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When I was young, I found it amazing when I realized that there were an infinite number of primitive Pythagorean triples that had one leg that was one less than the hypotenuse. This was one of those tricks that you learned in popular math books, take any odd integer to serve as one leg, then square it and divide as evenly as possible to get the other leg and hypotenuse. For one leg of 7, the square is 49, and 49 can be almost evenly divided into 24 and 25, so 7, 24, 25 is a primitive triple. On the Barning tree I noticed that each of these was created by starting with P=(4,3,5) and applying the first matrix operation, AP gives (12,5,13), A2P gives (24, 7,25) and so on.
In looking at the tree, I realized that each transformation matrix preserved one arithmetic difference in the three values. AP for any point P would preserve the difference between the first leg and hypotenuse. If we let P = (20,21,29) the difference is nine, and transformations by A preserve this so that AP = (80,39, 89) and A2P gives (176,57,185).
If we wish to preserve the difference between the second and third value of the point, we transform by matrix C. Keeping P = (20,21,29) the transformation CP gives (36,77,85), and C2P gives (52, 165, 173).
Matrix B then, preserves the difference between the legs. If we take (3,4,5) and repeatedly apply the transformation B, we get (21, 20, 29); (119, 120, 169); etc.
I’m sure there is more hidden in the fabric of Barning trees, so if you see what I didn’t, drop me a line.