Provided by: libmath-planepath-perl_129-1_all 
NAME
Math::PlanePath::PythagoreanTree -- primitive Pythagorean triples by tree
SYNOPSIS
use Math::PlanePath::PythagoreanTree;
my $path = Math::PlanePath::PythagoreanTree->new
(tree_type => 'UAD',
coordinates => 'AB');
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This path enumerates primitive Pythagorean triples by a breadth-first traversal of one of three ternary
trees,
"UAD" Berggren, Barning, Hall, and others
"FB" Firstov and Price
"UMT" Firstov
Each X,Y point is a pair of integer A,B sides of a right triangle. The points are "primitive" in the
sense that 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
(all integers)
<-A->
A primitive triple always has one of A,B odd and the other even. The trees here have 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. See the author's mathematical write-up for
more properties.
<http://user42.tuxfamily.org/triples/index.html>
UAD Tree
The UAD tree by Berggren (1934) and later independently by Barning (1963), Hall (1970), and 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.
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 UAD matrices are
/ 1 -2 2 \ / 1 2 2 \ / -1 2 2 \
U = | 2 -1 2 | A = | 2 1 2 | D = | -2 1 2 |
\ 2 -2 3 / \ 2 2 3 / \ -2 2 3 /
They're multiplied on the right of an (A,B,C) column vector, for example
/ 3 \ / 5 \
U * | 4 | = | 12 |
\ 5 / \ 13 /
The starting point is N=1 at X=3,Y=4 which is the well-known triple
3^2 + 4^2 = 5^2
From it three further points N=2, N=3 and N=4 are derived, then three more from each of those, etc,
tree_type => "UAD" coordinates A,B
______________ 3,4 _____________
/ | \
5,12 21,20 15,8
/ | \ / | \ / | \
7,24 55,48 45,28 39,80 119,120 77,36 33,56 65,72 35,12
rows depth = 0 N=1
depth = 1 N=2..4
depth = 2 N=5..13
depth = 3 N=14..
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
Nrow = 1 + (1 + 3 + 3^2 + ... + 3^(depth-1))
= (3^depth + 1) / 2
= 1, 2, 5, 14, 41, 122, 365, ... (A007051)
The level numbering is like a mixed-radix representation of N where the high digit is binary (so always
1) and the digits below are ternary.
+--------+---------+---------+-- --+---------+
N = | binary | ternary | ternary | ... | ternary |
+--------+---------+---------+-- --+---------+
1 0,1,2 0,1,2 0,1,2
The number of ternary digits is the "depth" and their value without the high binary 1 is the position in
the row.
U Repeatedly
Taking the upper "U" matrix repeatedly gives
3.4 -> 5,12 -> 7,24 -> 9,40 -> etc
with C=B+1 and A the odd numbers. 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 = any odd integer, so A^2 any odd square
B = (A^2-1)/2
C = (A^2+1)/2
/ A^2-1 \ / A^2+1 \
A^2 + | ------ |^2 = | ----- |^2
\ 2 / \ 2 /
This is also described by Fibonacci (Liber Quadratorum) 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. The length of the gnomon is odd and when it's an odd
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.
A Repeatedly
Taking the middle "A" matrix repeatedly gives
3,4 -> 21,20 -> 119,120 -> 697,696 -> etc A,B legs
which are the triples with legs A,B differing by 1 and so just above and below the X=Y leading diagonal.
The N values are 1,3,9,27,etc = 3^depth.
D Repeatedly
Taking the lower "D" matrix repeatedly gives
3,4 -> 15,8 -> 35,12 -> 63,16 -> etc A,B legs
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 k even >= 2 for primitives
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()".
UArD Tree
Option "tree_type => "UArD"" varies the UAD tree by applying a left-right reflection under each "A"
matrix. The result is ternary reflected Gray code order. The 3 children under each node are unchanged,
just their order.
tree_type => "UArD" coordinates A,B
______________ 3,4 _____________
/ | \
5,12 21,20 15,8
/ | \ / | \ / | \
7,24 55,48 45,28 77,36 119,120 39,80 33,56 65,72 35,12
Notice the middle points 77,36 and 39,80 are swapped relative to the UAD shown above. In general, the
whole tree underneath an "A" is mirrored left <->right. If there's an even number of "A"s above then
those mirrorings cancel out to be plain again.
