I like the 4-5-6 theorem:

    pi^4 + pi^5 = e^6

Well, to five decimal places, anyway. Some other good ones:

    e^pi - pi = 20

    sqrt(2) ln pi = phi

There are also famous "almost integers" such as this one discovered by Ramanujan:

    e^(pi sqrt(163))

Which is is almost an integer to 12 decimal places.

at this rate, pi square is close to 'g'

The second fact, pi^2 ~= g, is famous enough that it has a separate section in Wikipedia [1].

[1] https://en.wikipedia.org/wiki/Mathematical_coincidence#Gravi...

My first thought was "well of course it is, since pi is a little larger than 3" but it was cool to see an actual derivation of how much pi squared differs from 10 as a nice, closed form series.

I remember discovering that pi x 10^7 is very close to the number of seconds in a year while at uni.

One of my tutors was convinced this had to be more than coincidence, but I always figured it was just chance and a nice but sometimes useful shortcut...

  6 weeks * 7 days * 24 hours * 60 minutes * 60 seconds:

    6 * 7 * 24 * 60 * 60
  = 6 * 7 * (3 * 8) * (4 * 5 * 3) * (3 * 2 * 10)
  = 6 * 7 * 3 * 8 * 4 * 5 * (3 * 3) * 2 * 10
  = 6 * 7 * 3 * 8 * 4 * 5 * 9 * 2 * 10
  = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10

pi^2 ~ 10, well known to anyone who used slide rules.

This first became apparent to me when I got a slide rule. Pi is often marked on the various scales and an x^2 scale is often nearby the x scale.

987654321 / 123456789 = 8 (to the 7th decimal place) is another nice one

If you don't unblock scripts from cdn.jsdelivr.net.cdn.cloudflare.net, the math code won't work.

need a countdown for when it gets there

I was a little disappointed that the upper range of gravity on earth only goes to 9.8337. Just a little more and there would have been somewhere on earth that was an exact match.

It would have been the ideal (if chilly) place to start a cult.

Pi^0 is exactly 1.

As an ex-physicist, pi^2 is 10. Like g.

I get it that this is a nice calculation with the Zeta function and everything, but 3 and a small something squared will be near 10 so it is 10.

The author wants tau=2*pi, but in the Greek alphabet, tau has one vertical stroke, and pi has two.

So, visually in Greek, pi=2*tau would seem an improvement.

Oh, well.