39 problems

A hyperbolic strictly-convex polyhedron is a strictly-convex polyhedron in hyperbolic space, with dihedral angles measured along its edges. Stoker's conjecture. Every hyperbolic st…

In the setting of Theorem, concerning robustness classes of convex solids and platonic solids, let downward external robustness and downward full robustness be the corresponding do…

Let $\pp$ be a compact, oriented, irreducible, atoroidal $3$-manifold with boundary equipped with an angled block structure, and let $\tilde{\pp}$ denote its universal cover. The a…

Milnor's continuity and vanishing conjecture. The volume function $V$ admits a continuous extension to $\overline{\Omega}$. Furthermore, the points on $\partial\Omega$ where $V$ va…

$4_n$ existence and zigzag conjecture. (i) A z-knotted $4_n$ exists if and only if $n\geq 30$ and $n\equiv 2\pmod 4$. (ii) A tight $4_n$ exists if and only if $n\geq 8$ and…

Symmetry-group conjecture. (i) $3_n(D_{2h})$ exists if and only if $\frac{n}{4}$ is even and $n\geq 16$; there are no tight $3_n(D_{2h})$. (ii) $3_n(D_{2d})$ exists if and only if…

Curvature-graph conjecture. The graph of curvatures of any tight graph of type $3_n$ is the graph in the first of those three cases.

Let $\mathcal{P}_a$ be the stellated tetrahedron with vertices … … … … where edges join all $p_i,p_j$ and $p_i,q_j$ for $i\ne j$, and $a\in(0.5,0.57)$. A polyhedron is Rupert if it…

Let $\mathcal{P}$ be a polyhedron. It is Rupert if there exist $R_1,R_2\in\operatorname{SO}(3)$ and $t\in\mathbb{R}^2$ such that … where $P:\mathbb{R}^3\to\mathbb{R}^2$ drops the…

Let $\mathcal{P}$ be a convex polyhedron, meaning the convex hull of a finite set of points in $\mathbb{R}^3$ in convex position. A polyhedron is Rupert if there exist…

Domokos–Horváth–Goriely–Regős conjecture. Every polyhedral tiling can be completely softened.

Finite-subcomplex relaxation conjecture. Condition (3) can be relaxed so that it is required to hold only outside a finite sub-complex of $\mathcal{D}$.

Let $P$ be any “polytope” with a full-dimensional affine span in $\mathbb R^d$. Let $\mathbf p_{n+1}$ be placed anywhere in $\mathbb R^d$. Let $\Omega$ be any stress for the bar fr…

Soft polyhedric realization conjecture. There exists a combinatorially equivalent, soft polyhedric tiling $M'$.

Let $K$ be a regular $n$-sided polygon with $n\geq 3$, circumscribed by a circle of unit radius. For a positive even integer $p$, let $L_{22}^{(p)}$ denote the corresponding…

A geometric polyhedron here may be nonconvex and self-intersecting; its faces, dihedral angles, edge lengths, and partially-flat vertices are understood in the sense of the paper.…

Asymptotic count conjecture. As $v \to \infty$, the counts $\alpha(v)$ and $\beta(v)$ are respectively asymptotic to

Zero-condition conjecture. The conditions from the Clean Condition are satisfied only when $F(i)=0$.

Let $P_1$ and $P_2$ be polyhedra with the same $1$-skeleton $\Gamma$. Suppose that, for every edge $e$ of $\Gamma$, the dihedral angles of $P_1$ and $P_2$ at the corresponding edge…

Let $Q$ be a polyhedron and let $D$ be its dual. A polyhedron is locally Rupert if it has a local Rupert passage, and reverse locally Rupert if it has the corresponding reverse pas…

A polyhedron is locally Rupert if it admits a Rupert passage after an arbitrarily small alteration of a suitable orientation, and locally reverse Rupert is the analogous reverse-pa…

Polyhedron-bisecting curve conjecture. For each polyhedron $P$ in $\mathbb{R}^3$ there exists a smooth simple closed curve $\mathcal{C}$ such that

The Rhombicosidodecahedron is the point-symmetric Archimedean solid considered here, and Rupert's property means that a congruent copy can pass through a hole cut into the original…

A convex polyhedron is a polyhedron in $\operatorname{\mathbb{R}}^3$ whose faces are convex polygons. Such a polyhedron has Rupert's property if a hole can be cut into it through w…

Irreducible non-realizability conjecture. There is an irreducible SOD satisfying the axioms that is not the visibility map $S_\mathcal P$ of any polyhedron $\mathcal P$ with respec…