Pythagoras and Matrices

I first learned about Barning trees in 2008.  Over the years people much brighter than I am, chipped in to tell me more.  This is a consolidation of posts since then that I hope will be readable and informative.  I find the ideas here fascinating.  Hope you enjoy


You can click on any image to expand it and make it sharper

It’s been a good weekend for Geometry for me. Several notes from folks telling me about geometry stuff I never knew… While I’m waiting on permission from the author of one, I wanted to tell you about the other.. The graph (tree) below shows a set of Primitive Pythagorean triples… All the ones with hypotenuse less than 100. Primitive Pythagorean triples are right triangles that have all sides as integers and none of them have a common factor.

What was new (to me) was that any one of them (and all the others not shown here) can be found as transformations of (4,3,5) using only three transformations. Let me make that clearer. If you think of a primitive Pythagorean triple as a point in three-space, then any other primitive Pythagorean triple is a point in three space that is just a transformation of this one… but there are only three transformations needed to get ALL of them. The tree shows which ones are generated by which ones, but you need to crank out the calculator and do some of these to see how neat it really is.
This type of graph is called a Barning-Tree because it seems to have first been discovered by F. J. M. Barning, “On Pythagorean and quasi-Pythagorean triangles and a generation process with the help of unimodular matrices, (Dutch) Math. Centrum Amsterdam Afd. Zuivere Wisk, ZW-011 (1963) 37 pp..
If you take the three matrices below, and write any of the primitive triples as a column vector, then multiplying by any of the matrices will give you another unique triple.. they never duplicate one, and they don’t leave any out (OK, I’m taking that on faith as I haven’t proven it for myself yet). I think that is kind of wild, and am totally impressed with people who can notice stuff like that.
Here are the three transformations