Empirical units, π, 𝑒 and 𝜙 (pi, e, and phi)
Posted by jpluimers on 2026/01/09
Being a non-native English speaker and having monaural hearing¹, the first time visiting the USA I thought they mentioned Empirical Units² when they tried to explain miles, feet and other measurement units they use on their island.
Then I learned they are in fact United States customary units but in the USA, they actually name those Imperial Units, implying that the UK still has a very strong influence on the USA. In reality, there are differences³ between Imperial Units and United States customary units to keep things in the USA practical (or lazy if you want), so I will keep calling their system Empirical Units as it is more fit for purpose (can’t name them Freedom Units any more given their Project 2025 regime).
Anyway, quite a while ago there was this cool XKCD “The Maritime Approximation” (image on the right) including only Imperial Units holding for Empirical Units as well: π mph ≈ 𝑒 kn (let’s use ISO unit symbols here, shall we) which is correct to < 0.5%.
Recently, I learned that with the same accuracy, there is a golden ratio between metric and Imperial/Empirical Units: 𝜙 km = 1 mi, also correct to < 0.5%.
Kevlin Henney wrote two great blog posts on these explaining way more background information:
- [Wayback/Archive] The Maritime Approximation. A transcendent measure of coincidence… | by Kevlin Henney | Jan, 2025 | Medium
- [Wayback/Archive] The Golden Mile Conversion. An aesthetic and coincidental alignment… | by Kevlin Henney | Jan, 2025 | Medium
Recommended reading!
Further reading
Unicode code points used based on Mathematical operators and symbols in Unicode – Wikipedia -> Mathematical Alphanumeric Symbols – Wikipedia:
- [Wayback/Archive] Unicode Character ‘MATHEMATICAL ITALIC SMALL E’ (U+1D452)
- [Wayback/Archive] Unicode Character ‘MATHEMATICAL ITALIC PHI SYMBOL’ (U+1D719)
- [Wayback/Archive] Unicode Character ‘MATHEMATICAL ITALIC SMALL PI’ (U+1D70B)
XCKD 3023:
- [Wayback/Archive] xkcd: The Maritime Approximation
[Wayback/Archive] the_maritime_approximation.png (265×126)Title text: It works because a nautical mile is based on a degree of latitude, and the Earth (e) is a circle.
- [Wayback/Archive] 3023: The Maritime Approximation – explain xkcd
Via
[Wayback/Archive] Kevlin Henney on X: “New essay on the blog: The Golden Mile Conversion “This time I want to dive into the delightful and coincidental relation that ϕ km ≈ 1 mi, i.e., the ratio of statute miles to kilometres, 1 mi:1 km ≈ ϕ:1, is approximated by the golden ratio.” …”
--jeroen
¹ many people refer to this as single sided deafness or unilateral hearing loss, but I like monaural hearing more, not just because it describes what I can’t versus what I cannot, but also now that binaural recording having become popular this century and that a query for it returns way more good research than the other terms, for instance [Wayback/Archive] Understanding sound direction estimation in monaural hearing | ScienceDaily.
² Apparently, I am not alone in this: [Wayback/Archive] Empirical units?
3 Comparison of the imperial and US customary measurement systems – Wikipedia
Queries:
The Wiert Corner - irregular stream of stuff 
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