Speed Limits and the Fibonacci Series

(By Amateria1121 - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=31513168)

Travelers between the United States and Canada have to adjust to speed limits from miles per hour to kilometres per hour. Astute drivers usually learn a few quick approximations for converting between the two systems. Some common limits and their approximate conversions are shown below.

km/h	30	50	80	130
mile/h	20	30	50	80

The more precise conversion is 1 mile = 1.61 km or 1 km = 0.62 mile.

If the numbers above seem familiar, you may have noticed the conversion between the two measures is very close to the golden ratio or 1.618. This ratio has the property that it's inverse equals itself minus one: 1/1.618 = 0.618 = 1.618 -1.

The calculation for the golden ratio is:

To be clear, the relationship between kilometres and miles being near the golden ratio is only a coincidence.

A common approximation to the golden ratio is given by ratio of successive elements of the Fibonacci Series. The series begins as:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...

Each entry of the series is the sum of the previous two entries. Beginning with 1/1, the ratio of successive elements of the series is:

1, 2, 1.5, 1.667, 1.6, 1.625, 1.615, 1.619, 1.618 

This ratio converges to the golden ratio of 1.6180339887...

Finally, return to the approximate kilometre-mile conversion table above, as we noted about successive entries of the Fibonacci series, each entry in both rows is the sum of the previous two entries.