Kevin Ryde > Math-PlanePath > Math::PlanePath::DiagonalRationals

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NAME ^

Math::PlanePath::DiagonalRationals -- rationals X/Y by diagonals

SYNOPSIS ^

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

DESCRIPTION ^

This path enumerates positive rationals X/Y with no common factor, going in diagonal order from Y down to X.

    17  |    96...
    16  |    80
    15  |    72 81
    14  |    64    82
    13  |    58 65 73 83 97
    12  |    46          84
    11  |    42 47 59 66 74 85 98
    10  |    32    48          86
     9  |    28 33    49 60    75 87
     8  |    22    34    50    67    88
     7  |    18 23 29 35 43 51    68 76 89 99
     6  |    12          36    52          90
     5  |    10 13 19 24    37 44 53 61    77 91
     4  |     6    14    25    38    54    69    92
     3  |     4  7    15 20    30 39    55 62    78 93
     2  |     2     8    16    26    40    56    70    94
     1  |     1  3  5  9 11 17 21 27 31 41 45 57 63 71 79 95
    Y=0 |
        +---------------------------------------------------
         X=0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16

The order is the same as the Diagonals path, but only those X,Y with no common factor are numbered.

    1/1,                      N = 1
    1/2, 1/2,                 N = 2 .. 3
    1/3, 1/3,                 N = 4 .. 5
    1/4, 2/3, 3/2, 4/1,       N = 6 .. 9
    1/5, 5/1,                 N = 10 .. 11

N=1,2,4,6,10,etc at the start of each diagonal (in the column at X=1) is the cumulative totient,

    totient(i) = count numbers having no common factor with i

                             i=K
    cumulative_totient(K) =  sum   totient(i)
                             i=1

Direction Up

Option direction => 'up' reverses the order within each diagonal to count upward from the X axis.

    direction => "up"

     8 |   27
     7 |   21 26
     6 |   17
     5 |   11 16 20 25
     4 |    9    15    24
     3 |    5  8    14 19
     2 |    3     7    13    23
     1 |    1  2  4  6 10 12 18 22
    Y=0|
       +---------------------------
       X=0  1  2  3  4  5  6  7  8

N Start

The default is to number points starting N=1 as shown above. An optional n_start can give a different start with the same shape, For example to start at 0,

    n_start => 0

     8 |   21
     7 |   17 22
     6 |   11
     5 |    9 12 18 23
     4 |    5    13    24
     3 |    3  6    14 19
     2 |    1     7    15    25
     1 |    0  2  4  8 10 16 20 26
    Y=0|
       +---------------------------
       X=0  1  2  3  4  5  6  7  8

Coprime Columns

The diagonals are the same as the columns in CoprimeColumns. For example the diagonal N=18 to N=21 from X=0,Y=8 down to X=8,Y=0 is the same as the CoprimeColumns vertical at X=8. In general the correspondence is

   Xdiag = Ycol
   Ydiag = Xcol - Ycol

   Xcol = Xdiag + Ydiag
   Ycol = Xdiag

CoprimeColumns has an extra N=0 at X=1,Y=1 which is not present in DiagonalRationals. (It would be Xdiag=1,Ydiag=0 which is 1/0.)

The points numbered or skipped in a column up to X=Y is the same as the points numbered or skipped on a diagonal, simply because X,Y no common factor is the same as Y,X+Y no common factor.

Taking the CoprimeColumns as enumerating fractions F = Ycol/Xcol with 0 < F < 1 the corresponding diagonal rational 0 < R < infinity is

           1         F
    R = -------  =  ---
        1/F - 1     1-F

           1         R
    F = -------  =  ---
        1/R + 1     1+R

which is a one-to-one mapping between the fractions F < 1 and all rationals.

FUNCTIONS ^

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

$path = Math::PlanePath::DiagonalRationals->new ()
$path = Math::PlanePath::DiagonalRationals->new (direction => $str, n_start => $n)

Create and return a new path object. direction (a string) can be

    "down"     (the default)
    "up"
($x,$y) = $path->n_to_xy ($n)

Return the X,Y coordinates of point number $n on the path. Points begin at 1 and if $n < 1 then the return is an empty list.

BUGS ^

The current implementation is fairly slack and is slow on medium to large N. A table of cumulative totients is built and retained for the diagonal d=X+Y.

OEIS ^

This enumeration of rationals is in Sloane's Online Encyclopedia of Integer Sequences in the following forms

http://oeis.org/A020652 (etc)

    direction=down, n_start=1  (the defaults)
      A020652   X, numerator
      A020653   Y, denominator
      A038567   X+Y sum, starting from X=1,Y=1
      A054431   by diagonals 1=coprime, 0=not
                  (excluding X=0 row and Y=0 column)

      A054430   permutation N at Y/X
                  reverse runs of totient(k) many integers

      A054424   permutation DiagonalRationals -> RationalsTree SB
      A054425     padded with 0s at non-coprimes
      A054426     inverse SB -> DiagonalRationals
      A060837   permutation DiagonalRationals -> FactorRationals

    direction=down, n_start=0
      A157806   abs(X-Y) difference

direction=up swaps X,Y.

SEE ALSO ^

Math::PlanePath, Math::PlanePath::CoprimeColumns, Math::PlanePath::RationalsTree, Math::PlanePath::PythagoreanTree

HOME PAGE ^

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

LICENSE ^

Copyright 2011, 2012, 2013, 2014 Kevin Ryde

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/>.

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