Pythagorean Triples--Convergent Sequences
by Richard Brown
In the the previous section, we looked much more carefully at the side angles of Pythagorean triples. The side angles were the next step in the development of angle measure after the straight line and the right angle in ancient Sumer and Babylon. In this section we will look at one of the results of that research, convergent sequences of Pythagorean Triples. The first such sequences is a table of triples that converge to the (3,4,5) triangle from above and below.
Exhibit I. Pythagorean Triples Convergent Sequences for the (3,4,5) Triple from Above and Below
| Number | Sequence | Generators u,v | Side A | Side B | Diagonal | Angle |
|---|
| 1 | 1 | 1 | 2 | 3 | 4 |
5 | 53.130102 |
| 5 | 2 | 2 | 5 | 21 | 20 | 29 | 43.602820 |
| 12 | 3 | 3 | 8 | 55 | 48 | 73 | 41.112091 |
| 21 | 4 | 4 | 11 | 105 | 88 | 137 | 39.966214 |
| 95 | 8 | 8 | 23 | 465 | 368 | 593 | 38.358017 |
| 392 | 16 | 16 | 47 | 1953 | 1504 | 2465 | 37.599771 |
| 1599 | 32 | 32 | 95 | 8001 | 6080 | 10,049 | 37.231385 |
| 106,799,254 | 8192 | 8192 | 24,575 | 536,821,761 | 402,636,800 | 671,039,489 | 36.871297 |
|
|
|
| | 1 | 2 | 4 | 3 | 5 | 36.869898 |
|
|
|
| 106,814,899 | 8192 | 8192 | 24,577 | 536,920,065 | 402,669,568 | 671,137,793 | 36.868500 |
| 1661 | 32 | 32 | 97 | 8385 | 6208 | 10,433 | 36.515127 |
| 423 | 16 | 16 | 49 | 2145 | 1568 | 2657 | 36.166891 |
| 109 | 8 | 8 | 25 | 561 | 400 | 689 | 35.489344 |
| 29 | 4 | 4 | 13 | 153 | 104 | 185 | 34.205459 |
| 18 | 3 | 3 | 10 | 91 | 60 | 109 | 33.398489 |
| 8 | 2 | 2 | 7 | 45 | 28 | 53 | 31.890792 |
| 3 | 1 | 1 | 4 | 15 | 8 | 17 | 28.072487 |
I found this result in my initial studies of side angles of Pythagorean Triples inspired by Dr. David Joyce's paper on the Plymton 322 tablet(see preceding sections for more on this). When I sorted the triples by small side angle, I observed that the smaller the Pythagorean triple number(or alternatively, the smaller the diagonal value), the larger the angular measure or "halo" around the the triple's small angle. The (3,4,5) Triple having the lowest number(1)and the smallest diagonal value(5) had the largest anglar measure or "halo" around it for any sorting of the Pythagorean triples by small angle. This explained the anomaly observed in Dr. Joyce's paper of having to fill a hole in the space next to the (3,4,5) triangle.
My finding was a pair of equations that defined for each Primitive Pythagorean triple, the size of the "halo" for each size of table of triples(Pythagorean triple number or diagonal measure)! For the(3,4,5) triangle the equation from above is (N, 3N-1) as the,expression of the generators of the "halo" triples
from sequence numbers. The equation from below for the (3,4,5) triple is (N, 3N+1) for the generators of "halo" triples. N is a sequence number. The parameters of the equations are related to the sides of the initial triangle.
Exhibit II. Pythagorean Triples and Convergent
Sequence Equation Parameters from Above and Below
| Number | Generators u,v | Side A | Side B | Diagonal | Parameters:Above,Below |
|---|
| 1 | 1 | 2 | 3 | 4 | 5 |
N;3N-1 | N;3N+1 |
| 2 | 2 | 3 | 5 | 12 | 13 | N;5N-1 | N;5N+1 |
| 3 | 1 | 4 | 15 | 8 | 17 | 3N-2;5N-3 | 3N+2;5N+3 |
| 4 | 3 | 4 | 7 | 24 | 25 | N;7N-1 | N;7N+1 |
| 5 | 2 | 5 | 21 | 20 | 29 | 3N-1;7N-2 | 3N+1;7N+2 |
| 6 | 1 | 6 | 35 | 12 | 37 | 5N-3;7N-4 | 5N+3;7N+4 |
| 7 | 4 | 5 | 9 | 40 | 41 | N;9N-1 | N;9N+1 |
| 8 | 2 | 7 | 45 | 28 | 53 | 5N-4;9N-7 | 5N+4;9N+7 |
| 9 | 5 | 6 | 11 | 60 | 61 | N;11N-1 | N;11N+1 |
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Last update 12/2/1998