Parent/child Relationships
By a result of Berggren (1934), all primitive Pythagorean triples can be generated from the (3, 4, 5) triangle by using the three linear transformations T1, T2, T3 below, where a, b, c are sides of a triple:
| new side a | new side b | new side c | |
| T1: | a − 2b + 2c | 2a − b + 2c | 2a − 2b + 3c |
| T2: | a + 2b + 2c | 2a + b + 2c | 2a + 2b + 3c |
| T3: | −a + 2b + 2c | −2a + b + 2c | −2a + 2b + 3c |
If one begins with 3, 4, 5 then all other primitive triples will eventually be produced. In other words, every primitive triple will be a “parent” to 3 additional primitive triples. Starting from the initial node with a = 3, b = 4, and c = 5, the next generation of triples is
| new side a | new side b | new side c |
| 3 − (2×4) + (2×5) = 5 | (2×3) − 4 + (2×5) = 12 | (2×3) − (2×4) + (3×5) = 13 |
| 3 + (2×4) + (2×5) = 21 | (2×3) + 4 + (2×5) = 20 | (2×3) + (2×4) + (3×5) = 29 |
| −3 + (2×4) + (2×5) = 15 | −(2×3) + 4 + (2×5) = 8 | −(2×3) + (2×4) + (3×5) = 17 |
The linear transformations T1, T2, and T3 have a geometric interpretation in the language of quadratic forms. They are closely related to (but are not equal to) reflections generating the orthogonal group of x2 + y2 − z2 over the integers. A different set of three linear transformations is discussed in Pythagorean triples by use of matrices and_linear transformations. For further discussion of parent-child relationships in triples, see: Pythagorean triple (Wolfram) and (Alperin 2005).
Read more about this topic: Pythagorean Triple
Famous quotes containing the words parent and/or child:
“Parenting can be established as a time-share job, but mothers are less good “switching off” their parent identity and turning to something else. Many women envy the father’s ability to set clear boundaries between home and work, between being an on-duty and an off-duty parent.... Women work very hard to maintain a closeness to their child. Father’s value intimacy with a child, but often do not know how to work to maintain it.”
—Terri Apter (20th century)
“Every child has an inner timetable for growth—a pattern unique to him. . . . Growth is not steady, forward, upward progression. It is instead a switchback trail; three steps forward, two back, one around the bushes, and a few simply standing, before another forward leap.”
—Dorothy Corkville Briggs (20th century)