I have a question at the very first assumption in the two squares case:
How did you make the assumption (assuming precalculus and calculator days and all that) that
2.8
Hi Ashwin, the comment form may have eaten your comment.
1/sqrt(2) comes from the Pythagorean theorem – it’s actually sqrt(2)/2 (which is the same thing), and sqrt(2) can be approximated using various algorithms: it’s more than 1, less than 2. It’s more than 5/4 (5/4 squared = 25/16 which is less than 2), and less than 6/4 (6/4 squared is 36/16 which is more than 2).
Not sure if that was the question but feel free to ask again, sorry about the form.
in your spreadsheet pi = 355/133 , u can knock this down to 22/7
Hi phyu, 22/7 is an approximation for pi, but it isn’t as accurate as 355/113. Check out the “Cranking the Formula” section for more details.
@ phyu
355/113 actually simplifies to 22/~7.0028169014084507042253521126761
the higher up in the fractions you go the farther from 7 the bottom number becomes, which falls right in line with the numeration given in Kalid’s chart.
Also @ Kalid very nice job in the compilation, always nice to see some interesting math facts!
such a great article! i was really happy that you included that little bit of life lesson at the end there. i have a tattoo of pi to remind myself that life doesn’t always make perfect sense :]
@Holy: Thanks for the comment and additional details!
@Anna: Glad you enjoyed it! Don’t think I’ve ever met anyone with a pi tattoo but that’s pretty intriguing :). Yep, I think math (or any subject) should enhance your outlook, not just teach facts.
Quite an enlightening article. The basics are all so clearly explained. Thank you very much.
Hi there. First of all, thanks for the article. I think I’ve got a silly question., but it’s driving me nuts!
My intuition keeps telling me that the inside perimeter (sin(x/2) above) and outside perimeter (tan(x/2) above) should be the same equation - it’s the same shape, just bigger, so the formula should be the same with larger values for x.
Can you tell me what I’m missing?
Hi Karl, that’s a great question! I had to think about it a bit.
You’re right, the two shapes (large and small square, large and small octagon) should have the same formula, scaled by some amount. The tricky thing is to realize that x/2 (the angle) should be the same in both cases; the angles don’t change no matter what size square you have.
You want to start with a formula (call it f(x) ) and scale it by some amount, called C: f(x) and C * f(x).
Looking closer, this is what’s happening: sin(x/2) is the basic formula, and tan(x/2) is really sin(x/2) / cos(x/2).
Since cosine is between 0-1, the division will actually be a multiplication or scaling. So tan(x/2) is always larger than sin(x/2), giving us the scaling factor we need.
Again, great question – sin(x/2) and tan(x/2) are really the same formula, but scaled by 1/cos(x/2). Phew :).
@Jo: Thanks, glad it was helpful.
[…] Archimedes “This site is a collection of Archimedean miscellanea under continual development.” See also: Archimedes’ Approximation of Pi, and The Archimedes Palimpsest. Edited to add: Prehistoric Calculus: Discovering Pi. […]
[…] Wow! No crazy formulas, no pi floating around — just multiply to convert rotational speed to linear speed. All because radians speak in terms of the mover. […]
Great articles Kalid, any similar insights or an intuitive approach you could share on eulers identity ? This explanation for pi is the one we were actually thought in school, and we were thought about e through continuous growth. (as in the article on e on this site). Both make perfect sense to me but I’m still blown away by eulers identity ( e(i.pi)+1 = 0 ). What is the meaning of this relation between e, i and pi … is there an intuitive way to look at this you are aware of ?
[…] Prehistoric Calculus: Discovering Pi […]
@enki: Whoops, sorry about the late response, think I missed this. There is an intuitive way to approach Euler’s identity that I’d like to write about (the book Visual Complex Analysis has a take on it, which I highly recommend). Basically, you can view it as a linkage between growth and rotation – but I’ll be writing about this topic in the future.
The minimum number of side to get 100% accuracy is 4070364
@Geo: Yep, that’s the point at which the calculator can’t tell the difference :).
Great article. This is one of the reasons why I have heard the circle referred to as an “infinigon.”