Running with the Fibonacci Numbers

The surprising connection between distance measurements and a famous mathematical sequence

If you’ve ever done any long-distance running, you’ve probably noticed that there’s a mishmash of distance measures used: meters, kilometers, miles. But if you convert the lengths of different races into alternate units, you notice something peculiar:

• 5k race: 5km → 3.11mi ~ 3mi
• 8k race: 8km → 4.97mi ~ 5mi
• Half-marathon: 13mi → 20.92km ~ 21km

For one thing, all of the converted units are very close to exact integers. For another, all of these numbers, when rounded to the nearest integer, are part of the famous Fibonacci Sequence:

This is just super weird, right? What on earth could distance unit measurements have to do with the Fibonacci Sequence, which was created to model rabbit population growth?

Fibonacci Numbers and the Golden Ratio

The Fibonacci Sequence is generated using the simple rule that each number in the sequence is the sum of the two numbers that came before it. Assuming that the first two numbers in the sequence, Fib(1) and Fib(2), are both equal to 1, this gives:

• Fib(3) = Fib(1) + Fib(2) = 1 + 1 = 2
• Fib(4) = Fib(2) + Fib(3) = 1 + 2 = 3
• Fib(5) = Fib(3) + Fib(4) = 2 + 3 = 5
• etc

The Fibonacci numbers are closely tied to another famous mathematical constant, the golden ratio.

The golden ratio, denoted by 𝜙, is defined as the ratio such that for two positive numbers a & b, the ratio of a/b is the same as the ratio of (a+b)/a. When you do the algebra, you find that 𝜙 equals around 1.618.

This value turns out to have a special relationship with the Fibonacci numbers. Namely for a given pair of successive Fibonacci numbers, Fib(n) and Fib(n+1), the ratio of the two approximately equals the golden ratio. That is:

Or equivalently:

Converting from Miles to Kilometers

The mile and the kilometer have very different origins, with the mile being derived from the length of Roman paces, and the kilometer being derived from the distance from the north pole to the equator. Nonetheless there is a simple way to convert between them: Just multiply a distance in miles by ~1.609 to get the distance in kilometers.

Now coincidentally, this happens to be very close to the value of 𝜙≈1.618 described in the last section.

This therefore leads to the odd situation where, for no reason other than bizarre coincidence, you can multiply a distance in miles by the golden ratio, and approximately get the distance measured in kilometers.

Tying it all together

So now we have all of the pieces we need to understand why Fibonacci numbers keep popping up when we convert the lengths of races between miles and kilometers.

Piece #1: Multiplying the n’th Fibonacci number by 𝜙 gives the (n+1)’th Fibonacci number

Piece #2: Multiplying a distance in miles by ~1.609 gives the distance in kilometers

Piece #3: The value of 𝜙 (~1.618) is coincidentally very close to this 1.609 value

This means that if our distance starts out as a Fibonacci number when measured in miles or kilometers, then it will continue to be a Fibonacci number when converted to the other unit.

Writing it out concretely, suppose d is the n’th Fibonacci number. Then:

The same steps can be performed in reverse to go from Fib(n+1) kilometers to Fib(n) miles.

As for why so many long-distance races start out already being Fibonacci numbers (5km, 8km, 13mi), that is just a happy coincidence about the universe.