This tree form is primarily of interest for "Digit Order Low to High" described below since it gives each
row of points in order clockwise down from the Y axis.
In "PQ Coordinates" below, with the default digits high to low, UArD also makes successive steps across
the row either horizontal or 45-degrees NE-SW.
In all cases, the Gray coding is applied to N first, then the resulting digits are interpreted either
high to low (the default) or low to high ("LtoH" option).
FB Tree
Option "tree_type => "FB"" selects a tree independently by
V. E. Firstov, "A Special Matrix Transformation Semigroup of Primitive Pairs and the Genealogy of
Pythagorean Triples", Matematicheskie Zametki, 2008, volume 84, number 2, pages 281-299 (in Russian),
and Mathematical Notes, 2008, volume 84, number 2, pages 263-279 (in English)
H. Lee Price, "The Pythagorean Tree: A New Species", 2008, <http://arxiv.org/abs/0809.4324> (version
2)
Firstov finds this tree by semigroup transformations. Price finds it by 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 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.
tree_type => "FB" coordinates A,B
______________ 3,4 _____________
/ | \
5,12 15,8 7,24
/ | \ / | \ / | \
9,40 35,12 11,60 21,20 55,48 39,80 13,84 63,16 15,112
UMT Tree
Option "tree_type => "UMT"" is a third tree type by Firstov (reference above). It is matrices U, M2, and
a new third T = M1*D.
tree_type => "UMT" coordinates A,B children
U,M2,T
______________ 3,4 _____________
/ | \
5,12 15,8 21,20
/ | \ / | \ / | \
7,24 35,12 65,72 33,56 55,48 45,28 39,80 91,60 105,88
The first "T" child 21,20 is the same as the "A" matrix, but it differs at further levels down. For
example "T" twice is 105,88 (bottom most in the diagram) which is not the same as "A" twice 119,120.
Digit Order Low to High
Option "digit_order => 'LtoH'" applies matrices using the ternary digits of N taken from low to high.
The points in each row are unchanged, as is the parent-child N numbering, but the X,Y values are
rearranged within the row.
The UAD matrices send points to disjoint regions and the effect of LtoH is to keep the tree growing into
those separate wedge regions. The arms grow roughly as follows
tree_type => "UAD", digit_order => "LtoH"
Y=80 | 6 UAD LtoH
| /
| /
Y=56 | / 7 10 9
| / / / /
| / / | / 8
| / _/ / / /
| / / / / /
Y=24 | 5 / / | / _/ __--11
| / / _/ |/_/ __--
Y=20 | / / / __3 __-- _____----12
| |/_/ __-- __--- ____-----
Y=12 | 2 __-- _/___---- ____13
| / __-- __-- _____-----
| /_--_____---4-----
Y=4 | 1---
|
+--------------------------------------------------
X=3 X=15 X=20 X=35 X=76
Notice the points of the second row N=5 to N=13 are almost clockwise down from the Y axis, except
N=8,9,10 go upwards. Those N=8,9,10 go upwards because the A matrix has a reflection (its determinant is
-1).
Option "tree_type => "UArD"" reverses the tree underneath each A, and that plus LtoH gives A,B points
going clockwise in each row. P,Q coordinates likewise go clockwise.
AC Coordinates
Option "coordinates => 'AC'" gives the A and C legs of each triple as X=A,Y=C.
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 all above the X=Y diagonal. The "D Repeatedly" triples described
above have C=A+2 so they are the points Y=X+2 just above the diagonal.
For the FB tree the set of points visited is the same (of course), but 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
BC Coordinates
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"
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 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 have C=B+1 and are at the start of each
tree row.
For the FB tree, the set of points visited is the same of course, but a different N numbering.
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
As seen in the diagrams, 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
= (C+B)*(C-B)
Then gcd(A,B)=1 means also gcd(C+B,C-B)=1 and so since C+B and C-B have no common factor they must each
be squares to give A^2. Call them s^2 and t^2,
C+B = s^2 and conversely C = (s^2 + t^2)/2
C-B = t^2 B = (s^2 - t^2)/2
s = odd integer s >= 3
t = odd integer 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 for each s. As t
increases, the B,C coordinate combination makes a line upwards at 45-degrees from those t=1 positions,
C + B = s^2 anti-diagonal 45-degrees,
position along diagonal determined by t
All primitive triples start from a C=B+1 with C=(s^2+1)/2 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 gcd(s,t)!=1 are gaps in the
anti-diagonal lines.
