9 problems

Let $P$ be a triangulated polyhedron with distinguished vertices $A,B,A',B'$ satisfying … Form the twinned polyhedron $T$ of $P$ around the equator $ABA'B'$, as in the construction…

Let $n\ge 4$, and consider exotic flexible cross-polytopes in the sphere $\mathbb{S}^n$. These are flexible cross-polytopes whose dihedral-angle tangents have the exotic parametriz…

Let $P(u)$ be a flexible polyhedron in the sphere $\mathbb{S}^n$, where $n\ge 3$. Replacing a vertex by its antipode means replacing that vertex of $P(u)$ with its antipodal point…

Let $P$ be a flexible polyhedron in one of the constant-curvature spaces $\mathbb{E}^n$, $\mathbb{S}^n$, or $\mathbbl{\Lambda}^n$. Two polyhedra are scissors congruent if they can…

Let $n\ge 3$, and let a flexible polyhedron lie either in Lobachevsky space $\Lambda^n$ or in the open hemisphere $S^n_+$. Its generalized oriented volume is defined in the relevan…

Let $P\colon K\to\mathbb{R}^n$ be a flexible, not necessarily embedded polyhedron, and let $V_K(P)$ denote its generalized oriented volume, defined by integrating its characteristi…

A flexor is a flexible polyhedron in $\mathbb{R}^3$, and a flexion is a continuous deformation preserving its combinatorial type and the shapes of its faces. The bellows conjecture…

Let $n\ge 3$. A flexible cross-polytope is a flexible polyhedral cross-polytope in the indicated ambient space. Self-intersection conjecture. All flexible cross-polytopes in…

Let $P$ be a flexible polyhedron in Euclidean space, and let its generalized volume be the volume invariant associated with its possibly self-intersecting realization. A bending is…