Summary
We describe special kinds of polygons, called Fano polygons or reflexive polygons, and their higher dimensional generalizations, called reflexive polytopes. Pairs of reflexive polytopes are related by an operation called polar duality. This combinatorial relationship has a deep and surprising connection to string theory: One may use reflexive polytopes to construct “mirror” pairs of geometric spaces called Calabi-Yau manifolds that could represent extra dimensions of the universe. Reflexive polytopes remain a rich source of examples and conjectures in mirror symmetry.
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Notes on contributors
Charles F. Doran
CHARLES F. DORAN received his Ph.D. from Harvard University in 1999. He is now an associate professor at the University of Alberta. As site director for the Pacific Institute of the Mathematical Sciences, he runs the Alberta Summer Math Institute for talented high school students.
Ursula A. Whitcher
URSULA A. WHITCHER received her Ph.D. from the University of Washington in 2009, and is now an assistant professor at the University of Wisconsin-Eau Claire. She enjoys working with undergraduates to research reflexive polytopes and the geometric spaces they describe.