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:
“Discipline is a symbol of caring to a child. He needs guidance. If there is love, there is no such thing as being too tough with a child. A parent must also not be afraid to hang himself. If you have never been hated by your child, you have never been a parent.”
—Bette Davis (1908–1989)
“There must be a solemn and terrible aloneness that comes over the child as he takes those first independent steps. All this is lost to memory and we can only reconstruct it through analogies in later life....To the child who takes his first steps and finds himself walking alone, this moment must bring the first sharp sense of the uniqueness and separateness of his body and his person, the discovery of the solitary self.”
—Selma H. Fraiberg (20th century)