PQ Coordinates
Primitive Pythagorean triples can be parameterized as follows for A odd and B even. This is per Euclid,
Diophantus, and anonymous Arabic manuscript for constraining it to primitive triples (Dickson's History
of the Theory of Numbers, start of chapter IV).
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, for either "tree_type". Because P>Q>=1 the values
fall in the eighth of the plane below the X=Y diagonal,
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 all 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
(leg "A" is already too close to matrix "A"!).
SM Coordinates
Option "coordinates => 'SM'" gives the small and medium legs from each triple as X=small,Y=medium. This
is like "AB" except that if A>B then they're swapped to X=B,Y=A so X<Y always. The effect is to mirror
the AB points below the X=Y diagonal up to the upper half quadrant,
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
SC Coordinates
Option "coordinates => 'SC'" gives the small leg and hypotenuse from each triple,
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 < sqrt(2)*Y since with X as the smaller leg must have X^2 < Y^2/2 so
X < Y*1/sqrt(2).
MC Coordinates
Option "coordinates => 'MC'" gives the medium leg and hypotenuse from each triple,
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 sqrt(2)*Y < X < Y. X is the bigger leg and X^2 > Y^2/2 so X > Y*1/sqrt(2).
UAD Coordinates AB, AC, PQ -- Turn Right
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.
UAD Coordinates BC -- Turn Left
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.
FUNCTIONS
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 => $str)"
Create and return a new path object. The "tree_type" option can be
"UAD" (the default)
"UArD" UAD with Gray code reflections
"FB"
"UMT"
The "coordinates" option can be
"AB" odd, even legs (the default)
"AC" odd leg, hypotenuse
"BC" even leg, hypotenuse
"PQ"
"SM" small, medium legs
"SC" small leg, hypotenuse
"MC" medium leg, hypotenuse
The "digit_order" option can be
"HtoL" high to low (the default)
"LtoH" low to high (the default)
"$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 = $path->xy_to_n ($x,$y)"
Return the point number for coordinates "$x,$y". If there's nothing at "$x,$y" then return "undef".
The return is "undef" if "$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.
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, $n_hi is roughly "3^max(x/2)", or for
FB smaller at roughly "3^log2(x)".
Descriptive Methods
"$x = $path->x_minimum()"
"$y = $path->y_minimum()"
Return the minimum X or Y occurring in the path. The value goes according to the "coordinates"
option,
coordinate minimum
---------- -------
A,S 3
B,M 4
C 5
P 2
Q 1
Tree Methods
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).
This is simply "3*$n-1, 3*$n, 3*$n+1". This is appending an extra ternary digit 0, 1 or 2 to the
mixed-radix form for N described above. Or staying all in ternary then appending to N+1 rather than
N and adjusting back.
"$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).
This is simply "floor(($n+1)/3)", reversing the "tree_n_children()" calculation above.
"$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 $depth in the path, or "undef" if nothing at that depth or
not a tree. The top of the tree is depth=0.
Tree Descriptive Methods
"$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.
FORMULAS
UAD Matrices
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 ( 2 -1 )
Q -> P ( 1 0 )
A P -> 2P+Q ( 2 1 )
Q -> P ( 1 0 )
D P -> P+2Q ( 1 2 )
Q -> Q unchanged ( 0 1 )
The advantage of P,Q for the calculation is that it's 2 values instead of 3. The transformations can be
written with the 2x2 matrices shown, 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
FB Transformations
The FB tree is calculated in P,Q and converted to A,B at the end as necessary. Its three transformations
are
M1 P -> P+Q ( 1 1 )
Q -> 2Q ( 0 2 )
M2 P -> 2P ( 2 0 )
Q -> P-Q ( 1 -1 )
M3 P -> 2P ( 2 0 )
Q -> P+Q ( 1 1 )
Price's paper shows rearrangements of a set of four values q',q,p,p'. Just the p and q are enough for
the calculation. The set of four has some attractive geometric interpretations though.
