Ciro Santilli @cirosantilli 35

Take the group of all Translation in ${\mathbb{R}}^1$.

Let's see how the generator of this group is the derivative operator:

The way to think about this is:

So let's take the exponential map:and we notice that this is exactly the Taylor series of $f(x)$ around the identity element of the translation group, which is 0! Therefore, if $f(x)$ behaves nicely enough, within some radius of convergence around the origin we have for finite $x_0$:

This example shows clearly how the exponential map applied to a (differential) operator can generate finite (non-infinitesimal) Translation!

E.g.: thecollegestore.co.uk/products/ladies-oxford-college-puffer-jacket?variant=40590030864549 Black with 5 rows, on left chest "colege name", logo, "Oxford", and right chest optional initials (or sometimes other identifiers/nicknames) to help distinguish from all the other people's identical clothes.

This has a whitelabel version: www.workweargiant.co.uk/product/result-urban-holkham-down-feel-jacket/, the name appears to be "Holkham Down Feel Jacket".

Circa 2020, these are likely given out by each college for free, and are widely used.

If you look 20 and wear one of those, it's almost an ID, you can get anywhere that does not require a key card, porters won't look at you twice!

This single experimental observation/idea is the basis for all of special relativity.

Special relativity is the direct result of people bending their backs to accommodate for this really weird fact.