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 "UAD" or "FB". Each point is an integer X,Y=A,B which has 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, no common factor X=A, Y=B ^ * ^ / /| | right triangle C / | B A side, odd / / | | B side, even v *---* v C hypotenuse <-A->
A primitive triple always has one of A,B odd and the other even. The trees here give triples ordered as A odd and B even.
The trees are traversed breadth-first and tend to go out to rather large A,B values while yet to complete smaller ones. The UAD tree goes out further than the FB.
The UAD tree by Berggren (1934) and later independently by Barning (1963), Hall (1970), and several other authors, uses three matrices U, A and D which can be multiplied onto an existing primitive triple to form three further new primitive triples. See "UAD Matrices" below for details of the 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=2,3,4 are derived, then three more from each of those, etc,
depth=0 depth=1 depth=2 depth=3 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 N=1 as depth=0, each level has 3^depth many points and the first N of a level tree_depth_to_n() is at
tree_depth_to_n()
Nrow = 1 + (1 + 3 + 3^2 + ... + 3^(depth-1)) = (3^depth + 1) / 2
The levels are like a mixed-radix representation of N where the high digit is binary and the digits below are ternary.
+--------+---------+---------+-- --+---------+ N = | binary | ternary | ternary | ... | ternary | +--------+---------+---------+-- --+---------+ 1 0,1,2 0,1,2 0,1,2
The high digit must be non-zero so is always 1. The number of ternary digits is the "depth" and their value without the high binary 1 is the position within the row.
Taking the middle "A" matrix repeatedly gives
3,4 -> 21,20 -> 119,120 -> 697,696 -> etc
which are the triples with legs A,B differing by 1 and therefore just above or below the X=Y leading diagonal. The N values are 1,3,9,27,etc = 3^depth.
Taking the lower "D" matrix repeatedly gives
3,4 -> 15,8 -> 35,12 -> 63,16 -> etc
which is the primitives among a sequence of triples known to the ancients (Dickson's History of the Theory of Numbers, start of chapter IV),
A = k^2-1 B = 2*k C = k^2+1 so C=A+2
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 at N=(3^(depth+1)-1)/2 which is tree_depth_to_n_end().
tree_depth_to_n_end()
Taking the upper "U" matrix repeatedly gives
3.4 -> 5,12 -> 7,24 -> 9,40 -> etc
with C=B+1. These are the first of each level so at Nrow described above. The resulting triples are a sequence known to Pythagoras (Dickson's History of the Theory of Numbers, start of chapter IV).
A = k k any odd integer B = (k^2-1)/2 so A^2 any odd square C = (k^2+1)/2 / k^2-1 \ / k^2+1 \ k^2 + | ------ |^2 = | ----- |^2 \ 2 / \ 2 /
This is also described by Fibonacci in his Liber Quadratorum (Book of Squares) in terms of sums of odd numbers
s = any odd square = A^2 B^2 = 1 + 3 + 5 + ... + s-2 = ((s-1)/2)^2 C^2 = 1 + 3 + 5 + ... + s-2 + s = ((s+1)/2)^2 so C^2 = A^2 + B^2 eg. s=25=A^2 B^2=((25-1)/2)^2=144 so A=5,B=12
The geometric interpretation is that an existing square of side B is extended by a "gnomon" around two sides making a new larger square of side C=B+1. If the length of the gnomon is a square then the new total area is the sum of two squares.
*****gnomon******* gnomon length an odd square = A^2 +--------------+ * | | * so new bigger square area | square | * C^2 = A^2 + B^2 | with side B | * | | * +--------------+ *
See Math::PlanePath::Corner for a path following such gnomons.
Option tree_type => "FB" selects the Fibonacci boxes tree by H. Lee Price
tree_type => "FB"
"The Pythagorean Tree: A New Species", 2008, http://arxiv.org/abs/0809.4324
This 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 so each is the sum of the preceding two in a fashion similar to the Fibonacci sequence. A box where p and q have no common factor corresponds to a primitive triple. See "PQ Coordinates" and "FB Transformations" 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 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
Option coordinates => 'AC' gives the A and C legs of each triple as X=A,Y=C.
coordinates => 'AC'
coordinates => "AC" 85 | 122 10 | | 73 | 6 | 65 | 11 40 61 | 41 | | 7 | | 41 | 14 | 13 35 | | 3 25 | 5 | 17 | 4 13 | 2 | Y=5 | 1 | +------------------------------------------- X=3 7 9 21 35 45 55 63 77
Since A<C the coordinates are X<Y so all above the X=Y diagonal. The "D Repeatedly" triples described have C=A+2 so Y=X+2 just above the diagonal.
