Report

A fundamental theme of work in "Discrete Differential Geometry" is the construction of representations and algorithms suitable for numerical modeling on the computer in such a way that facts holding true in the idealized setting of pure mathematics are also fulfilled in the practical computational setting. One thing that researchers in this area have discovered over the years is that this is best accomplished by carrying geometric relationships over to the discrete setting.

An example of this approach is given by the modeling of the intricate structures formed when a drop of ink falls into quiescent water. The so-called "Ink Chandeliers" can be modeled as curves with thickness (effectively tubes of ink immersed in water), so called vortex filaments. While being mathematical abstractions of infinitely thin tubes which rotate, creating motion in the surrounding fluid due to friction, this model no less captures the intricate deformation dynamics of the ink (see attachment). Remarkably the same geometric principles can explain the equally subtle dynamics of bubble rings moving underwater as created playfully by dolphins, for example (see attachment).

This theme of curves with attendant physical dynamics also applies in other more exotic settings. Consider the dramatic images of our sun's corona with tube like glowing structures (see attachment). In that case the curves carry a magnetic field which interacts with the plasma of ionized particles creating the structures we can see emanating from the surface of the sun. The dynamics of these structures forming and evolving can once again be described using elementary geometry. Another example, even further removed is the motion of a snake. Consider the spine of the snake as a curve which undulates with a wavelike motion (see attachment).

How does that "wiggling" result in forward motion? As it turns out experiencing less friction when moving along its length versus moving sideways, together with the principle that nature seeks the path of least resistence is enough to reproduce the forward motion of a snake purely from its undulating shape change. This same principle of creating motion from shape change also applies to many other settings ranging from falling cats landing on their feet to spring board divers with their fanciful twists and turns or Jellyfish swimming under water (see attachment). Coming up with simple geometric models suitable for computation are but a few examples of the scientific work performed by the present Einstein Foundation sponsored research group.