"One Mile Is About Phi Kilometers" And Other Mathematical Coincidences

There's an amusing coincidence that one mile is about phi kilometers (too big by about 5%, where phi is the golden ratio = (1 + √5 )/2 ≈ 1.61803...)

From the Wikipedia page: Since the golden ratio is the limit of the ratio of successive terms of the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...), this gives a sequence of approximations F_n miles = F_n+1 kilometers, e.g. 5 miles = 8 km, 8 miles = 13 km, 13 miles = 21 km, ...

By the way, 1 mile = 1.609344 kilometers. My rule of thumb is 60 miles ≈ 100 km (too big by about 4%.)

There are also a lot of curious coincidences on this page. For example, do you know that π⁴ + π⁵ = e⁶ to 5 parts in a hundred millions?