For the FB tree the set of points visited is the same, but with a different N numbering.
tree_type => "FB", coordinates => "AC" 85 | 11 35 | | 73 | 9 | 65 | 23 12 61 | 7 | | 17 | | 41 | 5 | 6 35 | | 8 25 | 4 | 17 | 3 13 | 2 | Y=5 | 1 | +------------------------------------------- X=3 7 9 21 35 45 55 63 77
Option coordinates => 'BC' gives the B and C legs of each triple as X=B,Y=C. This is the B=even and C=long legs of all primitive triples. This combination has points on 45-degree straight lines.
coordinates => 'BC'
coordinates => "BC" 101 | 121 97 | 12 | 89 | 8 85 | 10 122 | | 73 | 6 | 65 | 40 11 61 | 41 | | 7 | | 41 | 14 | 13 35 | | 3 25 | 5 | 17 | 4 13 | 2 | Y=5 | 1 | +-------------------------------------------------- X=4 12 24 40 60 84
Since B<C the coordinates are X<Y and therefore above the X=Y leading diagonal. N=1,2,5,14,41,etc along the X=Y-1 diagonal are the "U Repeatedly" triples described above which are at the start of each depth level and have C=B+1.
tree_type => "FB", coordinates => "BC" 101 | 15 97 | 50 | 89 | 10 85 | 35 11 | | 73 | 9 | 65 | 12 23 61 | 7 | | 17 | | 41 | 5 | 6 35 | | 8 25 | 4 | 17 | 3 13 | 2 | Y=5 | 1 | +---------------------------------------------- X=4 12 24 40 60 84
The B,C points fall on 45-degree straight lines going up from X=Y-1. This occurs because a primitive triple A,B,C with A odd and B even can be written
A^2 = C^2 - B^2 A^2 = (C+B)*(C-B) gcd(A,B)=1 means gcd(C+B,C-B)=1 in this product, and therefore gcd(B,C)=1 so C+B = s^2 C = (s^2 + t^2)/2 C-B = t^2 B = (s^2 - t^2)/2 s = odd integer >= 3 t = odd integer, and s > t >= 1 with gcd(s,t)=1 so that gcd(C+B,C-B)=1
When t=1 this is C=(s^2+1)/2 and B=(s^2-1)/2 which is the "U"-repeated points at Y=X+1. As t increases the B,C coordinate combination makes a line upwards at 45-degrees,
C + B = s^2 anti-diagonal 45-degrees, position along diagonal determined by t
All primitive triples start from a C=B+1 for C=(s^2+1)/2, which is half an odd square, and go up from there. To ensure the triple is primitive must have gcd(s,t)=1. Values of t where that's not so are gaps in the lines.
Primitive Pythagorean triples can be parameterized as follows for A odd and B even. This is per Diophantus, and anonymous Arabic manuscript for constraining it to primitive triples.
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 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.
Option coordinates => 'PQ' gives these as X=P,Y=Q as (for either tree_type). Because P>Q>=1 the values fall in the eighth of the plane below the X=Y diagonal,
coordinates => 'PQ'
tree_type
tree_type => "UAD", coordinates => "PQ" 10 | 9842 9 | 3281 8 | 1094 23 7 | 365 32 6 | 122 38 5 | 41 8 4 | 14 11 12 15 3 | 5 6 16 2 | 2 3 7 10 22 1 | 1 4 13 40 121 Y=0 | +-------------------------------------------------------- X=0 1 2 3 4 5 6 7 8 9 10 11
The diagonal N=1,2,5,14,41,etc is P=Q+1 as per "U Repeatedly" above.
The one-to-one correspondence between P,Q and A,B means both tree types visit all P,Q pairs, so all X,Y with no common factor and one odd one even. There's other ways to iterate through such coprime pairs and any such method would generate Pythagorean triples too, in a different order from the trees here.
The letters P and Q here are a little bit arbitrary. This parameterization is often written m,n or u,v but don't want "n" to be confused that with N point numbering or "u" to be confused with the U matrix.
Option coordinates => 'SM' gives the small and medium values from each triple as X=small,Y=medium. This is like "AB" except that if A>B they're swapped to X=B,Y=A so that X<Y always. The effect is to fold the AB points below the X=Y diagonal up to the upper eighth,
coordinates => 'SM'
coordinates => "SM" 91 | 16 84 | 122 | 8 | 10 72 | 12 | | 60 | 41 40 | 11 55 | 6 | | 7 40 | 14 | 35 | 13 | 24 | 5 21 | 3 | 12 | 2 4 | Y=4 | 1 | +---------------------------------------- X=3 8 20 33 48 60 65
Option coordinates => 'SC' gives the small leg and hypotenuse from each triple,
coordinates => 'SC'
coordinates => "SC" 85 | 122 10 | | 73 | 6 | | 40 11 61 | 41 | 53 | 7 | | 41 | 14 37 | 13 | | 3 25 | 5 | | 4 13 | 2 | Y=5 | 1 | +----------------------------- X=3 8 20 33 48
The points are all X < 0.7*Y since with X as the smaller leg must have X^2 no more than half the hypotenuse Y^2 so X<Y*1/sqrt(2).
