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 integer hypotenuse A^2+B^2=C^2 and primitive because A and B have 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.
my $path = Math::PlanePath::PythagoreanTree->new (tree_type => 'UAD');
Starting from A=3,B=4,C=5, the well-known 3^2 + 4^2 = 5^2, the tree visits all and only primitive triples.
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
For the path the starting point N=1 is X=3,Y=4 and from it three further N=2,3,4 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
N = 1 + 3 + 3^2 + ... + 3^(level-1) N = (3^level + 1) / 2
Taking the middle "A" direction at each node, ie. 21,20 then 119,120 then 697,696, etc, gives the triples with legs differing by 1, so just below the X=Y leading diagonal. These are at N=3^level.
Taking the lower "D" direction at each node, ie. 15,8 then 35,12 then 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,
http://arxiv.org/abs/0809.4324
is based on expressing triples in certain "Fibonacci boxes" with 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).
my $path = Math::PlanePath::PythagoreanTree->new (tree_type => 'FB');
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.
Y=40 | 5 | | | | 17 Y=24 | 4 | | 8 | Y=12 | 2 6 | | 3 Y=4 | 1 | +---------------------------------------------- X=3 X=15 x=21 X=35
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, taking 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
And 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 coordinates,
coordinates
my $path = Math::PlanePath::PythagoreanTree->new (tree_type => 'UAD', # or 'FB' coordinates => 'PQ'); my ($p,$q) = $path->n_to_xy(1); # P=2,Q=1
Since P>Q>=1, the values fall in an octant below the X=Y diagonal,
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. Of course there's other ways to iterate through such P,Q, such as simply P=2,3,etc, and which 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 to confuse that with the N numbered points or the U matrix in UAD.
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.
$path = Math::PlanePath::PythagoreanTree->new ()
Create and return a new path object.
($x,$y) = $path->n_to_xy ($n)
Return the X,Y coordinates of point number $n on the path. Points begin at 0 and if $n < 0 then the return is an empty list.
$n
$n < 0
Fractional positions give an X,Y position along a straight line between the integer positions. Integer positions are always just 1 apart either horizontally or vertically, so the effect is that the fraction part appears either added to or subtracted from X or Y.
$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 with the PQ option if 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 visit large X,Y coordinates while yet to finish values closer 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)
The three UAD matrices are as follows
/ 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)
But internally the code uses P,Q and calculates an A,B at the end as necessary. The transformations in P,Q coordinates are
U P -> 2P-Q Q -> P A P -> 2P+Q Q -> P D P -> P+2Q 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 an A,B calculated at the end. The three transformation 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 four values q',q,p,p', but just the p and q are enough for a calculation.
An A,B or P,Q point can be reversed up the tree to its parent as follows,
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 and the number of steps is the level and a starting N for the digits. If at any stage the P,Q aren't one odd the other even and P>Q then it means the original point, either an A,B or a P,Q, was not a primitive triple. For a primitive triple the endpoint is always P=2,Q=1.
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 rather similar to the binary greatest common divisor algorithm, but designed for one value odd and the other even. As for the UAD ascent above if that opposite parity doesn't hold at any stage 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 in 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 Q=2^(level-1)
and then the 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
H. Lee Price, "The Pythagorean Tree: A New Species", 2008, <http://arxiv.org/abs/0809.4324>.
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011 Kevin Ryde
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