I’ve lost count of the number of times I’ve watched the short video “Möbius transformations revealed” by Douglas Arnold and Jonathan Rogness.
It is a beautiful tribute to beautiful functions. As a complex analysis and hyperbolic geometry fangirl, I am contractually required to get a little misty-eyed when I think about Möbius transformations.
I thought “Möbius transformations revealed” was the pinnacle of Möbius transformation-related video until Henry Segerman posted this one last month.
(It took me a while to realize that if you’re on a computer and your browser supports it, you can push the arrows in the compass at the top left of the frame to look around the sphere. For phones it’s a little different. My iPhone shows the full frame, but I think on Android devices, you can see different parts of the video by moving your phone around. Your mileage may vary.)
Möbius transformations turn out to be the answer to some tricky problems that come up in spherical video, namely, how to zoom or rotate the way you would with rectangular videos. Like Frank Farris’ book Creating Symmetry, the mathematics inspires the creation of beautiful and interesting images. Segerman wrote a guest post about using Möbius transformations in spherical video editing on the eleVR blog. EleVR is the virtual reality research group of Emily Eifler, Vi Hart, Andrea Hawksley, and Elijah Butterfield, the team behind Hypernom, a game where you try to eat the cells of four-dimensional Platonic solids and other VR endeavors. Their blog documents the fascinating challenges, both mathematical and logistical, of creating and interacting with spherical videos. I love the Sphere-a-day post in which Eifler talks about how different spherical videos feel to her as a creator:
Many of the daily spherical videos so far feel more like ‘Come hang out with me while I do this thing I would be doing anyway,’ the largest number falling into the ‘I am making art and the camera is running’ category which reveals a lot about how I spend my life, something I never let happen when I was making flat video. In flat video I kept video Emily and RL Emily very separate, but not so in spherical. Its feels like documentation without translation into textual or verbal language and the non-framed non-presentational all seeing eye of the camera makes me feel more relaxed, more open, and frankly just more adventurous and laissez faire about what I can shoot with it. Nothing gets cropped out, nothing gets left behind.
Spherical videos already offer viewers the opportunity to interact with videos in new ways, but using Möbius transformations on them gives creators new ways to tell stories. With the help of the eleVR-ers, Segerman illustrates this beautifully in a spherical Droste video where you feel like you’re taking an endless scroll through a time loop.