According to Wikipedia:

Although it was known to Gauss by 1801 that the regular 65537-gon was constructible, the first explicit construction of a regular 65537-gon was given by Johann Gustav Hermes (1894). The construction is very complex; Hermes spent 10 years completing the 200-page manuscript.

As Greg noted (somewhere, years ago), the appearance of is a bit of a red herring here. The ‘generalized Viète formula” which lets us compute pi starting from any n-gon is equivalent to Euler’s formula

and this formula is probably the most elegant distillation of what’s going on.

]]>I saw this video about how if we draw a spiral from a circle to a bigger circle in the ration of 1.68, then the LENGTH of that spiral is pi itself.

The maker of this video doesn’t have any contact. Can someone please confirm this??

And if yes, what is the unit of measuring the 3.14??

]]>That makes me very happy to have stumbled on this vaguely similar—but much less profound—formula:

Thanks for telling me about Carr’s work!

]]>I guess it isn’t so surprising that Carr gives this proof but it’s fun to think about how much it would have delighted Ramanujan.

BTW, I first learned about Ramanujan from your lecture about the number 5, so thanks for that.

]]>While it’s true that the golden number is a special case, the way it enters into the formula i.e. cos(pi/5)=phi/2 is elegant. This relates a simple trig function of pi with phi, and indeed a simple trig function can be related to a simple shape like a regular pentagon (and vice versa). The golden number is a simple expression in terms of a radical [sqrt(5)] unlike pi; in this sense pi is the greater enigma. The open question is: what other elegant ways are there of proving this connection? ]]>