Mind blown, I never realized this coincidence:

You can use the next number in the Fibonacci sequence (1 1 2 3 5 8 13 21 34) to convert from miles to kilometers.

For example, 13 miles is 21 kilometers.



I would assume it gets more accurate the further into the sequence you go?

@Scrith @fribbledom
Actually it does not. Have a look at the sequence of ratios:

2/1 = 2.0
3/2 = 1.5
5/3 = 1.666...
8/5 = 1.6
13/8 = 1.625
21/13 = 1.615...
34/21 = 1.619...

You'll see that the sequence of ratios approximates the golden ratio (1.618...). But it does so non-monotonically, in a chaotic or oscillating fashion, sometimes closer to the conversion factor 1.6, sometimes further away from it.

The sequence of fibonacci _ratios_ can be split into two: One sequence with all elements with an odd index and the other only with even. Based on the numbers above:

2.0, 1.666..., 1.625, 1.619..., ...
1.5, 1.6, 1.615..., ...

This is just a guess, but maybe both sequences approach the golden ratio monotonically. (A proof for or against is possible.)

@Scrith Then indeed the first sequence above would only get closer 1.6, i.e. the observation by @fribbledom would only increase in accuracy.

Hmm...interesting. Man, mathematics has so many fascinating facets

@Scrith The disparity settles rather quickly to be 0.56% away from the actual answer

@Scrith @fribbledom The ratio between consecutive numbers is oscillating, but converges to 1.6180339887.. which is further away from say 55/34 = 1.61764..

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