Slow Research: Disk and Sphere Projections / by Siobhán Cronin

I have begun studying differential geometry, and have been looking for points of departure in everyday life to motivate my studies. Tonight while stirring a pot of mushroom soup, I noticed something interesting about the light's reflection in the beads of fat on the surface as I moved my head from side to side. I found that if I focussed on the surrounding surface, and allowed my eyes to take in the beads less directly, I could perceive each disk as a sphere resting on the surface. 

After I finished eating, I served up another bowl of soup to observe this phenomena more closely. I found that if I imagined the lamps reflection were actually a floating light "projected" by each of the spheres appearing on the soup's surface, my eye could stick with the trick more faithfully. It seemed assigning the imagined spheres this agency helped my mind hold onto the mirage. 

I can imagine the optics explanation of why my mind might be able to switch from seeing light reflected off a disk as thought it were reflecting off spheres would rest somewhere in the special relationship circles and spheres share, with no visible edges to break the illusion of an aerial view observed as a askance perspective. However I can't help wonder how these two different scenarios relate - the light reflecting off individual beads of fat on the soup's surface, and the imagined equivalently plausible scenario of suspended spheres (with our without the special light-producing agency). 

Here's the soup:

How might we define the relationship from the light to the beads and compare this to the light to the spheres? What traditions in geometry or topology might helps us do that? 

On a side note, I observed this evening that the reverse optic trick could be applied to the sun's reflection off the the surface of the moon. A quick web search for a geometric rendering of that led me to I discover that there are people who strongly believe we are circumambulating a flat earth. Wowzers!