1Math::PlanePath::DigitGUrsoeurpsC(o3n)tributed Perl DocuMmaetnht:a:tPiloannePath::DigitGroups(3)
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NAME

6       Math::PlanePath::DigitGroups -- X,Y digits grouped by zeros
7

SYNOPSIS

9        use Math::PlanePath::DigitGroups;
10
11        my $path = Math::PlanePath::DigitGroups->new (radix => 2);
12        my ($x, $y) = $path->n_to_xy (123);
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DESCRIPTION

15       This path splits an N into X,Y by digit groups separated by a 0.  The
16       default is binary so for example
17
18           N = 110111001011
19
20       is split into groups with a leading high 0 bit, and those groups then
21       go to X and Y alternately,
22
23           N = 11 0111 0 01 011
24                X   Y  X  Y  X
25
26           X = 11      0    011 = 110011
27           Y =    0111   01     =  11101
28
29       The result is a one-to-one mapping between numbers N>=0 and pairs
30       X>=0,Y>=0.
31
32       The default binary is
33
34           11  |   38   77   86  155  166  173  182  311  550  333  342  347
35           10  |   72  145  148  291  168  297  300  583  328  337  340  595
36            9  |   66  133  138  267  162  277  282  535  322  325  330  555
37            8  |  128  257  260  515  272  521  524 1031  320  545  548 1043
38            7  |   14   29   46   59  142   93  110  119  526  285  302  187
39            6  |   24   49   52   99   88  105  108  199  280  177  180  211
40            5  |   18   37   42   75   82   85   90  151  274  165  170  171
41            4  |   32   65   68  131   80  137  140  263  160  161  164  275
42            3  |    6   13   22   27   70   45   54   55  262  141  150   91
43            2  |    8   17   20   35   40   41   44   71  136   81   84   83
44            1  |    2    5   10   11   34   21   26   23  130   69   74   43
45           Y=0 |    0    1    4    3   16    9   12    7   64   33   36   19
46               +-------------------------------------------------------------
47                  X=0    1    2    3    4    5    6    7    8    9   10   11
48
49       N=0,1,4,3,16,9,etc along the X axis is X with zero bits doubled.  For
50       example X=9 is binary 1001, double up the zero bits to 100001 for N=33
51       at X=9,Y=0.  This is because in the digit groups Y=0 so when X is
52       grouped by its zero bits there's an extra 0 from Y in between each
53       group.
54
55       Similarly N=0,2,8,6,32,etc along the Y axis is Y with zero bits
56       doubled, plus an extra zero bit at the low end coming from the first
57       X=0 group.  For example Y=9 is again binary 1001, doubled zeros to
58       100001, and an extra zero at the low end 1000010 is N=66 at X=0,Y=9.
59
60   Radix
61       The "radix => $r" option selects a different base for the digit split.
62       For example radix 5 gives
63
64           radix => 5
65
66           12  |  60  301  302  303  304  685 1506 1507 1508 1509 1310 1511
67           11  |  55  276  277  278  279  680 1381 1382 1383 1384 1305 1386
68           10  | 250 1251 1252 1253 1254 1275 6256 6257 6258 6259 1300 6261
69            9  |  45  226  227  228  229  670 1131 1132 1133 1134 1295 1136
70            8  |  40  201  202  203  204  665 1006 1007 1008 1009 1290 1011
71            7  |  35  176  177  178  179  660  881  882  883  884 1285  886
72            6  |  30  151  152  153  154  655  756  757  758  759 1280  761
73            5  | 125  626  627  628  629  650 3131 3132 3133 3134  675 3136
74            4  |  20  101  102  103  104  145  506  507  508  509  270  511
75            3  |  15   76   77   78   79  140  381  382  383  384  265  386
76            2  |  10   51   52   53   54  135  256  257  258  259  260  261
77            1  |   5   26   27   28   29  130  131  132  133  134  255  136
78           Y=0 |   0    1    2    3    4   25    6    7    8    9   50   11
79               +-----------------------------------------------------------
80                 X=0    1    2    3    4    5    6    7    8    9   10   11
81
82   Real Line and Plane
83       This split is inspired by the digit grouping in the proof by Julius
84       König that the real line is the same cardinality as the plane.
85       (Cantor's original proof was a "ZOrderCurve" style digit interleaving.)
86
87       In König's proof a bijection between interval n=(0,1) and pairs
88       x=(0,1),y=(0,1) is made by taking groups of digits stopping at a non-
89       zero.  Non-terminating fractions like 0.49999... are chosen over
90       terminating 0.5000... so there's always infinitely many non-zero digits
91       going downwards.  For the integer form here the groupings are digit
92       going upwards and there's infinitely many zero digits above the top,
93       hence the grouping by zeros instead of non-zeros.
94

FUNCTIONS

96       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
97       classes.
98
99       "$path = Math::PlanePath::DigitGroups->new ()"
100       "$path = Math::PlanePath::DigitGroups->new (radix => $r)"
101           Create and return a new path object.  The optional "radix"
102           parameter gives the base for digit splitting (the default is
103           binary, radix 2).
104
105       "($x,$y) = $path->n_to_xy ($n)"
106           Return the X,Y coordinates of point number $n on the path.  Points
107           begin at 0 and if "$n < 0" then the return is an empty list.
108

OEIS

110       Entries in Sloane's Online Encyclopedia of Integer Sequences related to
111       this path include
112
113           <http://oeis.org/A084471> (etc)
114
115           radix=2 (the default)
116             A084471    N on X axis, bit 0->00
117             A084472    N on X axis, in binary
118             A060142    N on X axis, sorted into ascending order
119

SEE ALSO

121       Math::PlanePath, Math::PlanePath::ZOrderCurve,
122       Math::PlanePath::PowerArray
123

HOME PAGE

125       <http://user42.tuxfamily.org/math-planepath/index.html>
126

LICENSE

128       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
129       Kevin Ryde
130
131       This file is part of Math-PlanePath.
132
133       Math-PlanePath is free software; you can redistribute it and/or modify
134       it under the terms of the GNU General Public License as published by
135       the Free Software Foundation; either version 3, or (at your option) any
136       later version.
137
138       Math-PlanePath is distributed in the hope that it will be useful, but
139       WITHOUT ANY WARRANTY; without even the implied warranty of
140       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
141       General Public License for more details.
142
143       You should have received a copy of the GNU General Public License along
144       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
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148perl v5.38.0                      2023-07-20   Math::PlanePath::DigitGroups(3)
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