Searching for Beauty Reveals Deeper Truths
May 15-19, 2006
Cleveland State University
FOR THIS WORKSHOP IS NOW CLOSED
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:
This ellipse has an area given by a formula working for ordinary
ellipses, too. Almost any fact about conics whose statement shows a
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.
- 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
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/.