Searching for monostatic polyhedra with optimization

Venue: Corvinus University, Building C, Room: TBA 

Date: 19 January 2024 

Time: 11:00-13:15 

Language: English 

More details 

Speakers: 

• Gábor Domokos (Budapest University of Technology and Economics) 
• Krisztina Regős (Budapest University of Technology and Economics) 
• Sándor Bozóki (HUN-REN Institute for Computer Science and Control and Corvinus University of Budapest) 
• Dávid Papp (North Carolina State University, Non-resident research fellow at CIAS-CCOR) 
• Gergő Almádi (Budapest University of Technology and Economics) 

Abstract: The stability of polyhedra was investigated first by Conway and Guy, and independently by Heppes in the 60s. A convex body is called mono-(un)stable if it has a unique (un)stable equilibrium. It is monostatic if it belongs to either of the two classes, and mono-monostatic, if it belongs to both. The Gömböc, constructed by Domokos and Várkonyi in 2006, is the first known mono-monostatic, homogeneous convex body. Several monostable polyhedra have also been found, but the minimal number of vertices (faces, edges) is unknown. Mono-unstability is even less understood; mono-monostatic homogenous polyhedra have not been explicitly constructed. Our four short talks will cover the geometry behind the stability of polyhedra; its algebraic reformulation into systems of polynomial inequalities; solvability of such systems using optimization techniques; and finally the inhomogenous case.