X,Y to N -- UAD
"xy_to_n()" works in P,Q coordinates. An A,B or other input is converted to P,Q per the formulas in "PQ
Coordinates" above. P,Q can be reversed up the UAD tree to its parent point
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,
however it was converted, was not a primitive triple. For a primitive triple the endpoint is always
P=2,Q=1.
The conditions P>3Q or P>2Q work because each matrix sends its parent P,Q to one of three disjoint
regions,
Q P=Q P=2Q P=3Q
| * U ---- A ++++++
| * ---- ++++++
| * ---- ++++++
| * ---- ++++++
| * ---- ++++++
| * ---- ++++++
| * ---- ++++++ D
| * ----++++++
| * ----++++
| ----++
|
+------------------------------------------------- P
So U is the upper wedge, A the middle, and D the lower. The parent P,Q can be anywhere in P>Q>=1, the
matrices always map to these regions.
X,Y to N -- FB
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 M1 P -> P-Q
(Q even) Q -> Q/2
if P > 2Q reverse M2 P -> P/2
(P even) Q -> P/2 - Q
otherwise reverse M3 P -> P/2
(P even) 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.
The M1 reversal works because M1 sends any parent P,Q to a child which has P odd. All odd P,Q come from
M1. The M2 and M3 always make children with P even. Those children are divided between two disjoint
regions above and below the line P=2Q.
Q P=Q P=2Q
| * M3 P=even ----
| * ----
| * ----
| * ----
| * ---- M2 P=even
| * ----
| * ----
| * ----
| * ---- M1 P=odd anywhere
| ----
|
+------------------------------------------------- P
X,Y to N -- UMT
After converting to P,Q as necessary, a P,Q point can be reversed up the UMT tree to its parent
if P > 2Q reverse "U" P -> Q
digit=0 Q -> 2Q-P
if P even reverse "M2" P -> P/2
(Q odd) Q -> P/2 - Q
otherwise reverse "T" P -> P - 3 * Q/2
(Q even) Q -> Q/2
These reversals work because U sends any parent P,Q to a child P>2Q whereas the M2 and T go below that
line. M2 and T are distinguished by M2 giving P even whereas T gives P odd.
Q P=Q P=2Q
| * U ----
| * ----
| * ----
| * ----
| * ---- M2 for P=even
| * ---- T for P=odd
| * ----
| * ----
| * ----
| ----
|
+------------------------------------------------- P
Rectangle to N Range -- UAD
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.
Rectangle to N Range -- FB
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 M1 steps half-way then a M2 step, then M1 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 M1 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 M2 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.
OEIS
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 row middle line, 3^n
A058529 possible values taken by abs(A-B),
being integers with all prime factors == +/-1 mod 8
UAD Tree HtoL
A321768 leg A
A321769 leg B
A321770 hypot C
A321782 P (and the same for LtoH)
A321783 Q
A321784 P+Q
A321785 P-Q
UAD Tree
A001542 row total p (even Pells)
A001653 row total q (odd Pells)
A001541 row total p + total q
A002315 row total p - total q
"U" repeatedly
A046092 leg B, 2n(n+1) = 4*triangular numbers
A099776 \ hypot C, being 2n(n+1)+1
A001844 / which is the "centred squares"
"A" repeatedly
A046727 \ leg A
A084159 / "Pell oblongs"
A046729 leg B
A001653 hypot C, numbers n where 2*n^2-1 is square
A000129 P and Q, the Pell numbers
A001652 leg S, the smaller
A046090 leg M, the bigger
"D" repeatedly
A000466 leg A, being 4*n^2-1 for n>=1
"M1" repeatedly
A028403 leg B, binary 10..010..000
A007582 leg B/4, binary 10..010..0
A085601 hypot C, binary 10..010..001
"M2" repeatedly
A015249 \ leg A, binary 111000111000...
A084152 |
A084175 /
A054881 leg B, binary 1010..1010000..00
"M3" repeatedly
A106624 P,Q pairs, 2^k-1,2^k
"T" repeatedly
A134057 leg A, binomial(2^n-1,2)
binary 111..11101000..0001
A093357 leg B, binary 10111..111000..000
A052940 \
A055010 | P, 3*2^n-1
A083329 | binary 10111..111
A153893 /
SEE ALSO
Math::PlanePath, Math::PlanePath::Hypot, Math::PlanePath::RationalsTree, Math::PlanePath::CoprimeColumns
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
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