Or as an arctangent identity: pi/2 = atan(+inf) = atan(13/84) + atan(5/12) + atan(8/15) + atan(33/56) or as a complex factorization pi/2 = Imag( log( (84+13I) (12+5I) (15+8I) (56+33I) )). The factorization looks hard-to-find, but can be built up from the complex factors of 5, 13, and 17 -- 2+-I, 3+-2I, 4+-I. Rich ------ Quoting Tom Rokicki <rokicki@gmail.com>:
[Can't believe I'm responding to a Gosper post . . .]
Pi/2 == ArcSin[13/85] + ArcSin[5/13] + ArcSin[8/17] + ArcSin[33/65]
This one is geometrically obvious, from four right triangles in the 2D lattice, following the sequence of points
(0,56) (33,56) (48,36) (60,45) (84,13) (84,0)
I wonder how many similar identities can be derived just from pulling right triangles from the lattice.
-tom
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