16 problems

A flexible polyhedron is a polyhedron whose faces remain rigid while its dihedral angles vary continuously. During such a flexion, its enclosed volume may in principle change. Bell…

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

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…

A polyhedron is embedded if its surface has no self-intersections. A polyhedron is rigid if its edge lengths determine its shape up to Euclidean motions, and generic if its geometr…

A polyhedron is a solid bounded by finitely many polygonal faces, and an isometric deformation is a deformation preserving the intrinsic metric of the polyhedron. Bellows conjectur…

A bi-Bennett is an arrangement of two Bennett mechanisms, and the classes (I2, II2, III2) and (I3, II3, III3) are the complete classes of flexible bipyramids and biprisms, respecti…

A flexible polyhedron is a polyhedron admitting a continuous Euclidean isometric deformation through noncongruent realizations. The bellows conjecture. The volume of a flexible pol…

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 a Kokotsakis polyhedron be a polyhedral surface in $\mathbb{R}^3$ consisting of one $n$-gon, quadrilaterals attached to its sides, and triangles between consecutive quadrilater…

Weak Strong Bellows Conjecture. The Dehn invariant of any flexible polyhedron remains constant during the flexion. This is a weaker form of the Strong Bellows Conjecture, since sci…

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…

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…

Strong Bellows Conjecture. If an embedded polyhedron $P_1$ is obtained from an embedded polyhedron $P_0$ by a continuous flex, then $P_1$ and $P_0$ are scissors congruent: there ex…