No it’s not. It’s close though: overestimates by about .00005.

]]>Pi is equal to ( 9 + 3*(5^(1/2)) )/5

]]>I’m a little late to this party. Neat! Another way of proving the identity in Puzzle 2 would be to use the Weierstrass infinite product formula for the sine and combine that with to get the infinite product for . With that, the Puzzle 2 identity basically follows from the fact that any integer has a unique representation as a power of 2 times an odd number (how it follows could be left as a puzzle).

]]>It’s correct, and mildly amusing, but not terribly interesting to a mathematician. The YouTube comment by Linus Brendel from 2 years ago is exactly right. The narrator (who oddly pronounces it “Fibonicci” and not “Fibonacci”, and who obscurely refers to “cubes”) is adding up lengths of just six quarter-circle arcs to get from A to B. The length of the largest quarter circle arc is , and each successive arc has the length of the previous. So the claim is that . A good high school mathematics student, given some time, can prove that with a little algebra and knowing that .

]]>That case is left as an exercise for the reader, along with the 257-gon and 65537-gon.

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??

]]>I’m glad to have introduced you to Ramanujan—he’s a really fascinating mathematician. My lecture about the number 5 mentioned this formula of his:

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!

]]>