Changes

# The Theorema Egregium - that there exists a quality inherent to a surface no matter how it is "bent."

# The Theorema Egregium - that there exists a quality inherent to a surface no matter how it is "bent."

# What happens if we remove some the Euclidean postulates taught in high school?  Specifically, the parallel postulate?

# What happens if we remove some the Euclidean postulates taught in high school?  Specifically, the parallel postulate?

# Double-bubble conjecture (a nice result, easy to state and wave hands about "curvature", and with a connection to CP!)

 

==== The Double Bubble Conjecture ====

You've probably noticed that when you blow a soap bubble, it quickly stretches into a nearly spherical shape.  But why should a bubble tend towards a spherical conformation, instead of a cube or an icosahedron?  According to physics, the bubble will tend toward a shape with the lowest energy, and the physical effect of surface tension is that this lowest energy is achieved when the bubble is in the shape with the least possible surface area enclosing a given volume.  For example, if we want to enclose one cubic inch of air, the smallest surface that could contain it is a perfect sphere.  The proof of this case, for a simple bubble, involves techniques from the mathematical study of [[differential geometry]], but turns out not to be too difficult.  Related problems in finding minimal surfaces, however, turn out to be quite fiendish.  The double bubble conjecture is one such.

 

*insert a picture here!

 

=== Topology ===

=== Topology ===