Kevin Ryde > Math-PlanePath-116 > Math::PlanePath::FractionsTree

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Module Version: 116   Source   Latest Release: Math-PlanePath-117

NAME ^

Math::PlanePath::FractionsTree -- fractions by tree

SYNOPSIS ^

 use Math::PlanePath::FractionsTree;
 my $path = Math::PlanePath::FractionsTree->new (tree_type => 'Kepler');
 my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION ^

This path enumerates fractions X/Y in the range 0 < X/Y < 1 and in reduced form, ie. X and Y having no common factor, using a method by Johannes Kepler.

Fractions are traversed by rows of a binary tree which effectively represents a coprime pair X,Y by subtraction steps of a subtraction-only form of Euclid's greatest common divisor algorithm which would prove X,Y coprime. The steps left or right are encoded/decoded as an N value.

Kepler Tree

The default and only tree currently is by Kepler.

Johannes Kepler, "Harmonices Mundi", Book III. Excerpt of translation by Aiton, Duncan and Field at http://ndirty.cute.fi/~karttu/Kepler/a086592.htm

In principle similar bit reversal etc variations as in RationalsTree would be possible.

    N=1                             1/2
                              ------   ------
    N=2 to N=3             1/3               2/3
                          /    \            /   \
    N=4 to N=7         1/4      3/4      2/5      3/5
                       | |      | |      | |      | |
    N=8 to N=15     1/5  4/5  3/7 4/7  2/7 5/7  3/8 5/8

A node descends as

          X/Y
        /     \
    X/(X+Y)  Y/(X+Y)

Kepler described the tree as starting at 1, ie. 1/1, which descends to two identical 1/2 and 1/2. For the code here a single copy starting from 1/2 is used.

Plotting the N values by X,Y is as follows. Since it's only fractions X/Y<1, ie. X<Y, all points are above the X=Y diagonal. The unused X,Y positions are where X and Y have a common factor. For example X=2,Y=6 have common factor 2 so is never reached.

    12  |    1024                  26        27                1025
    11  |     512   48   28   22   34   35   23   29   49  513     
    10  |     256        20                  21       257          
     9  |     128   24        18   19        25  129               
     8  |      64        14        15        65                    
     7  |      32   12   10   11   13   33                         
     6  |      16                  17                              
     5  |       8    6    7    9                                   
     4  |       4         5                                        
     3  |       2    3                                             
     2  |       1                                                  
     1  |
    Y=0 |   
         ----------------------------------------------------------
          X=0   1    2    3    4    5    6    7    8    9   10   11

The X=1 vertical is the fractions 1/Y at the left end of each tree row, which is

    Nstart=2^level

The diagonal X=Y-1, fraction K/(K+1), is the second in each row, at N=Nstart+1. That's the maximum X/Y in each level.

The N values in the upper octant, ie. above the line Y=2*X, are even and those below that line are odd. This arises since X<Y so the left leg X/(X+Y) < 1/2 and the right leg Y/(X+Y) > 1/2. The left is an even N and the right an odd N.

FUNCTIONS ^

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

$path = Math::PlanePath::FractionsTree->new ()

Create and return a new path object.

$n = $path->n_start()

Return 1, the first N in the path.

($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)

Return a range of N values which occur in a rectangle with corners at $x1,$y1 and $x2,$y2. The range is inclusive.

For reference, $n_hi can be quite large because within each row there's only one new 1/Y fraction. So if X=1 is included then roughly $n_hi = 2**max(x,y).

Tree Methods

Each point has 2 children, so the path is a complete binary tree.

@n_children = $path->tree_n_children($n)

Return the two children of $n, or an empty list if $n < 1 (before the start of the path).

This is simply 2*$n, 2*$n+1. The children are $n with an extra bit appended, either a 0-bit or a 1-bit.

$num = $path->tree_n_num_children($n)

Return 2, since every node has two children, or return undef if $n<1 (before the start of the path).

$n_parent = $path->tree_n_parent($n)

Return the parent node of $n, or undef if $n <= 1 (the top of the tree).

This is simply floor($n/2), stripping the least significant bit from $n (undoing what tree_n_children() appends).

$depth = $path->tree_n_to_depth($n)

Return the depth of node $n, or undef if there's no point $n. The top of the tree at N=1 is depth=0, then its children depth=1, etc.

The structure of the tree with 2 nodes per point means the depth is simply floor(log2(N)), so for example N=4 through N=7 are all depth=2.

Tree Descriptive Methods

$num = $path->tree_num_children_minimum()
$num = $path->tree_num_children_maximum()

Return 2 since every node has 2 children, making that both the minimum and maximum.

$bool = $path->tree_any_leaf()

Return false, since there are no leaf nodes in the tree.

OEIS ^

The trees are in Sloane's Online Encyclopedia of Integer Sequences in the following forms

http://oeis.org/A020651 (etc)

    tree_type=Kepler
      A020651  - X numerator (RationalsTree AYT denominators)
      A086592  - Y denominators
      A086593  - X+Y sum, and every second denominator
      A020650  - Y-X difference (RationalsTree AYT numerators)

The tree descends as X/(X+Y) and Y/(X+Y) so the denominators are two copies of X+Y time after the initial 1/2. A086593 is every second, starting at 2, eliminating the duplication. This is also the sum X+Y, from value 3 onwards, as can be seen by thinking of writing a node as the X+Y which would be the denominators it descends to.

SEE ALSO ^

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

Math::NumSeq::SternDiatomic, Math::ContinuedFraction

HOME PAGE ^

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

LICENSE ^

Copyright 2011, 2012, 2013, 2014 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/>.

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