Abstract: This is a joint project with A. Kanel-Belov, Ph. Rukhovich, and V. Zgurskii. A Euclidean outer billiard on a convex figure in the plane is the map sending a point outside the figure to the other endpoint of a segment touching the figure at the middle. Iterating such a process was suggested by J. Moser as a crude model of planetary motion. Polygonal outer billiards are arguably the principal examples of Euclidean piecewise rotations, which serve as a natural generalization of interval exchange maps. They also found applications in electrical engineering. Previously known rigorous results on outer billiards on regular N-polygons are, apart from “trivial” cases of N=3,4,6, based on dynamical self-similarities (this approach was originated by S. Tabachnikov). Dynamical self-similarities have been found so far only for N=5,7,8,9,10,12. In his ICM 2022 address, R. Schwartz asked whether “outer billiard on the regular N-gon has an aperiodic orbit if N is not 3, 4, 6”. We answer this question in affirmative for N not divisible by 4. Our methods are not based on self-similarity. Rather, scissor congruence invariants (including those of Sah-Arnoux-Fathi and Hadwiger-Glur) play a key role.