10170

# Primitive Pythagorean Triples 4: Ordered Tree Matrices

Given a primitive Pythagorean triple (PPT), three new PPTs, which we call children of the original, are generated by multiplying the original on the right by three fixed matrices called Hall matrices. Moreover, any PPT can be generated by a series of such right-multiplications starting from the PPT . Equivalently, we can generate the three new PPTs by right-multiplying "redundant forms" of the original PPT by a single matrix. The PPT has a total of eight redundant forms, , but only the four redundant forms, , are required.
In this Demonstration, the single matrix is denoted . The " derivation" button shows how the Hall matrices are computed as a combination of and other matrices described in Details, and the " button shows the generation of children using Hall matrices.

### DETAILS

Let be a PPT, that is, , , and are positive integers such that with . A primitive right triangle has sides that are a PPT.
Redundant form means that the PPT definition is relaxed to allow negative values.
Each PPT can be represented by a unique base-3 number called the PPT's ID number. The three children of a PPT have ID numbers formed by appending 0, 1, or 2 to the ID number of the PPT. The parent of a PPT has an ID number formed by dropping the last digit of the PPT's ID number.
Let be a PPT. When post-multiplied by any one of four 3x3 matrices , , , or , a new PPT is the result. These four matrices are:
, , , .
If a suitably selected redundant form of is used, then is the only matrix needed, as shown by clicking the matrix control "P". Also, gives a redundant form of the parent PPT, because .
However, the proper redundant form can be calculated by first multiplying by , , where
, , ;
that is, , , and . Set , , and . These are shown with the matrix control set to " and derivation".
The Demonstration "Primitive Pythagorean Triples 2: Ordered Pairs" shows that adding to each element of accomplishes the same thing as multiplication by .
Define ; then and , where is the 3×3 identity matrix.
Any PPT can be obtained by post-multiplying or by a unique chain of matrices. Their subscripts spell out a unique PPT ID number, as shown in "Primitive Pythagorean Triples 3: Ordered Tree Graph".
MathWorld's "Pythagorean Triple" and other sources use , , and , as in [1].
References
[1] B. Berggren, "Pytagoreiska trianglar" (in Swedish), Tidskrift för elementär matematik, fysik och kemi, 17, 1934 pp. 129–139.
[2] A. Hall, "Genealogy of Pythagorean Triads," Mathematical Gazette, LIV(390), 1970 pp. 377–379.
[4] H. Lee Price, "The Pythagorean Tree: A New Species", 2008 http://arxiv.org/abs/0809.4324.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.