# Harmonic Numbers and Non-Uniform Milestones

2019-03-21

Spending some time on a treadmill got me thinking about the experience of tracking the passage of a fixed period of time. Sometimes people like to note numeric milestones, like when the period is halfway over. These numeric milestones can come in different flavors depending on whether you're thinking about percentages, or time units like days or minutes. But it occurred to me that the experience of noting these milestones can be more engaging when the milestones aren't uniformly spaced.

Due to my affinity for harmonic numbers, I would start noticing when I had finished 1/5 of the distance I was running, then 1/4, 1/3, and 1/2, and then for symmetry also 2/3, 3/4, 4/5, etc. This meant that the milestones were clustered around the beginning and end of the period, which was kind of interesting. But this had the downside that the time between the more central milestones could be rather long. Instead of breaking these up with other fractions like 3/5, it occurred to me that I could recursively track the spacing between milestones. For example, if I was waiting for the time period to progress from 1/2 to 2/3, but this progression would itself take a long time, I could notice when I'm halfway from 1/2 to 1/3, and then 2/3 of the way from 1/2 to 2/3, etc. This is obviously difficult to take very far using mental arithmetic, but I realized a computer could take it as far as you like. This means that very long time periods, of years or more, can be decomposed into non-uniform intervals such that there's always some kind of arguably-interesting milestone happening, and it's generally going to be somewhat different from the ones that have happened recently and the ones that will be happening shortly.

So I put together a thing that visualizes this for you, using a series of clock-like things:

You can set it up with a specific time interval, or just start a new one from the current time. This is it right here.

Thanks to Carissa Miller and Tim Galeckas for valuable feedback.