Math::PlanePath::PythagoreanTree -- primitive Pythagorean triples by tree
use Math::PlanePath::PythagoreanTree; my $path = Math::PlanePath::PythagoreanTree->new (tree_type => 'UAD', coordinates => 'AB'); my ($x, $y) = $path->n_to_xy (123);
This path enumerates primitive Pythagorean triples by a breadth-first traversal of a ternary tree, either a "UAD" or "FB" tree. Each point is an integer X,Y=A,B with an integer hypotenuse, and primitive in the sense that A and B have no common factor.
A^2 + B^2 = C^2 gcd(A,B)=1 ie. no common factor
Such a triple always has one of A,B odd and the other even. The trees here give them ordered as A odd and B even.
The breadth-first traversal goes out to rather large A,B values while smaller ones have yet to be reached. The UAD tree goes out further than the FB.
The UAD tree by Berggren (1934) and later independently by Barning (1963), Hall (1970), and a number of others, uses three matrices U, A and D which can be multiplied onto an existing primitive triple to form three new primitive triples. See "UAD Matrices" below for details of the tree descent.
tree_type => "UAD" (the default Y=40 | 14 | | | | 7 Y=24 | 5 | Y=20 | 3 | Y=12 | 2 13 | | 4 Y=4 | 1 | +-------------------------------------------------- X=3 X=15 X=20 X=35 X=45
The starting point is N=1 at X=3,Y=4 which is the well-known 3^2 + 4^2 = 5^2. From there further N=2, N=3, N=4 are derived, then three more from each of those, etc,
N=1 N=2..4 N=5..13 N=14... +-> 7,24 +-> 5,12 --+-> 55,48 | +-> 45,28 | | +-> 39,80 3,4 --+-> 21,20 --+-> 119,120 | +-> 77,36 | | +-> 33,56 +-> 15,8 --+-> 65,72 +-> 35,12
Counting the N=1 point as level 1, each level has 3^(level-1) many points and the first N of the level is at
Nstart = 1 + 3 + 3^2 + ... + 3^(level-1) = (3^level + 1) / 2
These levels are like a mixed-radix representation of N where the high digit is binary, and so since the high is non-zero thus always 1, and further digits below in ternary. The number of digits is the level and the ternary digits are the position within the level.
N=1,3,9,27,etc 3^level is the middle "A" matrix at each node, giving 3,4 then 21,20 then 119,120 then 697,696, etc, which are the triples with legs differing by 1, and thus just below the X=Y leading diagonal.
Taking the lower "D" matrix at each node, ie. 3,4 to 15,8 to 35,12 to 63,16, etc, is the primitives among a sequence of triples known to the ancients,
A = k^2-1, B = 2*k, C = k^2+1
When k is even these are primitive. (If k is odd then A and B are both even, ie. a common factor of 2, so not primitive.) These points are the last of each level, so N=(3^(level+1)-1)/2.
The FB tree by H. Lee Price
"The Pythagorean Tree: A New Species", 2008 http://arxiv.org/abs/0809.4324
is based on expressing triples in certain "Fibonacci boxes" with a box of four values q',q,p,p' having p=q+q' and p'=p+q, each the sum of the preceding two in a fashion similar to the Fibonacci sequence. Any box where p and q have no common factor corresponds to a primitive triple (see "PQ Coordinates" below).
tree_type => "FB" Y=40 | 5 | | | | 17 Y=24 | 4 | | 8 | Y=12 | 2 6 | | 3 Y=4 | 1 | +---------------------------------------------- X=3 X=15 x=21 X=35
For a given box three transformations can be applied to go to new boxes corresponding to new primitive triples. This visits all and only primitive triples, but in a different order and different tree structure to the UAD above.
The first point N=1 is again at X=3,Y=4, from which three further points N=2,3,4 are derived, then three more from each of those, etc.
