Arrangements of Curves

Geometry has dramatically evolved from metrical work (measuring lengths, angles, areas and perimeters) involving polygons and curves (circles and ellipses) that was initiated in many cultures but is often associated with the names Euclid, Archimedes, and Pappus. More attention has been given by geometers in recent times to combinatorial ideas. Though in many ways there is less overhead in working with the point and line diagrams known as graphs (see Figure 1a) than using the axiomatic approaches one sees in Euclid, combinatorial ideas entered mathematics much more recently. The problems discussed below involve 3-dimensional bounded convex (no holes, notches, tunnels) polyhedra, but grow out of a graph theory (dots and lines) point of view).