Date: Wed, 15 Feb 2012 17:38:06 -0800 From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] Curses, anticipated again. Neil has just uncovered fq.math.ca/Scanned/6-6/hoggatt.pdf predating my http://mathworld.wolfram.com/SquareDissection.html of a square into ten acute isosceles triangles. Neil also found in a Gardner book a reference to a Monthly paper (June-July 1962, pp550-552) claiming that *any* obtuse triangle can be cut into eight acute isosceles triangles, implying at most nine for dissecting a right isosceles, in contradiction of my round(tan(69)) solutions assertion. Has anyone a picture of this dissection? --rwg This is weird: Calculating with pen and paper for a change, it seems that sunlight falling on the diagrams and equations actually makes them *easier* to read. http://mathworld.wolfram.com/SquareTiling.html M. Laczkovich has shown that there are exactly three shapes of non-right triangles that tile the square with similar copies, corresponding to angles (pi/8,pi/4,5pi/8), (pi/4,pi/3,5pi/12), and (pi/12,pi/4,2pi/3) (Stein and Szabó 1994). in degrees that is (22.5, 45, 112.5), (45,60,75), and (15, 45, 120) The Gosper-Hoggatt tilings use these angles. Stuart