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:
“Being a parent is a form of leadership. . . . Parents make a mistake, along with leaders of organizations, when they are unwilling to recognize the power inherent in the positions they occupy and when they are unwilling to use this power. . . . I do not mean a figure who is irrational, autocratic, or sadistic. I mean leaders who have the strength of character to stand up for what they believe.”
—Abraham Zaleznik. In Support of Families, ed. Michael W. Yogman and T. Berry Brazelton, ch. 8 (1986)
“Although my parents have never been the kind to hint around about grandchildren, I can think of no better tribute to them than giving them some.... I can’t help thinking that the cycle is not complete until I can introduce them to a child of their child. And I can think of no better comfort when they are gone than to know that something of them lives on, not only in me but in my children.”
—Anne Cassidy. “Every Child Should Have a Father But....,” McCall’s (March 1985)