Sadly the public has very little idea of what mathematicians do. Do they add bigger and bigger numbers? Do they balance big checkbooks? In this talk I will try to give you a taste of mathematical research through polyhedra. These are very pretty objects and are familiar to everyone from a young age: Polygons, triangles, squares, cubes, pyramids, Platonic solids have captivated humans attention for thousands of years and are familiar shapes from childhood. Polyhedra in high-dimensional versions turn out to be widely used in applied mathematics. Their beauty and simplicity appeal to all, but very few people know of the many easy-to-state but difficult-to-solve mathematical problems that hide behind their beauty. They illustrate well what a modern mathematician does everyday. This lecture will have lots of nice pictures and will introduce the audience to some fascinating unsolved questions at the frontier of mathematical research and its applications. No prior knowledge beyond your faint memory of high school geometry will be assumed.