Abstract

To a first approximation, the behavior of light appears quite simple: it always travels in a straight line from the source to the viewer. But upon deeper inspection - various surprising effects - from mirrors to mirages, show the need for a more sophisticated theory.

These seemingly disparate phenomena are consequences of a single underlying principle: in every circumstance light endeavors to take the most efficient path between two points, or the path of least flight time. From this observation springs forth a deep connection of the theory of ray optics to differential geometry, modeling the trajectory of light in a varying medium as the shortest path - or geodesic- in an abstract curved space.

In this talk we will demonstrate the power of this viewpoint, producing computer simulations of lenses and mirages by solving for geodesics in the appropriate metric. Finally, we investigate telltale signs of curvature in the real world — including ring-like mirages appearing in images from the Hubble space telescope, and consider Einstein’s great insight: that gravity is not a force, but just a consequence of living in a curved world.

Occurrences

I originally wrote this talk upon the invitation of the National Museum of Mathematics, and have since had the opportunity to give it as a public lecture and high school colloquium in various instances.
I’ve attempted to list these here:

• Bay Area Mathematical Adventures
• Stanford Online High School
• National Museum of Mathematics
• MAA Golden Sectional Meeting
• 76th Harry S. Kieval Lecture
• Connecting the Young World International STEM Fair

Here’s a recording of the MoMath lecture.

And here’s a recording from my talk at BAMA

Slides

My current slide deck for this talk contains several live-running simulations and videos, and as such is difficult to post here. I will be making a PDF version in the future however, when I transition the slides from RevealJS to Keynote.