and so on.

Notice this means in each part the base-9 points 11, 21, 31, points are directed away from the middle in the same way, relative to the sub-part locations. This gives a reasonably simple way to characterize points on the boundary of a given expansion level.

Working through the directions and boundary sides gives a state machine for which unit squares are on the boundary. For level >= 1 a given unit square has one of both of two sides on the boundary.

       B
    +-----+         
    |     |            unit square with expansion direction,   
    |     |->  A       one or both of sides A,B on the boundary    
    |     |
    +-----+

A further low base-9 digit expands the square to a block of 9, with squares then boundary or not. The result is 4 states, which can be expressed by pairs of digits

    write N in base-9 using level many digits,
    delete all 2s in 2nd or later digit
    non-boundary =
      0 anywhere
      5 or 6 or 7 in 2nd or later digit
      pair 13,33,53,73, 14,34,54,74 anywhere
      pair 43,44, 81,88 at 2nd or later digit

Pairs 53,73,54,74 can be checked just at the start of the digits, since 5 or 7 anywhere later are non-boundary alone irrespective of what (if any) pair they might make.

Numbering Rotate 8

Parameter numbering_type => 'rotate-8' applies a rotation to 8 directions for each sub-part according to its position around the preceding level.

     ^       ^       ^
      \      |      /
       +---+---+---+
       | 4 | 3 | 2 |
       +---+---+---+
    <--| 5 | 0>| 1 |-->
       +---+---+---+
       | 6 | 7 | 8 |
       +---+---+---+
      /      |      \
     v       v       v

The effect can be illustrated again by N in base-9.

    41 48-47 32-31 38 23-22-21
     |\    |  |  |  |  |   /
    42 40 46 33 30 37 24 20 28      numbering_type => 'rotate'
     |     |  |     |  |     |          N shown in base-9
    43-44-45 34-35-36 25-26-27

    58-57-56  4--3--2 14-13-12
           |  |     |  |     |
    51-50 55  5  0--1 15 10-11
     |     |  |        |
    52-53-54  6--7--8 16-17-18

    67-66-65 76-75-74 85-84-83
     |     |  |     |  |     |
    68 60 64 77 70 73 86 80 82
      /    |  |  |  |  |   \ |
    61-62-63 78 71-72 87-88 81

Notice this means in each part the 11, 21, 31, etc, points are directed away from the middle in the same way, relative to the sub-part locations.

FUNCTIONS

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

$path = Math::PlanePath::SquareReplicate->new ()

Create and return a new path object.

($x,$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, 9**$level - 1).

SEE ALSO

Math::PlanePath, Math::PlanePath::CornerReplicate, Math::PlanePath::LTiling, Math::PlanePath::GosperReplicate, Math::PlanePath::QuintetReplicate

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/>.