On 12/12/05, David Gale <gale@math.berkeley.edu> wrote:

...

Moving on a little, if D is superposed onto A, then its image D' = A lies on AB, and the "triangle" referred to appears to have zero area. Might this submission have benefitted from more careful presentation?

Gosh, I was hoping someone would grab a piece of 8.5 by 11 and actually do it. The image of BD (under the folding map) is the broken line forming the hypotenuse and one side of the triangle. This of course only gives the "shape" of the Pyth. triple, which is all one could expect.

The light dawns. Around 1955 (ouch!) I visited a secondary school in Hamburg, where I attended (amongst others) a mathematics class. The subject was the (quadratic) relations between the sides of a right-angled triangle and those of the smaller (similar) triangles formed by dropping an altitude to the hypoteneuse. I have to admit I was quite fascinated, and went away to iterate the procedure manically and generate cubics and quartics by the bucketful. But subsequently I'd never seen the idea mentioned again, until now. I somehow had the impression then that the Germans found geometry fun, in a way that the British did not. [\end of sweeping generalisation]