Option coordinates => 'MC' gives the medium leg and hypotenuse from each triple,
coordinates => 'MC'
coordinates => "MC" 65 | 11 40 61 | 41 | 53 | 7 | | 41 | 14 37 | 13 | 29 | 3 25 | 5 | 17 | 4 13 | 2 | Y=5 | 1 | +----------------------------------- X=4 15 24 35 40 56 63
The points are in a wedge 0.7*Y < X < Y. X is the bigger leg and X^2 is at least half the hypotenuse Y^2 so X>Y*1/sqrt(2).
In the UAD tree with coordinates AB, AC or PQ the path always turns to the right. For example in AB coordinates at N=2 the path turns to the right to go towards N=3.
coordinates => "AB" 20 | 3 N X,Y | -- ------ 12 | 2 1 3,4 | 2 5,12 | 3 21,20 4 | 1 | turn towards the +------------------------- right at N=2 3 5 21
This can be proved from the transformations applied to seven cases, a triplet U,A,D, then four crossing a gap within a level, then two wrapping around at the end of a level. The initial N=1,2,3 can be treated as a wrap-around from the end of depth=0 (the last case D to U,A).
U triplet U,A,D A D U.D^k.A crossing A,D to U U.D^k.D across U->A gap A.U^k.U k>=0 A.D^k.A crossing A,D to U A.D^k.D across A->D gap D.U^k.U k>=0 U.D^k.D crossing D to U,A U.U^k.U across U->A gap A.U^k.A k>=0 A.D^k.D crossing D to U,A A.U^k.U across A->D gap D.U^k.A k>=0 D^k .A wraparound A,D to U D^k .D k>=0 U^(k+1).U D^k wraparound D to U,A U^k.U k>=0 U^k.A (k=0 is initial N=1,N=2,N=3 for none,U,A)
The powers U^k and D^k are an arbitrary number of descents U or D. In P,Q coordinates these powers are
U^k P,Q -> (k+1)*P-k*Q, k*P-(k-1)*Q D^k P,Q -> P+2k*Q, Q
For AC coordinates squaring to stretch to P^2,Q^2 doesn't change the turns. Then a rotate by -45 degrees to A=P^2-Q^2, C=P^2+Q^2 also doesn't change the turns.
In the UAD tree with coordinates BC the path always turns to the left. For example in BC coordinates at N=2 the path turns to the right to go towards N=3.
coordinates => "BC" 29 | 3 N X,Y | -- ------ | 1 4,5 | 2 12,13 13 | 2 3 20,29 | 5 | 1 turn towards the | left at N=2 +--------------- 4 12 20
As per above A,C turns to the right, which squared is A^2,C^2 to the right too, which equals C^2-B^2,C^2. Negating the X coordinate to B^2-C^2,C^2 mirrors to be a left turn always, and addition shearing to X+Y,Y doesn't change that, giving B^2,C^2 always left and so B,C always left.
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
"UAD" (the default) "FB"
The coordinates option can be
coordinates
"AB" odd, even legs (the default) "AC" odd leg, hypotenuse "BC" even leg, hypotenuse "PQ" "SM" small, medium legs "SC" small leg, hypotenuse "MC" medium leg, hypotenuse
$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, per the coordinates option.
$rsquared = $path->n_to_radius ($n)
Return the radial distance R=sqrt(X^2+Y^2) of point $n. If there's no point $n then return undef.
For coordinates=AB or SM this is the hypotenuse C and therefore an integer, for integer $n.
($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)
Each point has 3 children, so the path is a complete ternary tree.
@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 appending an extra ternary digit 0, 1 or 2 to the mixed-radix form for N described above. Or staying all in ternary then appending to N+1 rather than N and 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 above.
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
$num = $path->tree_num_children_minimum()
$num = $path->tree_num_children_maximum()
Return 3 since every node has 3 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 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 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.
Repeatedly applying "U" gives
U 2P-Q, P U^2 3P-2Q, 2P-Q U^3 4P-3Q, 3P-2Q ... U^k (k+1)P-kQ, kP-(k-1)Q = P+k(P-Q), Q+k*(P-Q)
If there's a run of k many high zeros in the Nrem = N-Nrow position in the level then they can be applied to the initial P=2,Q=1 as
U^k P=k+2, Q=k+1 start for k high zeros
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. An A,B or other input is converted per the formulas in "PQ Coordinates" above. 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 digit=2 Q -> unchanged if P > 2Q reverse "A" P -> Q digit=1 Q -> P-2Q otherwise reverse "U" P -> Q digit=0 Q -> 2Q-P
This gives a ternary digit 2, 1, 0 respectively from low to high. Those plus a high "1" bit make N. The number of steps is the "depth" level.
If at any stage P,Q doesn't satisfy P>Q>=1, 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 to P,Q as 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>=1, 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.
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include,
http://oeis.org/A007051 (etc) A007051 N start of depth=n, (3^n+1)/2, ie. tree_depth_to_n() A003462 N end of depth=n-1, (3^n-1)/2, ie. tree_depth_to_n_end() A000244 N of "A repeatedly", 3^n
Math::PlanePath, Math::PlanePath::Hypot, Math::PlanePath::RationalsTree, Math::PlanePath::CoprimeColumns
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013 Kevin Ryde
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