/ 
math-image-78
River stage zero No dependents
/ Math::PlanePath::MathImageDigitGroups

NAME

Math::PlanePath::MathImageDigitGroups -- 2x2 self-similar Z shape digits

SYNOPSIS

use Math::PlanePath::MathImageDigitGroups;

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

# or another radix digits ...
my $path3 = Math::PlanePath::MathImageDigitGroups->new (radix => 3);

DESCRIPTION

This is a split of N into X and Y values by groups of digits with a leading 0. For example,

N = 120345007089

is grouped and split to X and Y as

12 0345 0 07 089
 X   Y  X  Y  X

X = 12 0 089 = 120089
Y = 0345 07  = 34507

This is a one-to-one mapping between N>=0 and pairs X>=0,Y>=0.

In decimal,

1000 10001 10002 10003 10004 10005 10006 10007 10008 10009 10100
  90  901  902  903  904  905  906  907  908  909 1090
  80  801  802  803  804  805  806  807  808  809 1080
  70  701  702  703  704  705  706  707  708  709 1070
  60  601  602  603  604  605  606  607  608  609 1060
  50  501  502  503  504  505  506  507  508  509 1050
  40  401  402  403  404  405  406  407  408  409 1040
  30  301  302  303  304  305  306  307  308  309 1030
  20  201  202  203  204  205  206  207  208  209 1020
  10  101  102  103  104  105  106  107  108  109 1010
   0    1    2    3    4    5    6    7    8    9  100

In binary,

11  |   38   77   86  155  166  173  182  311  550  333  342  347
10  |   72  145  148  291  168  297  300  583  328  337  340  595
 9  |   66  133  138  267  162  277  282  535  322  325  330  555
 8  |  128  257  260  515  272  521  524 1031  320  545  548 1043
 7  |   14   29   46   59  142   93  110  119  526  285  302  187
 6  |   24   49   52   99   88  105  108  199  280  177  180  211
 5  |   18   37   42   75   82   85   90  151  274  165  170  171
 4  |   32   65   68  131   80  137  140  263  160  161  164  275
 3  |    6   13   22   27   70   45   54   55  262  141  150   91
 2  |    8   17   20   35   40   41   44   71  136   81   84   83
 1  |    2    5   10   11   34   21   26   23  130   69   74   43
Y=0 |    0    1    4    3   16    9   12    7   64   33   36   19
    +-------------------------------------------------------------
       X=0    1    2    3    4    5    6    7    8    9   10   11

R <-> RxR

This construction is inspired by the similar digit grouping used in the proof that the real line is the same cardinality as the plane (by Cantor was it?). In that case a bijection between interval z=(0,1) and pairs x=(0,1),y=(0,1) is made by groups of fraction digits stopping at a non-zero digit.

In that proof non-terminating fractions like 0.49999... are chosen over terminating 0.5000... so there's infinitely many non-zero digits. For the integer form here there's infinitely many zero digits at the high ends of N, X and Y, hence grouping by zero digits instead of non-zero.

FUNCTIONS

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

$path = Math::PlanePath::MathImageDigitGroups->new ()
$path = Math::PlanePath::MathImageDigitGroups->new (radix => $r)

Create and return a new path object. The optional radix parameter gives the base for digit splitting (the default is binary, radix 2).

($x,$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.

SEE ALSO

Math::PlanePath, Math::PlanePath::ZOrderCurve