Pi Day: Are you still using decimal notation?!

Today is Pi Day (3/14). You’re not still using decimal, are you?

The Madness of π

Many years ago, while happily working at Symbolics, I was exposed to the mathematical madness of Bill Gosper. He was fond of using continued fraction expansions for all sorts of odd things. At the time (in the 1980s) he held the world’s record for largest number of digits of π ever computed: first extracted as a continued fraction using some Ramanujan-like formula, then – in his words – “encrypted” into decimal digits.

So I was pleased to see one day that in some ways the most compact way to represent a rational approximation to a real number was as a continued fraction. It takes fewer bits than as a decimal (or binary floating point), or even as a rational.

π Day Last Year (Pre-Pandemic)

So when π day ^[1] came around last year, we had just shut down our research labs for the pandemic and I had begun working from home. I had a long (17 year) tradition of putting mathematical humor on my door to amuse passers-by. (This is what I had put there a week or so earlier; it sat there for 4 months of sad, empty hallways until I retired amid the pandemic summer.)

• Here you can see the first 12 convergents of the continued fraction for π this encodes π to many hundreds of digits, but with much greater economy than good ol’ decimal notation. (Just as a rational number is the root of a linear polynomial with integer coefficients, a finite continued fraction is the root of a quadratic polynomial with integer coefficients.)
• The bitmap below encodes the first 256 convergents of the continued fraction for π: each column is a binary number, with black denoting 1 and white denoting 0. The 0 bit is at the bottom. Can you read off the first 12 and compare with the continued fraction above? (I really wanted the bathroom tile around the base of our tub & shower to be this, but the Weekend Editrix objected.)
• And of course, the madness I saw one year when somebody baked a picosahedron: 20 trianguar pecan pies, arranged with magnets into the form of an icosahedron. Now, that’s the divine madness and the spirit of ϜΤΦ! (Why doesn’t that capital digamma render?!)

Anybody wanna bake a picosahedron? I’m available as sous-chef.

---

Notes & References

1: Just entertain the notion that τ might be better, because Vi Hart says so. ↩