Project EVA (Enhanced Visual Awareness)

Computer experiments which produce elementary and well structured visual output often enhance and inspire further theoretical considerations:

(a)

Motivated by the large computer-time otherwise needed to produce pictures of Costa's minimal surface, we have found a new parametric representation of that surface. The representation is directly given in terms of Weierstrass' zeta function, whereby complex integration (of rational expressions in Weierstrass' P function and its derivative) is circumvented and computer-time thus reduced considerably. Click here to display Costa's minimal surface.

(b)

An observation indicating a foliating property of the osculating circles for logarithmic spirals on the 2-sphere and in hyperbolic 2-space has been extended into a theorem concerning this phenomenon for all curves with monotone curvature.

We wish to accelerate this train of computer aided developments from "classical to modern" differential geometry. For example, the theory of curves in R³ can be lifted directly to Riemannian 3-manifolds via the notion of development due to E. Cartan.

The conceptual modifications needed for such generalizations are easily implemented and illustrated via computer. The computer may thus serve as a natural vehicle both for the teaching of advanced modern differential geometry and also for the revitalization of a wealth of interesting (but partly forgotten) classical insight into low dimensional geometry.

An example of the use of a Java applet to visualize a simple geometric concept can be seen by clicking here.