| Abstract: A tetrahedron is called
isosceles if its four triangular facets are congruent to one another.
We will show that if the four triangular facets of a 3-dimensional tetrahedron
can be partitioned into pairs with the same area, then those pairs must be
congruent. In particular, if the four faces of a tetrahedron have the
same area then the tetrahedron must be isosceles. This result can be
generalized to polyhedral surfaces, but many analogous assertions turn out
to be false in dimension 4 and greater. This talk will give an overview
of these questions as well as pose some new ones. It should be accessible
to graduate students and advanced undergraduate students. |