The Value of Pi
A post for Pi Day—but should it be a different day?
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Emma Haruka Iwao set an official Guinness world record in 2019 by calculating pi to 31,415,926,535,897 digits. If she had calculated it to 62,831,853,071,795 digits, she might still hold that record today.
Today, Pi Day (3/14), we ask why 
The main alternatives are the values we currently call 
Two things that stand out to us from Iwao’s story told on Google’s blogs, besides her commanding the power of Google’s cloud services to manage 170 terabytes of data over 121 days for the calculation, are:
- She was inspired while growing up in Japan by the current record holders being Japanese professors, one of whom she later worked for as a student.
- There was a quicker checking phase for the final product—in particular, for the 8,956,768,817,536 new digits. This used a mixture of two versions of the BBP method, which we covered a few times.
The new record of 50 trillion digits by Timothy Mullican have been added to Iwao’s available for delivery by Google. We do not know the extent to which they are concretely pseudorandom, despite coming from a simple rule, but that extent would be shared equally by
Why This Value?
Giorgia Fortuna in 2015 wrote a long article on the Wolfram blog about 
We want to pose a different question, one suggested to me today by a reference to the 3rd-century algorithm of the Chinese mathematician Liu Hui, which efficiently generated values far more precise than what was known in the west:
Why did several separate cultures fix upon the ratio of the circumference to the diameter, rather than the ratio of the circumference to the radius, or the diameter to half the circumference?
The last ratio, 1.5707963267948966…, strikes us as the most natural to think of in the great outdoors: You have a choice of rowing across a circular pond or walking around. How much more distance does the latter involve?
We don’t know of a name for partisans of
Sesquipidalians.
This branches off the word sesquipedalian in an appropriate way, since sesqui- is Latin for one-and-a-half (as in “sesquicentennial”) and 
Did the ancients favor 3.14… because it was the closest to an integer of three main options?
Open Problems
Is there any merit to our historical question? We leave it as a riddle until the next Pi Day—or at least until 6/28/2021. Note also our previous query about whether the Indian mathematician Aryabhata favored
[some word and format tweaks, added note on Aryabhata]
will you sacrifice e^{i \pi} =-1?
I view it as gaining the division operation and the integer 2, expanding the embrace of single-use perfection. 🙂
” Did the ancients favor 3.14… because it was the closest to an integer of three main options?”
At least according to wikipedia Euler initially used 6.28… and later changed his preference to 3.14… And before him it was custom to actually type a ratio perimeter/radius or perimeter/diameter (with pi, rho, and delta). Now, I do not know whether you consider Euler an ‘ancient’.
What is the reference on Wikipedia? I did not find it at his bio or the “Euler’s formula” article.
I have read it on the page about Pi: https://en.wikipedia.org/wiki/Pi#Adoption_of_the_symbol_%CF%80
Albert Eagle’s 1958 treatise The Elliptic Functions as they should be took a goodly number of idiosyncratic stances. One of them was to use \(\tau\) for half-\(\pi\), with a recommended pronunciation of “hi”.
As a young child I thought of a circle as a round thing, rather than a locus or something drawn with a compass. Perhaps the ancients carried around the idea of a circle like this: where diameter was more fundamental than radius.
I personally think Euler should’ve stayed with tau or an equivalent definition of Pi.
1) It is conceptually more fundamental. A circle is defined by its radius and central point.
2) Both it and 1.57.. are easier to accurately approximate with a simple rational fraction. 6.25 is only ~0.03 away from the true value, or over 99% accurate. Pi/2 or 1.57… isn’t too bad either; approximating it as 3/2 is within 5% accuracy. Pi itself isn’t totally terrible- 22/7 is a close approximation- but dividing by 7 is significantly harder than dividing by 2 or 4.
3) It makes things like trigonometry way more intuitive and straightforward. A quarter of the circle is “tau/4”, a whole circle is “tau”, etc. A lot of formulas in basic geometry, trigonometry, and related math would either simplify or at least be easier to reason about, remember, and teach.
I agree that the overall # of ‘improved’ formulas (adding multiples or divides by 2 vs removing them) is probably a wash. But NOT all formulae are created equal. Yeah you might add a net # of new multiplies and divides by 2, 4, etc. to many of the power series and fancy integrals that involve Pi. But those things are already complicated; the actual change in complexity would be negligible.
Looking at the history, it appears that our current Pi won out because it was the first and easiest “circle ratio” to both observe & measure. Calculating Pi can be done by simply measuring the perimeter and diameter of a naturally occurring (near) circle, then doing 1 divide. Same goes for the ancient algorithms like inscribed polygons that ancient mathematicians used to try and compute the value to N places, before the modern infinite series equations were known. Only later did people recognize that a circle’s radius is more fundamental, or that using the ‘tau’ ratio can simplify the abstract math. Real world circles and rods to measure distance came first. Pi and the early algorithms mathematicians invented to calculate it came first.
All that aside, Euler could’ve easily shifted the balance towards Pi=6.253… winning out if he’d just stuck to using it. Just as Leibnitz defined the winning, better symbol convention for calculus despite Newton’s being first.