Anyway, in my Saturday morning isolation boredom, I learned (on my own) a couple of neat math oddities I hadn’t already known. I’ll start with a few similar ones I already knew, and end up with something I have proven to myself but that @Ether might have made a math quiz.

Known for years (I love φ):

x = (√5 - 1)/2 = 0.61803398874989484820458683436564…

φ = 1/x = 1.61803398874989484820458683436564…

φ² = 2.61803398874989484820458683436564…

x = √2 - 1 = 0.4142135623730950488016887242097…

1/x = 2.4142135623730950488016887242097…

Learned today (though it feeds into a tile pattern I used for a bathroom floor about a year before the guy who pulled me into FRC (@gixxy) was born!):

2 * atan(1/3) = atan(3/4)

2 * atan(1/2) = atan(4/3)

atan(1/3) + atan(1/2) = atan(1)

And also learned today. I have proven to myself, but I’d like someone else to explain it. I can’t give “credit” like the old CD, but I will like the first clear statement why the digits in the second case match those in the first.

x = 5 - √8 = 2.1715728752538099023966225515806…

x² = 4.715728752538099023966225515806…

yup, the same set of digits, one place down.

Also, I invite other cool/weird math stuff in this topic!