Random Facts
Some random math facts that I came across last week.
Poincaré group
The Special Euclidean group combines rotations and translations in Euclidean space. The SE group is an isometry, which means it preserves angles and distances.
In special relativity, the 4-vectors are three space variables and one time variable. Rotations in this space can be generalized to the Poincaré group. Instead of having sines and cosines in the rotation part as in SE, they have hyperbolic sines and cosines.
The norm of a vector is defined through its inner product:
which in special relativity is
A friend pointed out that the Lorentz transformation was only recently proved to be linear.
Geodesics and Metrics
One of the most beautiful things I learnt in the last few weeks is the
connection between geodesics (shortest paths) and metrics in that
space. If 
is a geodesic path and 
is the
velocity of the path:
then the metric is defined by
Moreover, the path satisfies the Euler-Lagrange equation. This was first shown by Arnold in a hydrodynamical context, and subsequently borrowed by control systems and computer vision folks.
A Poor Joke
Two functions 
and 
were going in a
car. looked ahead and said, “Oh shit! There comes a
differential operator.” says with a smirk, “It can’t do
nothing to me!” On approaching the differential operator, it says
“Haha! I’m d/dy.”
As narrated to me by a random math major last week…