N=1 N=2..4 N=5..13 N=14... +-> 9,40 +-> 5,12 --+-> 35,12 | +-> 11,60 | | +-> 21,20 3,4 --+-> 15,8 --+-> 55,48 | +-> 39,80 | | +-> 13,84 +-> 7,24 --+-> 63,16 +-> 15,112
Primitive Pythagorean triples can be parameterized as follows, for A odd and B even.
A = P^2 - Q^2 B = 2*P*Q C = P^2 + Q^2 with P > Q >= 1, one odd, one even, and no common factor
Or conversely,
P = sqrt((C+A)/2) Q = sqrt((C-A)/2)
The first P=2,Q=1 is the triple A=3,B=4,C=5. The coordinates option on the path gives these P,Q values as the returned X,Y (for either tree type), Since P>Q>=1, the values fall in the eighth of the plane below the X=Y diagonal,
coordinates
coordinates => "PQ" 11 | * 10 | * 9 | * 8 | * * 7 | * * * 6 | * * 5 | * * * 4 | * * * * 3 | * * * 2 | * * * * * 1 | * * * * * * 0 | +------------------------ 0 1 2 3 4 5 6 7 8 9 ...
The correspondence between P,Q and A,B means the trees visit all P,Q pairs with no common factor and one of them even. There's other ways to iterate through such coprime pairs P,Q, and that would generate triples too, in a different order from the trees here.
Incidentally letters P and Q used here are a little bit arbitrary. This parameterization is often found as m,n or u,v, but don't want "n" to be confused that with the N point numbering or "u" confused with the U matrix in UAD.
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::PythagoreanTree->new ()
$path = Math::PlanePath::PythagoreanTree->new (tree_type => $str, coordinates => $str)
Create and return a new path object. The tree_type option can be
tree_type
"UAD" (the default) "FB"
The coordinates option can be
"AB" (the default) "PQ"
$n = $path->n_start()
Return 1, the first N in the path.
($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.
$n
$n<1
$n = $path->xy_to_n ($x,$y)
Return the point number for coordinates $x,$y. If there's nothing at $x,$y then return undef.
$x,$y
undef
The return is undef if $x,$y is not a primitive Pythagorean triple, or for the PQ option if $x,$y doesn't satisfy the PQ constraints described above ("PQ Coordinates").
($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.
$x1
$y1
$x2
$y2
Both trees go off into large X,Y coordinates while yet to finish values close to the origin which means the N range for a rectangle can be quite large. For UAD $n_hi is roughly 3**max(x/2), or for FB smaller at roughly 3**log2(x).
$n_hi
3**max(x/2)
3**log2(x)
@n_children = $path->tree_n_children($n)
Return the three children of $n, or an empty list if $n < 1 (ie. before the start of the path).
$n < 1
This is simply 3*$n-1, 3*$n, 3*$n+1. This is like appending an extra ternary digit to go to the next level, but onto N+1 rather than N and then adjusting back.
3*$n-1, 3*$n, 3*$n+1
$num = $path->tree_n_num_children($n)
Return 3, since every node has three children, or return undef if $n<1 (ie. 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).
$n <= 1
This is simply floor(($n+1)/3), reversing the tree_n_children() calculation.
floor(($n+1)/3)
tree_n_children()
$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 3 nodes per point means the depth is floor(log3(2N-1)), so for example N=5 through N=13 all have depth=2.
$n = $path->tree_depth_to_n($depth)
$n = $path->tree_depth_to_n_end($depth)
Return the first or last N at tree level $depth in the path, or undef if nothing at that depth or not a tree. The top of the tree is depth=0.
