Math::PlanePath::FractionsTree -- fractions by tree
use Math::PlanePath::FractionsTree; my $path = Math::PlanePath::FractionsTree->new (tree_type => 'Kepler'); my ($x, $y) = $path->n_to_xy (123);
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.
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
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.
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
$y2. The range is inclusive.
$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).
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
$n<1 (before the start of the path).
$n_parent = $path->tree_n_parent($n)
Return the parent node of
$n <= 1 (the top of the tree).
This is simply
floor($n/2), stripping the least significant bit from
$n (undoing what
$depth = $path->tree_n_to_depth($n)
Return the depth of node
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.
$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.
The trees are in Sloane's Online Encyclopedia of Integer Sequences in the following forms
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.
Copyright 2011, 2012, 2013, 2014, 2015, 2016 Kevin Ryde
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