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:
“Preoccupied with her self, the adolescent sees enormous changes, whereas the parent sees the child she knew all along. For the parent, new developments are superficial and evanescent. For the adolescent, they are thrilling and profound.”
—Terri Apter (20th century)
“To deny the need for comprehensive child care policies is to deny a reality—that there’s been a revolution in American life. Grandma doesn’t live next door anymore, Mom doesn’t work just because she’d like a few bucks for the sugar bowl.”
—Editorial, The New York Times (September 6, 1983)