Thursday, September 22, 2016

Entry 3

Ramanujan gave the beautiful continued fractions (with the second simplified by this author) 51/4ϕϕ=e2π/51+e2π1+e4π1+e6π1+
51+ϕ1(ϕ5+ϕ10+1)1/5ϕ=e2π/51+e2π51+e4π51+e6π51+
(ϕ5+ϕ10+1)1/5=e2π/(55)1+e2π/51+e4π/51+e6π/51+
with the golden ratio ϕ=1+52. In 1913, the British mathematician G.H. Hardy, after reading the letter Ramanujan sent to him (which included examples of these extraordinary continued fractions), remarked, “…the [theorems] defeated me completely; I had never seen anything in the least like them before.” He would have been even more amazed had he known that these were connected to geometry.

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