Provided by: libmath-planepath-perl_129-1_all bug

NAME

       Math::PlanePath::CornerReplicate -- replicating U parts

SYNOPSIS

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

DESCRIPTION

       This path is a self-similar replicating corner fill with 2x2 blocks.  It's sometimes
       called a "U order" since the base N=0 to N=3 is like a "U" (sideways).

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

       The pattern is the initial N=0 to N=3 section,

           +-------+-------+
           |       |       |
           |   3   |   2   |
           |       |       |
           +-------+-------+
           |       |       |
           |   0   |   1   |
           |       |       |
           +-------+-------+

       It repeats as 2x2 blocks arranged in the same pattern, then 4x4 blocks, etc.  There's no
       rotations or reflections within sub-parts.

       X axis N=0,1,4,5,16,17,etc is all the integers which use only digits 0 and 1 in base 4.
       For example N=17 is 101 in base 4.

       Y axis N=0,3,12,15,48,etc is all the integers which use only digits 0 and 3 in base 4.
       For example N=51 is 303 in base 4.

       The X=Y diagonal N=0,2,8,10,32,34,etc is all the integers which use only digits 0 and 2 in
       base 4.

       The X axis is the same as the "ZOrderCurve".  The Y axis here is the X=Y diagonal of the
       "ZOrderCurve", and conversely the X=Y diagonal here is the Y axis of the "ZOrderCurve".

       The N value at a given X,Y is converted to or from the "ZOrderCurve" by transforming
       base-4 digit values 2<->3.  This can be done by a bitwise "X xor Y".  When Y has a 1-bit
       the xor swaps 2<->3 in N.

           ZOrder X  = CRep X xor CRep Y
           ZOrder Y  = CRep Y

           CRep X  = ZOrder X xor ZOrder Y
           CRep Y  = ZOrder Y

       See Math::PlanePath::LCornerReplicate for a rotating corner form.

   Level Ranges
       A given replication extends to

           Nlevel = 4^level - 1
           0 <= X < 2^level
           0 <= Y < 2^level

   Hamming Distance
       The Hamming distance between two integers X and Y is the number of bit positions where the
       two values differ when written in binary.  In this corner replicate each bit-pair of N
       becomes a bit of X and a bit of Y,

              N      X   Y
           ------   --- ---
           0 = 00    0   0
           1 = 01    1   0     <- difference 1 bit
           2 = 10    1   1
           3 = 11    0   1     <- difference 1 bit

       So the Hamming distance is the number of base4 bit-pairs of N which are 01 or 11.
       Counting bit positions from 0 for least significant bit, this is the 1-bits in even
       positions,

           HammingDist(X,Y) = count 1-bits at even bit positions in N
                            = 0,1,0,1, 1,2,1,2, 0,1,0,1, 1,2,1,2, ... (A139351)

FUNCTIONS

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

       "$path = Math::PlanePath::CornerReplicate->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.

       "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
           The returned range is exact, meaning $n_lo and $n_hi are the smallest and biggest in
           the rectangle.

   Level Methods
       "($n_lo, $n_hi) = $path->level_to_n_range($level)"
           Return "(0, 4**$level - 1)".

FORMULAS

   N to dX,dY
       The change dX,dY is given by N in base 4 count trailing 3s and the digit above those
       trailing 3s.

           N = ...[d]333...333      base 4
                     \--exp--/

       When N to N+1 crosses between 4^k blocks it goes as follows.  Within a block the pattern
       is the same, since there's no rotations or transposes etc.

           N, N+1        X      Y        dX       dY       dSum     dDiffXY
           --------   -----  -------   -----  --------    ------    -------
           033..33       0    2^k-1      2^k  -(2^k-1)        +1    2*2^k-1
           100..00      2^k       0

           133..33      2^k    2^k-1       0       +1         +1       -1
           200..00      2^k    2^k

           133..33      2^k  2*2^k-1    -2^k     1-2^k   -(2^k-1)      -1
           200..00       0     2^k

           133..33       0   2*2^k-1   2*2^k -(2*2^k-1)       +1    4*2^k-1
           200..00    2*2^k      0

       It can be noted dSum=dX+dY the change in X+Y is at most +1, taking values 1, -1, -3, -7,
       -15, etc.  The crossing from block 2 to 3 drops back, such as at N=47="233" to N=48="300".
       Everywhere else it advances by +1 anti-diagonal.

       The difference dDiffXY=dX-dY the change in X-Y decreases at most -1, taking similar values
       -1, 1, 3, 7, 15, etc but in a different order to dSum.

OEIS

       This path is in Sloane's Online Encyclopedia of Integer Sequences as

           <http://oeis.org/A000695> (etc)

           A059906    Y coordinate
           A059905    X xor Y, being ZOrderCurve X
           A139351    HammingDist(X,Y), count 1-bits at even positions in N

           A000695    N on X axis, base 4 digits 0,1 only
           A001196    N on Y axis, base 4 digits 0,3 only
           A062880    N on diagonal, base 4 digits 0,2 only
           A163241    permutation base-4 flip 2<->3,
                        converts N to ZOrderCurve N, and back

           A048647    permutation N at transpose Y,X
                        base4 digits 1<->3

SEE ALSO

       Math::PlanePath, Math::PlanePath::LTiling, Math::PlanePath::SquareReplicate,
       Math::PlanePath::GosperReplicate, Math::PlanePath::ZOrderCurve, Math::PlanePath::GrayCode

HOME PAGE

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

LICENSE

       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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/>.