Oddball applied math and numerical project ideas

These are ideas for investigation of the natural world that I have not seen anywhere else. I have not had time to try them! They might make fun projects for any young or experienced applied mathematician...

Alex

Fractal dimension of lightning strike

The electrical discharge and subsequent explosion of high-temperature gas occurs in of order 1 millisecond or less, simultaneously along the lightning channel. However the thunder heard is spread out over often many seconds, due to the range of travel times from points along the channel to the listener (there are also reflections from buildings which prolong the sound). That thunder initially sounds like `tearing', especially when it strikes close by, is no coincidence: the discharge channel is a fractal, therefore the arrival time distribution will be fractal too. Tearing of paper or other sheets is a chaotic process with build-up of elastic energy followed by catastrophic releases of energy (rips) occuring on all scales (this self-similarity is also known in earthquakes). In principle by studying the acoustic thunder signal at a single location, one could compute the fractal dimension of the strike. How, and what is it? Does it depend on weather conditions? Note that we are projecting a curve in 3D down to 1D. This constrasts the visual estimation of fractal dimension which involves projecting from 3D to 2D.

By recording at three or more locations, one might be able to reconstruct the full 3D shape of the strike. How tractable an inverse problem is this? (It certainly seems harder than optical reconstruction!) How much of a problem are reflections off the ground or other objects?

Potential reference: LIGHTNING: PHYSICS AND EFFECTS (ENCYCLOPEDIA OF LIGHTNING) by V.A. RAKOV and M.A. UMAN (Cambridge University Press, 2003). Chapter 11 is on acoustics of lightning, ie thunder (I have not yet read).

Gaussian curvature of lettuce leaf

Estimate the negative radius of curvature of a lettuce leaf, using a method of your choosing. This probably changes as a function of distance to the edge. However, since almost all of the area is within the last couple of cm from the edge, the radius must be quite tight. How much area would there be if the leaf continued with this curvature, for another 10 cm? (I suspect this would be huge--exponentially large).