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Self-dual polygons

Self-dual polygons

Monday, November 27, 2023 12:00 pm
Monday, November 27, 2023 12:50 pm
Kidder Hall 237
Ana C. Chavez-Caliz
The Mathematical Institute of Heidelberg University

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 RP2.

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 A1, A3, ... A2n-1, then its dual polygon P* has vertices B*2, B*4, ... B*2n, where B*i is the line connecting the points Ai-1, Ai+1. Given an integer m, a polygon P is m-self-dual if there is a projective transformation f such that f(Ai) = BI+m.

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