depth:                    X x Y                             required memory:
  1:                      4 x 4                      ==      16 _b
  2:                     16 x 16                     ==     256 _b
  3:                     64 x 64                     ==       4 Kb
  4:                    256 x 256                    ==      64 Kb
  5:                   1024 x 1024                   ==       1 Mb
  6:                   4096 x 4096                   ==      16 Mb
  7:                  16384 x 16384                  ==     256 Mb
  8:                  65536 x 65536                  ==       4 Gb
  9:                 262144 x 262144                 ==      64 Gb
 10:                1048576 x 1048576                ==       1 Tb
 11:                4194304 x 4194304                ==      16 Tb
 12:               16777216 x 16777216               ==     256 Tb
 13:               67108864 x 67108864               ==       4 Pb
 14:              268435456 x 268435456              ==      64 Pb
 15:             1073741824 x 1073741824             ==       1 Eb
 16:             4294967296 x 4294967296             ==      16 Eb
 17:            17179869184 x 17179869184            ==     256 Eb
 18:            68719476736 x 68719476736            ==       4 Zb
 19:           274877906944 x 274877906944           ==      64 Zb
 20:          1099511627776 x 1099511627776          ==       1 Yb
 21:          4398046511104 x 4398046511104          ==      16 Yb
 22:         17592186044416 x 17592186044416         ==     256 Yb
 23:         70368744177664 x 70368744177664         ==    4 2^90b
 24:        281474976710656 x 281474976710656        ==   64 2^90b
 25:       1125899906842624 x 1125899906842624       ==   1 2^100b
 26:       4503599627370496 x 4503599627370496       ==  16 2^100b
 27:      18014398509481984 x 18014398509481984      == 256 2^100b
 28:      72057594037927936 x 72057594037927936      ==   4 2^110b
 29:     288230376151711740 x 288230376151711740     ==  64 2^110b
 30:    1152921504606847000 x 1152921504606847000    ==   1 2^120b
 31:    4611686018427387900 x 4611686018427387900    ==  16 2^120b
 32:   18446744073709552000 x 18446744073709552000   == 256 2^120b

##</code><code>##

if( X == Y ) {
    return depth;
} 
elsif ( ( X,Y ) is in a 4x4 block on the major diagonal ) {
    return $depth - 1;
}
else {
    ## Move the 4x4 block to the origin and treat as for depth -1
    X %= 4;
    Y %= 4;
    depth--;
    recurse;
}

##</code><code>##

#! perl -slw
use strict;

$|++;

our $DEPTH ||= 3;

sub fract2D {
    use integer;
    my( $x, $y, $depth ) = @_;

    if( $x < 16 and $y <  ) {        ## A block on the major diagonal
        return $x == $y ? $depth : $depth -1;
    }
    elsif( $depth > 1 ) {
        return fract2D( $x % 4**$depth, $y % 4**$depth, $depth -1 );
    }
    else {
        return fract2D( $x % 4**$depth, $y % 4**$depth, $depth );
    }
}

for my $depth ( $DEPTH ) {
    for my $y ( 0 .. 4**$depth-1 ) {
        my $c = 0;
        for my $x ( 0 .. 4 ** $depth -1 ) {
            printf +( ( ++$c % 4 ) == 0 ? " %2d " : " %2d" ),
                fract2D( $x, $y, $depth );
        }
        print +( $y % 4 ) == 3 ? "\n" : '';
    }
    print '';
}