$depth
The UAD matrices are
/ 1 2 2 \ U = | -2 -1 -2 | \ 2 2 3 / / 1 2 2 \ A = | 2 1 2 | \ 2 3 3 / / -1 -2 -2 \ D = | 2 1 2 | \ 2 2 3 /
They're multiplied on the right of an (A,B,C) vector, for example
(3, 4, 5) * U = (5, 12, 13)
Internally the code uses P,Q and calculates an A,B at the end as necessary. The UAD transformations in P,Q coordinates are
U P -> 2P-Q Q -> P A P -> 2P+Q Q -> P D P -> P+2Q Q -> Q unchanged
The advantage of P,Q for the calculation is that it's 2 values instead of 3. The transformations could be written as 2x2 matrix multiplications if desired, but explicit steps are enough for the code.
The FB tree is calculated in P,Q and converted to A,B at the end as necessary. Its three transformations are
K1 P -> P+Q Q -> 2Q K2 P -> 2P Q -> P-Q K3 P -> 2P Q -> P+Q
Price's paper shows rearrangements of a set of four values q',q,p,p', but just the p and q are enough for the calculation.
xy_to_n() works in P,Q coordinates and converts an A,B input. A P,Q point can be reversed up the UAD tree to its parent point
xy_to_n()
if P > 3Q reverse "D" P -> P-2Q Q -> unchanged if P > 2Q reverse "A" P -> Q Q -> P-2Q otherwise reverse "U" P -> Q Q -> 2Q-P
This gives a ternary digit 2, 1, 0 respectively for N (low to high), plus a high "1" digit. The number of steps is the level.
If at any stage P,Q doesn't satisfy P>Q, one odd, the other even, then it means the original point, whether it was an A,B or a P,Q, was not a primitive triple. For a primitive triple the endpoint is always P=2,Q=1.
After converting A,B to P,Q if necessary, a P,Q point can be reversed up the FB tree to its parent
if P odd reverse K1 P -> P-Q (so Q even) Q -> Q/2 if Q < P/2 reverse K2 P -> P/2 Q -> P/2 - Q otherwise reverse K3 P -> P/2 Q -> Q - P/2
This is a little like the binary greatest common divisor algorithm, but designed for one value odd and the other even. Like the UAD ascent above, if at any stage P,Q doesn't satisfy P>Q, one odd, the other even, then the initial point wasn't a primitive triple.
For the UAD tree, the smallest A,B within each level is found at the topmost "U" steps for the smallest A or the bottom-most "D" steps for the smallest B. For example in the table above of level=2 N=5..13 the smallest A is the top A=7,B=24, and the smallest B is in the bottom A=35,B=12. In general
Amin = 2*level + 1 Bmin = 4*level
In P,Q coordinates the same topmost line is the smallest P and bottom-most the smallest Q. The values are
Pmin = level+1 Qmin = 1
The fixed Q=1 arises from the way the "D" transformation sends Q->Q unchanged, so every level includes a Q=1. This means if you ask what range of N is needed to cover all Q < someQ then there isn't one, only a P < someP has an N to go up to.
For the FB tree, the smallest A,B within each level is found in the topmost two final positions. For example in the table above of level=2 N=5..13 the smallest A is in the top A=9,B=40, and the smallest B is in the next row A=35,B=12. In general,
Amin = 2^level + 1 Bmin = 2^level + 4
In P,Q coordinates a Q=1 is found in that second row which is the minimum B, and the smallest P is found by taking K1 steps half-way then a K2 step, then K1 steps for the balance. This is a slightly complicated
Pmin = / 3*2^(k-1) + 1 if even level = 2*k \ 2^(k+1) + 1 if odd level = 2*k+1 Q = 1
The fixed Q=1 arises from the K1 steps giving
P = 2 + 1+2+4+8+...+2^(level-2) = 2 + 2^(level-1) - 1 = 2^(level-1) + 1 Q = 2^(level-1) followed by K2 step Q -> P-Q = 1
As for the UAD above this means small Q's always remain no matter how big N gets, only a P range determines an N range.
Math::PlanePath, Math::PlanePath::Hypot, Math::PlanePath::RationalsTree, Math::PlanePath::CoprimeColumns
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012 Kevin Ryde
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