Conics

Conics: Searching for Beauty Reveals Deeper Truths

Conic sections have occupied a central position in geometry since antiquity.We’ll view conic sections from a perspective that makes the whole study more unified and more beautiful. Our principal guides—symmetry and eliminating exceptions—will lead us to the “natural habitat” of conics, the complex numbers. This will open our eyes to some startling facts that the traditional approach misses. Two examples:

• Any ellipse or hyperbola has four foci, not two— there’s a pair on each principal axis.
• There is a normally-unsees ellipse bridging the gap between the two branches of any hyperbola
  

This ellipse has an area given by a formula working for ordinary ellipses, too. Almost any fact about conics whose statement shows a “favoritism” of some sort (ellipse versus hyperbola, major axis versus minor, and so on) can be rewritten in a more consistent and beautiful way using the basic principles introduced in Conics, a recent publication from the MAA authored by the program director. This text is included with each participant’s registration materials. In it are effective tools for guessing at new results—tools valuable to the professional mathematician, and also to graduating math majors searching for an interesting and doable exit project.

Some selected readings will be assigned to provide a common base before we meet. Once we get together, there will be some introductory lectures, free-form discussions and idea-generating sessions. The aim is to provide inspiration that leads to new conjectures. Then, in smaller groups, participants will test out—and possibly prove—the new ideas in a nearby computer lab using Maple and the geometry-drawing program Cabri.

At some point after the workshop, we will produce a collection of write-ups by participants and the director, describing our results. This could take the form of a booklet. For more information, visit the website at http://www.csuohio.edu/math/kendig/.