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Pi and the Golden Ratio (johncarlosbaez.wordpress.com)
172 points by signa11 on Mar 28, 2017 | hide | past | web | favorite | 22 comments



I wanted to make in intelligent comment about this. But that comment turned out be: "Well holly crap, I didn't expect that".

Then again John Baez is the dude who very nearly explained General Relativity to my satisfaction, long after I got my PhD in physics.

    http://math.ucr.edu/home/baez/einstein/
So I am no longer surprised by what he (and apparently Greg Egan) can do.


That link was hard for me to copy on mobile. Here it is, unformatted: http://math.ucr.edu/home/baez/einstein/


Baez is also the author of this interesting paper:

Physics, Topology, Logic and Computation: A Rosetta Stone

http://math.ucr.edu/home/baez/rosetta.pdf


In case you missed it, John Baez's collaborator in that post is Greg Egan, who is well known for his brilliant and visionary science fiction. http://www.gregegan.net/


Example of his books (unfortunately the only one I have read...so far)

Permutation City: People can run simulations of themselves or entirely migrate their consciousness into computer programs, but existence is often limited to the amount of computation that you can afford, leading to slow existences that stretch time into fractions of realtime. That premise had me hooked, but the book has fantastic thought provoking plot points throughout. Highly recommended.


I feel in love with his novel Diaspora before I knew anything about Egan the mathematician.

Then in my grad school work, I randomly ran across his name and have subsequently read most of his books.

Diaspora, however, remains my all time favorite novel.


I always thought that Egan was primarily a SF writer who happened to be a proficient mathematician. Have I got that the wrong way around?


If you read The Clockwork Rocket, it becomes very, very obvious that Egan is a mathematician who will go to any length - including writing excellent fiction - to attempt to cram knowledge into your head.


That was my point, only my wording was suboptimal.


> Greg Egan and I came up with this formula last weekend.

He says with the same sort of casual tone that I would refer to my wife and I rearranging the living room last weekend.


> Greg Egan and I came up with this formula last weekend.

I always wonder what it feels like to be able to do that. Even if, as Baez says, the formula wasn't actually "new", to be able to dive down and come back up with new to you must be very satisfying.


Probably related to what some call "flow".


So the key here is that cos(pi/5) = phi/2. From there, you could relate pi and phi in many ways. The proof of cos(pi/5) is quite simple, neatly done here: http://faculty.wwu.edu/curgus/Courses/125/Pentagon.pdf


I have always disliked pi. Well, the 3.14 version. Ever since learning other number bases... http://turner.faculty.swau.edu/mathematics/materialslibrary/...


One advantage to the hexadecimal version is that you can compute [1] an arbitrary digit without finding all the digits before it.

[1] https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%9...


I had not seen that before. I wonder if any other interesting numbers are better calculated in alternate bases. Thanks


Decimal is so mainstream


Those people who want to ditch pi in favor of 2pi should read this article to learn why they're wrong.


I'm not sure I follow. Multiplying through by two to get tau would simply move the nested square roots from the denominator to the numerator in the relation with the golden ratio (as is explained in the link).

I certainly don't see any step in the outlined derivation that gains or loses the ability to be intuited simply due to a factor of two.


But... why? If you wanted to express their equation in terms of tau (that is, 2pi), you could just set the first term of the right hand side to 10/phi instead of 5/phi. In fact, throughout their derivation there are points where a tau-based version would be a bit cleaner (though of course there are other points where tau might be a bit messier).


i can't tell you how many failed trig tests happen as a direct result of a non-intuitive unit circle defined in terms of Pi [1].

Pi is fundamentally wrong. A circle is defined by a point and a radius. Diameter is just something that has always been easier to measure, so now we're stuck with it :( The fact that Tau makes some formulas less "elegant" is irrelevant.

[1] http://www.shelovesmath.com/wp-content/uploads/2012/11/Unit-...


I haven't read this article yet - but I just wantedto point out an interesting trivia tidbit about the golden ratio...

If you draw a typical star, the one you likely first learned to draw, each line is intersected by 1 to 1.618




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