At first I thought `gosh, an integer triangle with a 120° angle -- how amazing!'. Then I realised that it's basically just saying that |8 + 5w|² = 7², which is unremarkable when one considers that 8 + 5w is itself a square (of 3 + w), and the squared norm of 3 + w is 7. So basically we can create lots of these integer triangles. Starting from a + bw, we obtain: (a + bw)^2 = a^2 + 2abw + b^2w^2 = (a^2 - b^2) + (2ab - b^2)w which implies the existence of a 120° triangle with side lengths: a^2 - 2ab, 2ab - b^2, a^2 + b^2 - ab where we impose a > 2b to make everything positive. Your triad (3,5,7) is generated by setting (a,b) = (3,1). Incidentally, I seem to recall that the Ancient Egyptians marked out a loop of rope of length 12 units with marks at positions 0, 3, 7 and pulled it tight to function as a set-square. Your (3,5,7) triangle could presumably have been used as a similar set-hexagon...? Sincerely, Adam P. Goucher
Sent: Wednesday, October 08, 2014 at 9:36 PM From: "Bill Gosper" <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] Delayed reaction Hoohah? for waiting rooms
and other foci of boredom: gosper.org/3-5-7hex.pdf (font deficiency may require downloading) --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun