# Self-dual polygons

# Self-dual polygons

In his book "Arnold's Problems", Vladimir Arnold shares a collection of questions without answers formulated during seminars in Moscow and Paris for over 40 years. One of these problems, stated in 1994, goes as follows: *Find all projective curves projectively equivalent to their duals*. The answer seems to be unknown even in RP^{2}.

Motivated by this question, in their paper "Self-dual polygons and self-dual curves" from 2009, D. Fuchs and S. Tabachnikov explore a discrete version of Arnold's question in 2-dimensions. If P is an n-gon with vertices A_{1}, A_{3}, ... A_{2n-1}, then its dual polygon P* has vertices B*_{2}, B*_{4}, ... B*_{2n}, where B*_{i} is the line connecting the points A_{i-1}, A_{i+1}. Given an integer m, a polygon P is m-self-dual if there is a projective transformation f such that f(A_{i}) = B_{I+m}.

In this talk, I will generalize Fuchs and Tabachnikov's work to polygons in higher dimensions.