# When my diary mentioned 密率

## One year ago

TIL that Gregory (and then Euler) used 6.28… for their circle constant and that Euler only later switched to using 3.14…

Of course 密率 was found 1500 years before Euler was even born, and that’s in favor of the 3.14 camp, but still. It’s not obvious what the best circle constant should be.

Space aliens probably know how to play go at least to an extent that they would recognize the New Zeeland style rules. Square tesselation, cell-filling, liberty counting is something I could see them inventing.

But would they react to 3.14 is the question? Maybe they’d be like “huh…? only half a turn? do they mean to signal a 180°!? a grave insult!”

Obv today in modern day Earth we can use both. π is pretty well established as half a turn and τ can represent a full turn. When π is convenient, use it, and when a full turn is convenient, use it.

## One week ago

I am suddenly obsessed with the reciprocal of τ, probably better known as one radian. Tripping the 密率 dream, one radian is 113/710 of a circle and one circle is 710/113 of a radian. Kinda.

Get on outta here with that 3.14 stuff. If trig was based around “diametrians” (reciprocals of π) then maybe. The angle subtended from the center of a circle which intercepts an arc equal in length to the diameter of the circle. Maybe the math nerds should’ve thought of that. Then there’d be π diametrians in a circle instead of this waxing gibbous nonsense.

## Today

I am so obsessed with 密率, an irreducible fraction with a 112 period (in decimal). Maxing out its denominator, which is 113, a prime.
I’m in love with this number. I don’t even care about the circle divided by r squared anymore, schmancendental. I’ve found my Buddha on the road and I don’t wanna pull the trigger.

It’s still a mystery to math dorks why there’s a huge honking 292 smack dab in the fraction expansion of pi. A001203 in the OEIS. I’m dangerously close to getting into some woo mystic religion nonsense just to honor and celebrate this wonderful number (密率, that is, not 292).

I wonder if there are any circles in nature that use 密率. Any bubbles or flower stalks or tree-rings where God cut some corners (Matthew 10:29) and finally made some rational decisions for once ♥︎🙏

## Later today

I have been wondering—if pi was secretly 密率 all along, could the circle have been squared? Turns out that yes, it could have, and Ramanujan did it in 1913! That is absolutely baller! Er, I mean, approximately baller!

## 21 months later

Some might think 密率 a.k.a. 355/113 isn’t so special. But here is why:

It’s easy to get approximations of pi that are smaller, like 3/1 or 16/5 or 22/7 but going to 355/113 suddenly gets almost three orders of magnitude closer.

And then that’s as close as it gets for a long time. The next one up is 52163/16604, a much clumsier number, four more digits only to gain less than 0.000001 of accuracy, hardly worth even using an approximation at that point, and what’s worse, it’s not even that special because the next one up is just around the corner, 52518/16717, and so it goes, there’s a whole bunch in the 50000-something series. 355/113 really does stand out.