An extremely nerdy post on math & running

Distance running/walking/swimming is perfect for the nerd like me, because it gives me lots and lots of time to think.
I think about solving all the worlds problems.
I think about new verses to songs that haven't been written.
I think about birds and turtles and grass and sky.
I think about numbers.  A lot.

What's my pace?
How many laps have I swum, miles have I run, steps have I taken?
I do calculations in my head for hours (I've now swum 1/72 of the laps I need for a mile, I'm now at the 1/4 point in my run...)

So, while I was mulling over all of these numbers in my head the other day, I came across something pretty neat.

Most races are plotted in kilometers.  Most Americans think in miles.  So we runners are always thinking "Now, how many miles is that?"

For instance, a 5k is estimated at 3.1 miles. Meaning that each mile is approximately 1.6 kilometers.
(5k=3.10686m)

1.6 is very close to my favorite number.

My favorite number is an irrational number that is derived from dividing the sum of 1 and the square root of 5 by 2.

1.618033989

My favorite number is Phi, otherwise known as the Golden Ratio.

The Golden Ratio has long been hailed in art and science as the most beautiful number.  It is found in nature in the number of seeds in flowers, the spiral of a sea shell, the branches of a tree, and in pretty much anything we find attractive.

So, thinking about this golden ratio, lead me to thinking about the Fibonacci Sequence, which is based off of this beautiful, irrational number.  This sequence is recursive, wherein each subsequent term in the sequence is the sum of the two previous terms.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144....

Now, let me get back to my running thoughts.  I know that 5k≈3miles.  Rob is running an 8k next month.  8/1.6= 5miles.  Do you see those numbers in the sequence?  Cool, right?  So now I can estimate that if I was running a 13k, I'd be running approximately 8 miles.  Each time I want to compare kilometers to miles, I can just look at the previous term in the sequence.  

If I want to compare miles to kilometers I can just go the opposite direction. For instance, if I just passed the two mile mark, I can estimate that I've run 3 kilometers (And I will have, 2m≈3.2k).

Now what about numbers that are not part of the Fibonacci Sequence?  That's easy, too.  Just look for the sum of terms in the sequence. Like this:

On Friday, I earned a medal for 16k. 16/1.6=10 miles.  Let's check it with the sequence: 16=13+3. 13k=8m, 3k=2m. 8+2=10miles.  16k≈9.94m It works!

So now, I have to plan to run for Stations 5 and 6 of the Lost Virtual series, which are 23 and 42k, respectively.  How many miles is that?

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144....

23k=21k+2k

21k=13 miles

2k=1 mile

23k=14 miles according to the sequence

23k≈14.29m

Pretty close!

42k=34k+8k

34k=21 miles

8k=5 miles

42k=26 miles

42k≈=26.1m

 Math is Cool.  And calculating the Fibonacci Sequence distracts me from mile 10.  So it works for everyone!